Erdoğan, M. Burak; Green, William R.
2017-06-01
We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a t -1 decay rate as an operator from the Hardy space H 1 to BMO, the space of functions of bounded mean oscillation. This estimate, along with the L 2 conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold s-wave resonance does not destroy the t -1 decay rate. As a consequence of our analysis we obtain a limiting absorption principle in the neighborhood of the threshold, and show that there are only finitely many eigenvalues in the spectral gap.
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Trzetrzelewski, Maciej
2016-11-01
Starting with a Nambu-Goto action, a Dirac-like equation can be constructed by taking the square-root of the momentum constraint. The eigenvalues of the resulting Hamiltonian are real and correspond to masses of the excited string. In particular there are no tachyons. A special case of radial oscillations of a closed string in Minkowski space-time admits exact solutions in terms of wave functions of the harmonic oscillator.
Decoherence in the Dirac equation
Meyer, D A
1998-01-01
A Dirac particle is represented by a unitarily evolving state vector in a Hilbert space which factors as $H_{spin} \\otimes H_{position}$. Motivated by the similarity to simple models of decoherence consisting of a two state system coupled to an environment, we investigate the occurence of decoherence in the Dirac equation upon tracing over position. We conclude that the physics of this mathematically exact model for decoherence is closely related to Zitterbewegung.
Structure of Dirac matrices and invariants for nonlinear Dirac equations
2004-01-01
We present invariants for nonlinear Dirac equations in space-time ${\\mathbb R}^{n+1}$, by which we prove that a special choice of the Cauchy data yields free solutions. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. Related problems on the structure of Dirac matrices are studied.
M. Ko(c)ak; B. G(o)nül
2007-01-01
The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schr(o)dinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.
Dirac equations in n + 1 dimensions
Jiang Yu [Departamento de FIsica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, 09340 Mexico DF (Mexico)
2005-02-04
The Dirac equation in n + 1 dimensions is derived by a simple algebraic approach. The similarity in the structure of the arbitrary n-dimensional Dirac equations in a central field and their solutions is discussed.
Discrete Dirac equation on a finite half-integer lattice
Smalley, L. L.
1986-01-01
The formulation of the Dirac equation on a discrete lattice with half-integer spacing and periodic boundary conditions is investigated analytically. The importance of lattice formulations for problems in field theory and quantum mechanics is explained; the concept of half-integer Fourier representation is introduced; the discrete Dirac equation for the two-dimensional case is derived; dispersion relations for the four-dimensional case are developed; and the spinor formulation for the Dirac fields on the half-integer lattice and the discrete time variable for the four-dimensional time-dependent Dirac equation are obtained. It is argued that the half-integer lattice, because it takes the Dirac Lagrangian into account, is more than a mere relabeling of the integer lattice and may have fundamental physical meaning (e.g., for the statistics of fermions). It is noted that the present formulation does not lead to species doubling, except in the continuum limit.
Dirac and Maxwell equations in Split Octonions
Beradze, Revaz
2016-01-01
The split octonionic form of Dirac and Maxwell equations are found. In contrast with the previous attempts these equations are derived from the octonionic analyticity condition and also we use different basis of the 8-dimensional space of split octonions.
Hammer, René; Arnold, Anton
2013-01-01
A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly. Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in $l_2$ which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stabi...
Contemplations on Dirac's equation in quaternionic coordinates
Schuricht, Dirk; Greiter, Martin
2004-11-01
A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several peculiarities, which we investigate and attempt to interpret.
On the Dirac equation for a quark
Pestov, I B
2003-01-01
It is argued from geometrical, group-theoretical and physical points of view that in the framework of QCD it is not only necessary but also possible to modify the Dirac equation so that correspondence principle holds valid. The Dirac wave equation for a quark is proposed and some consequences are considered. In particular, it is shown that interquark potential expresses the Coulomb law for the quarks and, in fact, coincides with the known Cornell potential.
Krylov subspace methods for the Dirac equation
Beerwerth, Randolf; Bauke, Heiko
2015-03-01
The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard.
Krylov subspace methods for the Dirac equation
Beerwerth, Randolf
2014-01-01
The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The Dirac Hamiltonian's property of not being bounded does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard.
Dirac equation on a curved surface
Brandt, F. T.; Sánchez-Monroy, J. A.
2016-09-01
The dynamics of Dirac particles confined to a curved surface is examined employing the thin-layer method. We perform a perturbative expansion to first-order and split the Dirac field into normal and tangential components to the surface. In contrast to the known behavior of second order equations like Schrödinger, Maxwell and Klein-Gordon, we find that there is no geometric potential for the Dirac equation on a surface. This implies that the non-relativistic limit does not commute with the thin-layer method. Although this problem can be overcome when second-order terms are retained in the perturbative expansion, this would preclude the decoupling of the normal and tangential degrees of freedom. Therefore, we propose to introduce a first-order term which rescues the non-relativistic limit and also clarifies the effect of the intrinsic and extrinsic curvatures on the dynamics of the Dirac particles.
Gravitationally Coupled Dirac Equation for Antimatter
Jentschura, U D
2013-01-01
The coupling of antimatter to gravity is of general interest because of conceivable cosmological consequences ("surprises") related to dark energy and the cosmological constant. Here, we revisit the derivation of the gravitationally coupled Dirac equation and find that the prefactor of a result given previously in [D.R. Brill and J.A. Wheeler, Rev. Mod. Phys., vol. 29, p. 465 (1957)] for the affine connection matrix is in need of a correction. We also discuss the conversion the curved-space Dirac equation from East-Coast to West-Coast conventions, in order to bring the gravitationally coupled Dirac equation to a form where it can easily be unified with the electromagnetic coupling as it is commonly used in modern particle physics calculations. The Dirac equation describes anti-particles as negative-energy states. We find a symmetry of the gravitationally coupled Dirac equation, which connects particle and antiparticle solutions for a general space-time metric of the Schwarzschild type and implies that particl...
Quantum simulation of the Dirac equation
Gerritsma, Rene; Kirchmair, Gerhard; Zaehringer, Florian; Blatt, Rainer; Roos, Christian [Institut fuer Quantenoptik und Quanteninformation, 6020 Innsbruck (Austria); Solano, Enrique [Departamento de Quimica Fisica, Universidad del Pais Vasco - Euskal Herriko Unibertsitatea, Bilbao (Spain)
2010-07-01
The Dirac equation is a cornerstone in the history of physics, merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and predicting the existence of anti-matter. However, the Dirac equation also predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an unexpected quivering motion of a free relativistic quantum particle first examined by Schroedinger. In this talk, we report on a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion, which is set to behave as a free relativistic quantum particle. We measure as a function of time the particle position and study Zitterbewegung for different initial superpositions of positive and negative energy spinor states, as well as the cross-over from relativistic to nonrelativistic dynamics.
Dirac equation on coordinate dependent noncommutative space-time
Kupriyanov, V G
2014-01-01
We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properties of the star product we derive the corresponding probability current density and prove its conservation. The energy-momentum tensor for the free noncommutative spinor field is calculated. We solve the free noncommutative Dirac equation and show that the standard energy-momentum dispersion relation remains valid in the noncommutative case.
Dirac equation on coordinate dependent noncommutative space–time
Kupriyanov, V.G., E-mail: vladislav.kupriyanov@gmail.com
2014-05-01
In this paper we discuss classical aspects of spinor field theory on the coordinate dependent noncommutative space–time. The noncommutative Dirac equation describing spinning particle in an external vector field and the corresponding action principle are proposed. The specific choice of a star product allows us to derive a conserved noncommutative probability current and to obtain the energy–momentum tensor for free noncommutative spinor field. Finally, we consider a free noncommutative Dirac fermion and show that if the Poisson structure is Lorentz-covariant, the standard energy–momentum dispersion relation remains valid.
Dirac equation on coordinate dependent noncommutative space-time
Kupriyanov, V. G.
2014-05-01
In this paper we discuss classical aspects of spinor field theory on the coordinate dependent noncommutative space-time. The noncommutative Dirac equation describing spinning particle in an external vector field and the corresponding action principle are proposed. The specific choice of a star product allows us to derive a conserved noncommutative probability current and to obtain the energy-momentum tensor for free noncommutative spinor field. Finally, we consider a free noncommutative Dirac fermion and show that if the Poisson structure is Lorentz-covariant, the standard energy-momentum dispersion relation remains valid.
Quantum simulation of the Dirac equation.
Gerritsma, R; Kirchmair, G; Zähringer, F; Solano, E; Blatt, R; Roos, C F
2010-01-07
The Dirac equation successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter and is able to reproduce accurately the spectrum of the hydrogen atom. The realm of the Dirac equation-relativistic quantum mechanics-is considered to be the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects, such as Klein's paradox and 'Zitterbewegung', an unexpected quivering motion of a free relativistic quantum particle. These and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. In recent years, there has been increased interest in simulations of relativistic quantum effects using different physical set-ups, in which parameter tunability allows access to different physical regimes. Here we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion set to behave as a free relativistic quantum particle. We measure the particle position as a function of time and study Zitterbewegung for different initial superpositions of positive- and negative-energy spinor states, as well as the crossover from relativistic to non-relativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate textbook examples of relativistic quantum physics.
Building Atomic Nuclei with the Dirac Equation
Serot, Brian D.
2003-01-01
The relevance of the Dirac equation for computations of nuclear structure is motivated and discussed. Quantitatively successful results for medium- and heavy-mass nuclei are described, and modern ideas of effective field theory and density functional theory are used to justify them.
Quantum simulation of the Dirac equation
Gerritsma, R; Zähringer, F; Solano, E; Blatt, R; Roos, C F
2009-01-01
The Dirac equation is a cornerstone in the history of physics, merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and predicting the existence of anti-matter. Furthermore, it is able to reproduce accurately the spectrum of the hydrogen atom and its realm, relativistic quantum mechanics, is considered as the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an unexpected quivering motion of a free relativistic quantum particle first examined by Schr\\"odinger. These and other predictions would be difficult to observe in real particles, while constituting key fundamental examples to understand relativistic quantum effects. Recent years have seen an increased interest in simulations of relativistic quantum effects in different physical setups, where parameter tunability allows accessibility to different physical regimes. Here, we perform a proof-of...
Dirac equation and the Melvin metric
Santos, L.C.N.; Barros, C.C. [Universidade Federal de Santa Catarina, Depto de Fisica-CFM, CP. 476, Florianopolis, SC (Brazil)
2016-10-15
A relativistic wave equation for spin 1/2 particles in the Melvin space-time, a space-time where the metric is determined by a magnetic field, is obtained. The energy levels for these particles are obtained as functions of the magnetic field and compared with the ones calculated with the Dirac equation in the flat Minkowski space-time. The numeric values for some magnetic fields of interest are shown. With these results, the effects of very intense magnetic fields on the energy levels, as intense as the ones expected to be produced in magnetars or in ultra-relativistic heavy-ion collisions, are investigated. (orig.)
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
I.Yu. Krivsky
2010-01-01
Full Text Available We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration is proved. In order to provide the corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.
Steen-Ermakov-Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
Prykarpatskyy, Yarema
2017-01-01
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide.
Lorentz-Dirac equation and circularly moving charges
Comay, E.
1987-09-01
The Lorentz-Dirac equation of radiation reaction is tested in a system of circularly moving changes. It is shown that this equation together with the Lienard-Wiechert retarded fields is consistent with energy conservation. Therefore, in this particular experiment, any alternative expression of radiation reaction must agree with the Lorentz-Dirac equation.
New exactly solvable periodic potentials for the Dirac equation
Samsonov, B F; Pozdeeva, E O; Glasser, M L
2003-01-01
A new exactly solvable relativistic periodic potential is obtained by the periodic extension of a well-known transparent scalar potential. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Dirac Kronig-Penney model. The solutions of the Dirac equation are expressed in terms of elementary functions.
Time Delay for the Dirac Equation
Naumkin, Ivan; Weder, Ricardo
2016-10-01
We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator {intlimits0 ^{∞}e^{iH0t}ζ(\\vert x\\vert /R) e^{-iH0t}dt}, as {R → ∞}, is presented. Here, H 0 is the free Dirac operator and {ζ(t)} is such that {ζ(t) = 1} for {0 ≤ t ≤ 1} and {ζ(t) = 0} for {t > 1}. This approach allows us to obtain the time delay operator {δ {T}(f)} for initial states f in {{H} 2^{3/2+ɛ}({R}3;{C}4)}, {ɛ > 0}, the Sobolev space of order {3/2+ɛ} and weight 2. The relation between the time delay operator {δ{T}(f)} and the Eisenbud-Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.
Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation
A. A. Deriglazov
2011-01-01
Full Text Available We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. Classical equations for the spin tensor are the same as those of the Barut-Zanghi model of spinning particle.
Topological Insulators Dirac Equation in Condensed Matters
Shen, Shun-Qing
2012-01-01
Topological insulators are insulating in the bulk, but process metallic states around its boundary owing to the topological origin of the band structure. The metallic edge or surface states are immune to weak disorder or impurities, and robust against the deformation of the system geometry. This book, Topological insulators, presents a unified description of topological insulators from one to three dimensions based on the modified Dirac equation. A series of solutions of the bound states near the boundary are derived, and the existing conditions of these solutions are described. Topological invariants and their applications to a variety of systems from one-dimensional polyacetalene, to two-dimensional quantum spin Hall effect and p-wave superconductors, and three-dimensional topological insulators and superconductors or superfluids are introduced, helping readers to better understand this fascinating new field. This book is intended for researchers and graduate students working in the field of topological in...
Topological insulators Dirac equation in condensed matter
Shen, Shun-Qing
2017-01-01
This new edition presents a unified description of these insulators from one to three dimensions based on the modified Dirac equation. It derives a series of solutions of the bound states near the boundary, and describes the current status of these solutions. Readers are introduced to topological invariants and their applications to a variety of systems from one-dimensional polyacetylene, to two-dimensional quantum spin Hall effect and p-wave superconductors, three-dimensional topological insulators and superconductors or superfluids, and topological Weyl semimetals, helping them to better understand this fascinating field. To reflect research advances in topological insulators, several parts of the book have been updated for the second edition, including: Spin-Triplet Superconductors, Superconductivity in Doped Topological Insulators, Detection of Majorana Fermions and so on. In particular, the book features a new chapter on Weyl semimetals, a topic that has attracted considerable attention and has already b...
Geometric Structures and Field Equations of Dirac-Lu Space
REN Xin-An; ZHANG Li-You
2008-01-01
In this paper, a -invariant Lorentz metric on the Dirac-Lu space is given, and then the geodesic equation is investigated. Finally, we discuss the field equations and find their solutions by the method of separating variables.
Letter: On the Solutions of the Lorentz-Dirac Equation
Vogt, D.; Letelier, P. S.
2003-12-01
We discuss the unstable character of the solutions of the Lorentz-Dirac equation and stress the need of methods like order reduction to derive a physically acceptable equation of motion. The discussion is illustrated with the paradigmatic example of the non-relativistic harmonic oscillator with radiation reaction. We also illustrate the removal of the noncausal pre-acceleration with the introduction of a small correction in the Lorentz-Dirac equation.
Intertwining technique for the one-dimensional stationary Dirac equation
Nieto, L M; Samsonov, B F; Samsonov, Boris F.
2003-01-01
The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples.
Canonical conjugated Dirac equation in a curved space
Dzhunushaliev, Vladimir
2012-01-01
It is shown that the calculation of Dirac operator for the spherical coordinate system with spherical Dirac matrices and using the spin connection formalism is in the contradiction with the definition of standard Dirac operator in the spherical Minkowski coordinate system. It is shown that such contradiction one can avoid by introducing a canonical conjugated covariant derivative for the spinor field. The Dirac equation solution on the Reissner - Nordstr\\"om background is obtained. The solution describes a bound state of a charged particle.
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Lorz, Alexander
2011-01-17
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
Dirac mass dynamics in a multidimensional nonlocal parabolic equation
Lorz, Alexander; Perthame, Benoit
2010-01-01
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a structure of gradient flow. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models darwinian evolution.
ON GROUND STATE SOLUTIONS FOR SUPERLINEAR DIRAC EQUATION
张建; 唐先华; 张文
2014-01-01
This article is concerned with the nonlinear Dirac equations Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth.
The Lorentz-Dirac equation and the structure of spacetime
De Souza, M M
1995-01-01
A new interpretation of the causality implementation in the Lienard-Wiechert solution raises new doubts against the validity of the Lorentz-Dirac equation and the limits of validity of the Minkowski structure of spacetime.
The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials
Contreras-Astorga, Alonso, E-mail: aloncont@iun.edu; Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, E-mail: xbataxel@gmail.com [Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States)
2014-10-15
We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)].
Single-cone real-space finite difference schemes for the Dirac von Neumann equation
Schreilechner, Magdalena
2015-01-01
Two finite difference schemes for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian are presented. Both utilize a single-cone staggered space-time grid which ensures a single-cone energy dispersion to formulate a numerical treatment of the mixed-state dynamics within the von Neumann equation. The first scheme executes the time-derivative according to the product rule for "bra" and "ket" indices of the density operator. It therefore directly inherits all the favorable properties of the difference scheme for the pure-state Dirac equation and conserves positivity. The second scheme proposed here performs the time-derivative in one sweep. This direct scheme is investigated regarding stability and convergence. Both schemes are tested numerically for elementary simulations using parameters which pertain to topological insulator surface states. Application of the schemes to a Dirac Lindblad equation and real-space-time Green's function formulations are discussed.
Dirac equation in low dimensions: The factorization method
Sánchez-Monroy, J. A.; Quimbay, C. J.
2014-11-01
We present a general approach to solve the (1 + 1) and (2 + 1) -dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein-Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.
Dirac-like equations for barotropic FRW cosmologies
Rosu, H C; Reyes, M; Jimenez, D
2002-01-01
Simple Schroedinger-like equations have been written down for FRW cosmologies with barotropic fluids by Faraoni. His results have been extended by Rosu, who employed techniques belonging to nonrelativistic supersymmetry. Further extensions are presented herein using the known connection between Schroedinger-like equations and Dirac-like equations in the same supersymmetric context
The Dirac equation as one fourth-order equation for one function -- a general form
Akhmeteli, Andrey
2015-01-01
Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma-matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma-matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also much more symmetric than that of the previous work and should be useful in applications of the Dirac equation.
Generalized de Broglie Relations for Dirac Equations in Curved Spacetimes
Arminjon, Mayeul
2011-01-01
One may ask whether the special relativistic relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We do this by applying Whitham's Lagrangian method to derive covariant equations describing wave packet motion which preserve the symmetries of the Dirac Lagrangian, and in particular, conserve the probability current. We show that generalized de Broglie relations emerge from the Whitham equations after transforming each Dirac equation into a canonical form via a local similarity transformation of the type first introduced by Pauli. This gives the de Broglie relations a universal character for spin-half particles in a curved spacetime. We show that COW and Sagnac type terms also appear in the Whitham equations. We further discuss the classical-quantum correspondence in a curved spa...
Conjugated Molecules Described by a One-Dimensional Dirac Equation.
Ernzerhof, Matthias; Goyer, Francois
2010-06-08
Starting from the Hückel Hamiltonian of conjugated hydrocarbon chains (ethylene, allyl radical, butadiene, pentadienyl radical, hexatriene, etc.), we perform a simple unitary transformation and obtain a Dirac matrix Hamiltonian. Thus already small molecules are described exactly in terms of a discrete Dirac equation, the continuum limit of which yields a one-dimensional Dirac Hamiltonian. Augmenting this Hamiltonian with specially adapted boundary conditions, we find that all the orbitals of the unsaturated hydrocarbon chains are reproduced by the continuous Dirac equation. However, only orbital energies close to the highest occupied molecular orbital/lowest unoccupied molecular orbital energy are accurately predicted by the Dirac equation. Since it is known that a continuous Dirac equation describes the electronic structure of graphene around the Fermi energy, our findings answer the question to what extent this peculiar electronic structure is already developed in small molecules containing a delocalized π-electron system. We illustrate how the electronic structure of small polyenes carries over to a certain class of rectangular graphene sheets and eventually to graphene itself. Thus the peculiar electronic structure of graphene extends to a large degree to the smallest unsaturated molecule (ethylene).
Dirac equation in low dimensions: The factorization method
Sánchez-Monroy, J.A., E-mail: antosan@if.usp.br [Instituto de Física, Universidade de São Paulo, 05508-090, São Paulo, SP (Brazil); Quimbay, C.J., E-mail: cjquimbayh@unal.edu.co [Departamento de Física, Universidad Nacional de Colombia, Bogotá, D. C. (Colombia); CIF, Bogotá (Colombia)
2014-11-15
We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials. - Highlights: • The low-dimensional Dirac equation in the presence of static potentials is solved. • The factorization method is generalized for energy-dependent Hamiltonians. • The shape invariance is generalized for energy-dependent Hamiltonians. • The stability of the Dirac sea is related to the existence of supersymmetric partner Hamiltonians.
Digital quantum simulation of Dirac equation with a trapped ion
Shen, Yangchao; Zhang, Xiang; Zhang, Junhua; Casanova, Jorge; Lamata, Lucas; Solano, Enrique; Yung, Man-Hong; Zhang, Jingning; Kim, Kihwan; Department Of Physical Chemistry Collaboration
2014-05-01
Recently there has been growing interest in simulating relativistic effects in controllable physical system. We digitally simulate the Dirac equation in 3 +1 dimensions with a single trapped ion. We map four internal levels of 171Yb+ ion to the Dirac bispinor. The time evolution of the Dirac equation is implemented by trotter expansion. In the 3 +1 dimension, we can observe a helicoidal motion of a free Dirac particle which reduces to Zitterbewegung in 1 +1 dimension. This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540. KK acknowledge the support from the recruitment program of global youth experts.
Geometric Correlation between Dirac Equation and Yang-mills Equation/ Maxwell Equation
Yu, Xuegang
2011-01-01
The problem about geometric correspondence of Dirac particle and contain quality item of Yang-Mills equation has always not been solved.This paper introduced the hyperbolic imaginary unit in Minkowski space, established a classes of Dirac wave equations with t'Hooft matrices.In lightlike region of Minkowski space,we can discuss the hermitian conjugate transformation of Dirac positive particle and antiparticle, find the space-time corresponding points of Dirac particle,and draw Feynman clip-art though the geometrical relation between timelike region and lightlike region.The coupling of motion equation of Dirac positive particle and antiparticle can get Klein-Gordon equation, when it reach classical approximate we can get Schrodinger equation,and this illustrated that p meson or m meson may be composite particle. Using the relation of timelike region and lightlike region in Minkowski momentum space to renormalize the rest mass of particles,we can describe the geometric relation between rest mass and electromagn...
Moving potential for Dirac and Klein–Gordon equations
Hamil B; Chetouani L
2016-04-01
Using the Lorentz transformation, the Klein–Gordon and Dirac equations with moving potentials are reduced to one standard where the potential is time-independent. As application, the reflection and transmission coefficients are determined by considering the moving step with a constant velocity $v$. It has been found that $R \\pm T = 1$ only at $x = vt$. The problem of massless (2+1) Dirac particle is also considerered.
Klein-Gordon and Dirac Equations with Thermodynamic Quantities
Arda, Altuğ; Tezcan, Cevdet; Sever, Ramazan
2016-03-01
We study the thermodynamic quantities such as the Helmholtz free energy, the mean energy and the specific heat for both the Klein-Gordon, and Dirac equations. Our analyze includes two main subsections: (1) statistical functions for the Klein-Gordon equation with a linear potential having Lorentz vector, and Lorentz scalar parts (2) thermodynamic functions for the Dirac equation with a Lorentz scalar, inverse-linear potential by assuming that the scalar potential field is strong ( A ≫ 1). We restrict ourselves to the case where only the positive part of the spectrum gives a contribution to the sum in partition function. We give the analytical results for high temperatures.
Generalization of the Lorentz-Dirac equation to include spin
Barut, A. O.; Unal, Nuri
1989-11-01
For the classical point electron with Zitterbewegung (hence spin) we derive, after regularization, the radiation reaction force and covariant equations for the dynamical variables (xμ, πμ, vμ, and Sμν), which reduce to the Lorentz-Dirac equation in the spinless limit.
New applications of pseudoanalytic function theory to the Dirac equation
Castaneda, Antonio; Kravchenko, Vladislav V [Seccion de Posgrado e Investigacion, Escuela Superior de IngenierIa Mecanica y Electrica, Instituto Politecnico Nacional, CP07738 Mexico DF (Mexico)
2005-10-21
In the present work, we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potential in a two-dimensional case and a pair of decoupled Vekua equations. In general, these Vekua equations are bicomplex. However, we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain nonrestrictive condition. As an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions. One of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result. Consider the Dirac equation with a scalar potential depending on one variable with fixed energy and mass. In general, this equation cannot be solved explicitly even if one looks for wavefunctions of one variable. Nevertheless, for such Dirac equation, we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions (depending on two variables). These solutions generalize the system of powers 1, z, z{sup 2}, ... in complex analysis and are called formal powers. With their aid any regular solution of the Dirac equation can be represented by its Taylor series in formal powers.
Dirac equation on coordinate dependent noncommutative space–time
Kupriyanov, V. G.
2014-01-01
We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properti...
Axel Schulze-Halberg
2006-01-01
New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form of a Riccati equation. The present note generalizes a series of former articles.
On radiation reaction and the Abraham-Lorentz-Dirac equation
de Oca, Alejandro Cabo Montes
2013-01-01
It is underlined that the Lienard-Wiechert solutions indicate that after the external force is instantly removed from a small charged particle, the field in its close neighborhood becomes a Lorentz boosted Coulomb field. It suggests that the force of the self-field on the particle should instantaneously vanish after a sudden removal of the external force. A minimal modification of Abraham-Lorentz-Dirac equation is searched seeking to implement this property. A term assuring this behavior is added to the equation by maintaining Lorentz covariance and vanishing scalar product with the four-velocity. The simple Dirac constant force example does not show runaway acceleration.
Relativistic Lagrangians for the Lorentz–Dirac equation
Deguchi, Shinichi, E-mail: deguchi@phys.cst.nihon-u.ac.jp [Institute of Quantum Science, College of Science and Technology, Nihon University, Chiyoda-ku, Tokyo 101-8308 (Japan); Nakano, Kunihiko [Institute of Quantum Science, College of Science and Technology, Nihon University, Chiyoda-ku, Tokyo 101-8308 (Japan); Suzuki, Takafumi [Junior College Funabashi Campus, Nihon University, Narashinodai, Funabashi, Chiba 274-8501 (Japan)
2015-09-15
We present two types of relativistic Lagrangians for the Lorentz–Dirac equation written in terms of an arbitrary world-line parameter. One of the Lagrangians contains an exponential damping function of the proper time and explicitly depends on the world-line parameter. Another Lagrangian includes additional cross-terms consisting of auxiliary dynamical variables and does not depend explicitly on the world-line parameter. We demonstrate that both the Lagrangians actually yield the Lorentz–Dirac equation with a source-like term.
Stability problem for singular Dirac equation system on finite interval
Ercan, Ahu; Panakhov, Etibar
2017-01-01
In this study, we show the stability problem for the singular Dirac equation system respect to two spectra on finite interval. The meaning of the stability problem of differential operators is to estimate difference of the spectral functions which considered problems when a finite number of eigenvalues of these problems coincide. The method is based on work by Ryabushko in [12]. The author in [12] studied to what extent only finitely many eigenvalues in one or both spectra determine the potential. We obtain a bound on variation of difference of the spectral functions for singular Dirac equation system.
General spin and pseudospin symmetries of the Dirac equation
Alberto, P; Frederico, T; de Castro, A
2015-01-01
In the 70's Smith and Tassie, and Bell and Ruegg independently found SU(2) symmetries of the Dirac equation with scalar and vector potentials. These symmetries, known as pseudospin and spin symmetries, have been extensively researched and applied to several physical systems. Twenty years after, in 1997, the pseudospin symmetry has been revealed by Ginocchio as a relativistic symmetry of the atomic nuclei when it is described by relativistic mean field hadronic models. The main feature of these symmetries is the suppression of the spin-orbit coupling either in the upper or lower components of the Dirac spinor, thereby turning the respective second-order equations into Schr\\"odinger-like equations, i.e, without a matrix structure. In this paper we propose a generalization of these SU(2) symmetries for potentials in the Dirac equation with several Lorentz structures, which also allow for the suppression of the matrix structure of second-order equation equation of either the upper or lower components of the Dirac...
Analytic Representation of Relativistic Wave Equations I The Dirac Case
Tepper, L; Zachary, W W
2003-01-01
In this paper we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wave functions, as is done by the Foldy-Wouthuysen method, and reveals the nonlocal time behavior of the particle-antiparticle relationship. It is well known that the Foldy-Wouthuysen transformation leads to a diagonalization that is nonlocal in space. We interpret the zitterbewegung, and the result that a velocity measurement (of a Dirac particle) at any instant in time is +(-)c, as reflections of the fact that the Dirac equation makes a spatially extended particle appear as a point in the present by forcing it to oscillate between the past and future at speed c. This suggests that although the Dirac Hamiltonian and the square-root Hamiltonian, are mathematically, they are not physically, equivalent. Furthermore, we see that alt! ho! ugh the form of the Dirac equation serve...
Non-Relativistic Limit of the Dirac Equation
Ajaib, Muhammad Adeel
2016-01-01
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.
Generalized Lorentz-Dirac equation for a strongly coupled gauge theory.
Chernicoff, Mariano; García, J Antonio; Güijosa, Alberto
2009-06-19
We derive a semiclassical equation of motion for a "composite" quark in strongly coupled large-N_{c} N = 4 super Yang-Mills theory, making use of the anti-de Sitter space/conformal field theory correspondence. The resulting nonlinear equation incorporates radiation damping, and reduces to the standard Lorentz-Dirac equation for external forces that are small on the scale of the quark Compton wavelength, but has no self-accelerating or preaccelerating solutions. From this equation one can read off a nonstandard dispersion relation for the quark, as well as a Lorentz-covariant formula for its radiation rate.
Spinor-electron wave guided modes in coupled quantum wells structures by solving the Dirac equation
Linares, Jesus [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)], E-mail: suso.linares.beiras@usc.es; Nistal, Maria C. [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)
2009-05-04
A quantum analysis based on the Dirac equation of the propagation of spinor-electron waves in coupled quantum wells, or equivalently coupled electron waveguides, is presented. The complete optical wave equations for Spin-Up (SU) and Spin-Down (SD) spinor-electron waves in these electron guides couplers are derived from the Dirac equation. The relativistic amplitudes and dispersion equations of the spinor-electron wave-guided modes in a planar quantum coupler formed by two coupled quantum wells, or equivalently by two coupled slab electron waveguides, are exactly derived. The main outcomes related to the spinor modal structure, such as the breaking of the non-relativistic degenerate spin states, the appearance of phase shifts associated with the spin polarization and so on, are shown.
Qualitative Properties of the Dirac Equation in a Central Potential
Esposito, G; Esposito, Giampiero; Santorelli, Pietro
1999-01-01
The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a perturbation resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the squared Dirac operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The underlying motivation for this qualitative analysis is given by the possib...
Higher dimensional supersymmetric quantum mechanics and Dirac equation
L P Singh; B Ram
2002-04-01
We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering `mass' as a function of coordinates. Its usefulness in solving potential problems is discussed with speciﬁc examples. We also discuss the `physical' signiﬁcance of the supersymmetric states in this formalism.
An extended Dirac equation in noncommutative space-time
Mendes, R Vilela
2015-01-01
Stabilizing, by deformation, the algebra of relativistic quantum mechanics a non-commutative space-time geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed, as well as the effects of coupling the two solutions.
Dirac reduced radial equations and the Problem of Additional Solutions
Khelashvili, Anzor
2016-01-01
For spinless particles there appear additional solutions in the framework of Schrodinger and Klein-Gordon equations. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions. Because the question is to determine the asymptotic behaviour of wave function at the origin, using the radial equations, is natural. The s...
Dirac Equation in Four Time and Four Space Dimensions
Nieto, J A
2016-01-01
The Dirac equation in four time and four space dimensions (or (4+4)-dimensions) is considered. Step by step we show that such an equation admits Majorana and Weyl solutions. In order to obtain the Majorana or Weyl spinors we used a method based on the construction of Clifford algebra in terms of 2x2-matrices. We argue that our approach can be useful in supergravity, superstrings and qubit theory.
Adjunctation and Scalar Product in the Dirac Equation - II
Dima, M.
2017-02-01
Part-I Dima (Int. J. Theor. Phys. 55, 949, 2016) of this paper showed in a representation independent way that γ 0 is the Bergmann-Pauli adjunctator of the Dirac { γ μ } set. The distiction was made between similarity (MATH) transformations and PHYS transformations - related to the (covariant) transformations of physical quantities. Covariance is due solely to the gauging of scalar products between systems of reference and not to the particular action of γ 0 on Lorentz boosts - a matter that in the past led inadvertently to the definition of a second scalar product (the Dirac-bar product). Part-II shows how two scalar products lead to contradictions and eliminates this un-natural duality in favour of the canonical scalar product and its gauge between systems of reference. What constitutes a proper observable is analysed and for instance spin is revealed not to embody one (except as projection on the boost direction - helicity). A thorough investigation into finding a proper-observable current for the theory shows that the Dirac equation does not possess one in operator form. A number of problems with the Dirac current operator are revealed - its Klein-Gordon counterpart being significantly more physical. The alternative suggested is finding a current for the Dirac theory in scalar form j^{μ } = < ρ rangle _{_{ψ }}v^{μ }_{ψ }.
On integrable rational potentials of the Dirac equation
Stachowiak, Tomasz, E-mail: stachowiak@cft.edu.pl [Center for Theoretical Physics PAS, Al. Lotnikow 32/46, 02-668 Warszawa (Poland); Przybylska, Maria, E-mail: M.Przybylska@proton.if.uz.zgora.pl [Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra (Poland)
2013-05-03
The one-dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.
Qualitative properties of the Dirac equation in a central potential
Esposito, Giampiero; Santorelli, Pietro
1999-07-01
The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wavefunction are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a `perturbation' resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the second-order operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The same methods are applied to the analysis of coupled eigenvalue equations when the anomalous magnetic moment of the electron is not neglected.
A generalized advection dispersion equation
Abdon Atangana
2014-02-01
This paper examines a possible effect of uncertainties, variability or heterogeneity of any dynamic system when being included in its evolution rule; the notion is illustrated with the advection dispersion equation, which describes the groundwater pollution model. An uncertain derivative is defined; some properties of the operator are presented. The operator is used to generalize the advection dispersion equation. The generalized equation differs from the standard equation in four properties. The generalized equation is solved via the variational iteration technique. Some illustrative figures are presented.
Non-Grassmann mechanical model of the Dirac equation
Deriglazov, A. A.; Zamudio, G. P.; Castro, P. S. [Department de Matematica, ICE, Universidade Federal de Juiz de Fora, MG (Brazil); Rizzuti, B. F. [ISB, Universidade Federal do Amazonas, Coari-AM (Brazil)
2012-12-15
We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both {Gamma}{sup {mu}} and {Gamma}{sup {mu}{nu}}-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing Zitterbewegung of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wavelength. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.
On integrable rational potentials of the Dirac equation
Stachowiak, Tomasz
2012-01-01
The Dirac equation, when reducible to an ordinary second order linear equation, exhibits a form of quasi-integrability, i.e. exact solutions exist only for a particular subset of energies. The differential Galois theory can be used to identify the integrable cases, recover integrable rational potentials, explicit solutions and strictly rule out the remaining cases as non-integrable. The effectiveness of this approach is demonstrated by providing a new class of potentials for which the equation in question can be transformed to the Whittaker form.
Essence of Special Relativity, Reduced Dirac Equation and Antigravity
Ni, Guang-jiong; Lou, Senyue; Xu, Jianjun
2010-01-01
The essence of special relativity is hiding in the equal existence of particle and antiparticle, which can be expressed by two discrete symmetries within one inertial frame --- the invariance under the (newly defined) space-time inversion (${\\bf x}\\to -{\\bf x},t\\to -t$), or equivalently, the invariance under a mass inversion ($m\\to -m$). The problems discussed are: the evolution of the $CPT$ invariance into a basic postulate, an unique solution to the original puzzle in Einstein-Podolsky-Rosen paradox, the reduced Dirac equation for hydrogenlike atoms, and the negative mass paradox leading to the prediction of antigravity between matter and antimatter. {\\bf Keywords}: Special relativity, Reduced Dirac Equation, Antiparticle, Antigravity
Dirac Equation in Noncommutative Space for Hydrogen Atom
Adorno, T C; Chaichian, M; Gitman, D M; Tureanu, A
2009-01-01
We consider the energy levels of a hydrogen-like atom in the framework of $\\theta $-modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels $2S_{1/2}, 2P_{1/2}$ and $ 2P_{3/2}$ is lifted completely, such that new transition channels are allowed.
Dirac equation in noncommutative space for hydrogen atom
Adorno, T.C., E-mail: tadorno@nonada.if.usp.b [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, CEP 05508-090 Sao Paulo, SP (Brazil); Baldiotti, M.C., E-mail: baldiott@fma.if.usp.b [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, CEP 05508-090 Sao Paulo, SP (Brazil); Chaichian, M., E-mail: Masud.Chaichian@helsinki.f [Department of Physics, University of Helsinki and Helsinki Institute of Physics, PO Box 64, FIN-00014 Helsinki (Finland); Gitman, D.M., E-mail: gitman@dfn.if.usp.b [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, CEP 05508-090 Sao Paulo, SP (Brazil); Tureanu, A., E-mail: Anca.Tureanu@helsinki.f [Department of Physics, University of Helsinki and Helsinki Institute of Physics, PO Box 64, FIN-00014 Helsinki (Finland)
2009-11-30
We consider the energy levels of a hydrogen-like atom in the framework of theta-modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels 2S{sub 1/2}, 2P{sub 1/2} and 2P{sub 3/2} is lifted completely, such that new transition channels are allowed.
Solving Dirac equation using the tridiagonal matrix representation approach
Alhaidari, A.D. [Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438 (Saudi Arabia); Bahlouli, H., E-mail: bahlouli@kfupm.edu.sa [Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438 (Saudi Arabia); Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261 (Saudi Arabia); Assi, I.A. [Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261 (Saudi Arabia)
2016-04-22
The aim of this work is to find exact solutions of the one dimensional Dirac equation using the tridiagonal matrix representation. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. We are honored to dedicate this work to Prof. Hashim A. Yamani on the occasion of his 70th birthday. - Highlights: • We choose L2 basis such that the Dirac wave operator is tridiagonal matrix. • We use the tridiagonal-matrix-representation approach. • The wave equation becomes a symmetric three-term recursion relation. • We solve the associated three-term recursion relation exactly. • The energy spectrum formula is obtained.
Noncommutativity into Dirac Equation with mass dependent on the position
Bastos, Samuel Batista; Almeida, Carlos Alberto Santos [Universidade Federal do Ceara - UFC, Fortaleza, CE (Brazil); Nunes, Luciana Angelica da Silva [Universidade Federal Rural do Semi-arido - UFERSA, Mossoro, RN (Brazil)
2013-07-01
Full text: In recent years, there is growing interest in the study of theories in non-commutative spaces. Non-commutative fields theories are related with compactifications of M theory, string theory and the quantum Hall effect. Moreover, the role of the non-commutativity of theories of a particle finds large applications when analyzed in scenarios of quantum mechanics and relativistic quantum mechanics. In these contexts investigations on the Schrodinger and Dirac equations with mass depending on the position (MDP) has attracted much attention in the literature. Systems endowed with MDP models are useful for the study of many physical problems. In particular, they are used to study the energy density in problems of many bodies, determining the electronic properties of semiconductor heterostructures and also to describe the properties of heterojunctions and quantum dots. In particular, the investigation of relativistic effects it is important for systems containing heavy atoms or doping by heavy ions. For these types of materials, the study of the properties of the Dirac equation, in the case where the mass becomes variable is of great interest. In this paper, we seek for the non-relativistic limit of the Dirac Hamiltonian in the context of a theory of effective mass, through a Foldy-Wouthuysen transformation. We analyse the Dirac equation with mass dependent on the position, in a smooth step shape mass distribution, in non-commutative space (NC). This potential type kink was recently discussed by several authors in the commutative context and now we present our results in the non-commutative context. (author)
Solution of the Dirac Equation with Special Hulthen Potentials
郭建友; 孟杰; 徐辅新
2003-01-01
The Dirac equation for the special case of a spinor in the relativistic potential with the even and odd components related by a constraint is solved exactly when the even component is chosen to be the Hulthen potential.The corresponding radial wavefunctions for two-component spinor are obtained in terms of the hypergeometric function, and the energy spectrum of the bound states is obtained as a solution to a given equation by boundary constraints, in which the nonrelativistic limit reproduces the usual Hulthen potential.
Abstract Wave Equations and Associated Dirac-Type Operators
Gesztesy, Fritz; Holden, Helge; Teschl, Gerald
2010-01-01
We discuss the unitary equivalence of generators $G_{A,R}$ associated with abstract damped wave equations of the type $\\ddot{u} + R \\dot{u} + A^*A u = 0$ in some Hilbert space $\\mathcal{H}_1$ and certain non-self-adjoint Dirac-type operators $Q_{A,R}$ (away from the nullspace of the latter) in $\\mathcal{H}_1 \\oplus \\mathcal{H}_2$. The operator $Q_{A,R}$ represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of $A^*A$. In addition to the unitary equivalence results concerning $G_{A,R}$ and $Q_{A,R}$, we provide a detailed study of the domain of the generator $G_{A,R}$, consider spectral properties of the underlying quadratic operator pencil $M(z) = |A|^2 - iz R - z^2 I_{\\mathcal{H}_1}$, $z\\in\\mathbb{C}$, derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The...
Wave-equation dispersion inversion
Li, Jing
2016-12-08
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.
Quantum mechanics for three versions of the Dirac equation in a curved spacetime
Arminjon, Mayeul
2008-01-01
We present a recent work on the Dirac equation in a curved spacetime. In addition to the standard equation, two alternative versions are considered, derived from wave mechanics, and based on the tensor representation of the Dirac field. The latter considers the Dirac wave function as a spacetime vector and the set of the Dirac matrices as a third-order tensor. Having the probability current conserved for any solution of the Dirac equation gives an equation to be satisfied by the coefficient fields. A positive definite scalar product is defined and a hermiticity condition for the Dirac Hamiltonian is derived for a general coordinate system in a general curved spacetime. For the standard equation, the hermiticity of the Dirac Hamiltonian is not preserved under all admissible changes of the coefficient fields.
The square root of the Dirac operator on the superspace and the Maxwell equations
Bzdak, A; Bzdak, Adam; Hadasz, Leszek
2003-01-01
We re-consider the procedure of ``taking a square root of the Dirac equation'' on the superspace and show that it leads to the well known superfield W_\\alpha and to the proper equations of motion for the components, i.e. the Maxwell equations and the massless Dirac equation.
The square root of the Dirac operator on superspace and the Maxwell equations
Bzdak, Adam; Hadasz, Leszek
2004-02-26
We re-consider the procedure of 'taking a square root of the Dirac equation' on superspace and show that it leads to the well-known superfield W{sub {alpha}} and to the proper equations of motion for the components, i.e., the Maxwell equations and the massless Dirac equation.
Solution of One-dimensional Dirac Equation via Poincare Map
Bahlouli, Hocine; Jellal, Ahmed
2011-01-01
We solve the general one-dimensional Dirac equation using a "Poincare Map" approach which avoids any approximation to the spacial derivatives and reduces the problem to a simple recursive relation which is very practical from the numerical implementation point of view. To test the efficiency and rapid convergence of this approach we apply it to a vector coupling Woods--Saxon potential, which is exactly solvable. Comparison with available analytical results is impressive and hence validates the accuracy and efficiency of this method.
The supersymmetric Dirac equation the application to hydrogenic atoms
Hirshfeld, Allen
2012-01-01
The solution of the Dirac equation for an electron in a Coulomb field is systematically treated here by utilizing new insights provided by supersymmetry. It is shown that each of the concepts has its analogue in the non-relativistic case. Indeed, the non-relativistic case is developed first, in order to introduce the new concepts in a familiar context. The symmetry of the non-relativistic model is already present in the classical limit, so the classical Kepler problem is first discussed in order to bring out the role played by the Laplace vector, one of the central concepts of the whole book.
General Electromagnetic Nonminimal Couplings in the Dirac Equation
Araujo, J B; Ferreira, M M
2016-01-01
We examine a new class of CPT-even and dimension-five nonminimal interactions between fermions and photons, deprived of higher-order derivatives, yielding electric dipole moment and magnetic dipole moment in the context of the Dirac equation. These couplings are Lorentz-violating nonminimal structures, composed of a rank-2 tensor, the electromagnetic tensor, and gamma matrices, being addressed in its axial and non-axial hermitian versions. We use the electron's anomalous magnetic and electric dipole moment measurements to reach upper bounds of $1$ part in $10^{11}$ and $10^{16}$ GeV$^{-1}$.
Nonlocal separable potential in the one-dimensional Dirac equation
Calkin, M.G.; Kiang, D.; Nogami, Y.
1988-08-01
The one-dimensional Dirac equation is solved for a separable potential of the form of Lorentz scalar plus vector, (..beta..g+h)v(x)v(x'). Exact analytic solutions are obtained for bound and scattering states for arbitrary v(x). For a particular combination of the values of g and h, degeneracy of the bound state occurs, and total reflection also takes place for a certain incident energy. The limiting case, in which v(x) becomes a delta function, is discussed in detail.
Abraham-Lorentz-Dirac Equation in 5D Stuekelberg Electrodynamics
Land, Martin
2016-01-01
We derive the Abraham-Lorentz-Dirac (ALD) equation in the framework of the electrodynamic theory associated with Stueckelberg manifestly covariant canonical mechanics. In this framework, a particle worldline is traced out through the evolution of an event $x^\\mu(\\tau)$. By admitting unconstrained commutation relations between the positions and velocities, the associated electromagnetic gauge fields are in general dependent on the parameter $\\tau$, which plays the role of time in Newtonian mechanics. Standard Maxwell theory emerges from this system as a $\\tau$-independent equilibrium limit. In this paper, we calculate the $\\tau$-dependent field induced by an arbitrarily evolving event, and study the long-range radiation part, which is seen to be an on-shell plane wave of the Maxwell type. Following Dirac's method, we obtain an expression for the finite part of the self-interaction, which leads to the ALD equation that generalizes the Lorentz force. This third-order differential equation is then converted to an...
A novel quantum-mechanical interpretation of the Dirac equation
K-H Kiessling, M.; Tahvildar-Zadeh, A. S.
2016-04-01
A novel interpretation is given of Dirac’s ‘wave equation for the relativistic electron’ as a quantum-mechanical one-particle equation. In this interpretation the electron and the positron are merely the two different ‘topological spin’ states of a single more fundamental particle, not distinct particles in their own right. The new interpretation is backed up by the existence of such ‘bi-particle’ structures in general relativity, in particular the ring singularity present in any spacelike section of the spacetime singularity of the maximal-analytically extended, topologically non-trivial, electromagnetic Kerr-Newman (KN)spacetime in the zero-gravity limit (here, ‘zero-gravity’ means the limit G\\to 0, where G is Newton’s constant of universal gravitation). This novel interpretation resolves the dilemma that Dirac’s wave equation seems to be capable of describing both the electron and the positron in ‘external’ fields in many relevant situations, while the bi-spinorial wave function has only a single position variable in its argument, not two—as it should if it were a quantum-mechanical two-particle wave equation. A Dirac equation is formulated for such a ring-like bi-particle which interacts with a static point charge located elsewhere in the topologically non-trivial physical space associated with the moving ring particle, the motion being governed by a de Broglie-Bohm type law extracted from the Dirac equation. As an application, the pertinent general-relativistic zero-gravity hydrogen problem is studied in the usual Born-Oppenheimer approximation. Its spectral results suggest that the zero-G KN magnetic moment be identified with the so-called ‘anomalous magnetic moment of the physical electron,’ not with the Bohr magneton, so that the ring radius is only a tiny fraction of the electron’s reduced Compton wavelength.
Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation
Bogoslovsky, G Yu
2004-01-01
The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space-time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented.
Considerations concering the generalization of the Dirac equations to unstable fermions
Kniehl, Bernd A.; Sirlin, Alberto [Max-Planck-Institut fuer Physik (Werner-Heisenberg-Institut), Muenchen (Germany)
2014-08-15
We discuss the generalization of the Dirac equations and spinors in momentum space to free unstable spin-1/2 fermions taking into account the fundamental requirement of Lorentz covariance. We derive the generalized adjoint Dirac equations and spinors, and explain the very simple relation that exists, in our formulation, between the unstable and stable cases. As an application of the generalized spinors, we evaluate the probability density. We also discuss the behavior of the generalized Dirac equations under time reversal.
Relativistic Dirac equation for particles with arbitrary half-integral spin
Guseinov, I I
2008-01-01
The sets of 2(2s+1)-component matrices through the four-component Dirac matrices are suggested, where s=3/2, 5/2,.... Using these matrices sets the Dirac relativistic equation for a description of arbitrary half-integral spin particles is constructed. The new Dirac equation of motion leads to an equation of continuity with a positive-definite probability density.
Introduction to nonlinear dispersive equations
Linares, Felipe
2015-01-01
This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introdu...
Yang-Mills gauge fields conserving symmetry algebra of Dirac equation in homogeneous space
Breev, A I
2014-01-01
We consider the Dirac equation with external Yang-Mills gauge field in a homogeneous space with invariant metric. The Yang-Mills fields for which the motion group of the space serves as symmetry group of the Dirac equation are found by comparison of the Dirac equation with a invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The basis of eigenfunctions and corresponding spectrum are obtained for the Dirac equation in the space $\\mathbb{R}^2 \\times \\mathbb{S}^2$ in the framework of the noncommutative integration method.
Partially Ordered Sets of Quantum Measurements and the Dirac Equation
Knuth, Kevin H.
2012-02-01
Events can be ordered according to whether one event influences another. This results in a partially ordered set (poset) of events often referred to as a causal set. In this framework, an observer can be represented by a chain of events. Quantification of events and pairs of events, referred to as intervals, can be performed by projecting them onto an observer chain, or even a pair of observer chains, which in specific situations leads to a Minkowski metric replete with Lorentz transformations (Bahreyni & Knuth, 2011. APS B21.00007). In this work, we unify this picture with the Process Calculus, which coincides with the Feynman rules of quantum mechanics (Goyal, Knuth, Skilling, 2010, arXiv:0907.0909; Goyal & Knuth, Symmetry 2011, 3(2), 171), by considering quantum measurements to be events. This is performed by quantifying pairs of events, which represent transitions, with a pair of numbers, or a quantum amplitude. In the 1+1D case this results in the Feynman checkerboard model of the Dirac equation (Feynman & Hibbs, 1965). We further demonstrate that in the case of 3+1 dimensions, we recover Bialnycki-Birula's (1994, Phys. Rev. D, 49(12), 6920) body-centered cubic cellular automata model of the Dirac equation studied more recently by Earle (2011, arXiv:1102.1200v1).
Dirac and Higher-Spin Equations of Negative Energies
Dvoeglazov, Valeriy V
2011-01-01
It is easy to check that both algebraic equation Det (\\hat p - m) =0 and Det (\\hat p + m) =0 for 4-spinors u- and v- have solutions with p_0= \\pm E_p =\\pm \\sqrt{{\\bf p}^2 +m^2}. The same is true for higher-spin equations. Meanwhile, every book considers the p_0=E_p only for both u- and v- spinors of the (1/2,0)\\oplus (0,1/2)) representation, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of negative-energy solutions. Recent works of Ziino (and, independently, of several others) show that the Fock space can be doubled. We re-consider this possibility on the quantum field level for both s=1/2 and higher spins particles.
Two-body Dirac equation approach to the deuteron
Galeao, A.P.; Castilho A, J.A.; Ferreira, P. Leal
1996-06-01
The two-body Dirac (Breit) equation with potentials associated to one-boson-exchanges with cutoff masses is solved for the deuteron and its observables calculated. The 16-component wave-function for the J{sup {pi}} = 1{sup +} state contains four independent radial functions which satisfy a system of four coupled differential equations of firs order. This system is numerically integrated, from infinity towards the origin, by fixing the value of the deuteron binding energy and imposing appropriate boundary conditions at infinity. For the exchange potential of the pion, a mixture of direct plus derivative couplings to the nucleon is considered. We varied the pion-nucleon coupling constant, and the best results of our calculations agree with the lower values recently determined for this constant. The present treatment differs from the more conventional ones in that non-relativistic reductions up to the order c{sup -2} are not used. (author). 20 refs., 1 fig., 2 tabs.
A B-spline Galerkin method for the Dirac equation
Froese Fischer, Charlotte; Zatsarinny, Oleg
2009-06-01
The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y=-λy, that can also be written as a pair of first-order equations y=λz, z=-λy. Expanding both y(r) and z(r) in the B basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the B basis and z(r) in the dB/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method ( B,B) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.
Analytical solutions for Dirac and Klein-Gordon equations using Backlund transformations
Zabadal, Jorge R.; Borges, Volnei, E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Dept. de Engenharia Mecanica; Ribeiro, Vinicius G., E-mail: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio, E-mail: marciophd@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Centro de Estudos Interdisciplinares
2015-07-01
This work presents a new analytical method for solving Klein-Gordon type equations via Backlund transformations. The method consists in mapping the Klein-Gordon model into a first order system of partial differential equations, which contains a generalized velocity field instead of the Dirac matrices. This system is a tensor model for quantum field theory whose space solution is wider than the Dirac model in the original form. Thus, after finding analytical expressions for the wave functions, the Maxwell field can be readily obtained from the Dirac equations, furnishing a self-consistent field solution for the Maxwell-Dirac system. Analytical and numerical results are reported. (author)
Duarte, Celso de Araujo
2015-01-01
Traditionally, the electromagnetic theory dictates the well-known second order differential equation for the components of the scalar and the vector potentials, or in other words, for the four-vector electromagnetic potential $\\phi^{\\mu}$. But the second order is not obligatory at least with respect to the electromagnetic radiation fields: actually, a heuristic first order differential equation can be constructed to describe the electromagnetic radiation, supported on the phenomenology of its electric and magnetic fields. Due to a formal similarity, such an equation suggests a direct comparative analysis with Dirac's equation for half spin fermions, conducting to the finding that the Dirac's spinor field $\\Psi$ for massive or massless fermions is equivalent to a set of two potential-like four vector fields $\\psi^{\\mu}$ and $\\chi^{\\mu}$. Under this point of view, striking similarities with the electromagnetic theory emerge with a category of "pseudo electric'' and "pseudo magnetic'' vector fermionic fields.
Searching for an equation: Dirac, Majorana and the others
Esposito, S
2011-01-01
We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein-Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz-Pauli, Kemmer, Rarita-Schwinger and many others. A particular interest comes out for the general problem of the description of particles with \\textit{arbitrary} spin, introduced (and solved) by Majorana as early as 1932, and later reconsidered, within a different approach, by Dirac in 1936 and by Fierz-Pauli in 1939. The final settlement of the problem in 1945 by Bhabha, who came back to the general ideas introduced by Majorana in 1932, is discussed as well, and, by making recourse also to unpublished documents by Majorana, we are able to reconstruct the line of reasoning behind the Majorana and the Bhabha equations, ...
Smoothing Effects for the Classical Solutions to the Landau-Fermi-Dirac Equation
Shuangqian LIU
2012-01-01
The smoothness of the solutions to the full Landau equation for Fermi-Dirac particles is investigated.It is shown that the classical solutions near equilibrium to the Landau-Fermi-Dirac equation have a regularizing effects in all variables (time,space and velocity),that is,they become immediately smooth with respect to all variables.
Dirac equation in very special relativity for hydrogen atom
Maluf, R V; Cruz, W T; Almeida, C A S
2014-01-01
In this work, we study the modified Dirac equation in the framework of very special relativity (VSR). The low-energy regime is accessed and the nonrelativistic Hamiltonian is obtained. It turns out that this Hamiltonian is similar to that achieved from the Standard Model Extension (SME) via coupling of the spinor field to a Lorentz-violating term, but new features arise inherited from the non-local character of the VSR. In addition, the implications of the VSR-modified Lorentz symmetry on the spectrum of a hydrogen atom is determined by calculating the first-order energy corrections in the context of standard quantum mechanics. Among the results, we highlight that the modified Hamiltonian provides non-vanishing corrections which lift the degeneracy of the energy levels and allow us to find an upper bound upon the VSR-parameter.
Lorentz invariant CPT breaking in the Dirac equation
Fujikawa, Kazuo
2016-01-01
If one modifies the Dirac equation in momentum space to $[\\gamma^{\\mu}p_{\\mu}-m-\\Delta m(\\theta(p_{0})-\\theta(-p_{0})) \\theta(p_{\\mu}^{2})]\\psi(p)=0$, the symmetry of positive and negative energy eigenvalues is lifted by $m\\pm \\Delta m$ for a small $\\Delta m$. The mass degeneracy of the particle and antiparticle is thus lifted in a Lorentz invariant manner since the combinations $\\theta(\\pm p_{0})\\theta(p_{\\mu}^{2})$ with step functions are manifestly Lorentz invariant. We explain an explicit construction of this CPT breaking term in coordinate space, which is Lorentz invariant but non-local at a distance scale of the Planck length. The application of this Lorentz invariant CPT breaking mechanism to the possible mass splitting of the neutrino and antineutrino in the Standard Model is briefly discussed.
Dirac equation in very special relativity for hydrogen atom
R.V. Maluf
2014-11-01
Full Text Available In this work, we study the modified Dirac equation in the framework of very special relativity (VSR. The low-energy regime is accessed and the nonrelativistic Hamiltonian is obtained. It turns out that this Hamiltonian is similar to that achieved from the Standard Model Extension (SME via coupling of the spinor field to a Lorentz-violating term, but new features arise inherited from the non-local character of the VSR. In addition, the implications of the VSR-modified Lorentz symmetry on the spectrum of a hydrogen atom are determined by calculating the first-order energy corrections in the context of standard quantum mechanics. Among the results, we highlight that the modified Hamiltonian provides non-vanishing corrections which lift the degeneracy of the energy levels and allow us to find an upper bound upon the VSR-parameter.
J J THOMSON'S ELECTRON: The Dirac equation. Cosmic implications of a tidy electron
Miller, David
1997-07-01
Dirac devised the quantum theory of the electron itself, which required him to generalize SchrÃ¶dinger's famous equation to cover relativistic motion. He interpreted the resulting equation as showing that an antiparticle to the electron must exist.
Pathology-Free Modification of the Lorentz-Dirac Equation
Blinder, S. M.
2001-04-01
The Lorentz-Dirac equation for the force on an accelerating electron is conventionally written in covariant form F_ext^λ=ma^λ-2 e^2\\over 3 c^3(dot a^λ+1\\over c^2 a^2 v^λ) However, this equation has fallen into disfavor in recent years because it admits pathological solutions representing runaway behavior or preacceleration violating classical causality. For example, force-free motion can exhibit unphysical runaway solutions of the form a(t)= a(0)exp(t/ τ_0), where τ_0≡ 2e^2/3mc^3≈ 6.26× 10-24 sec. Note that the first two terms of the L-D equation could originate from expansion of ma^λ(τ-τ_0) in powers of τ_0. We propose the following differential-difference equation as a compact non-pathologial alternative to the L-D equation: F^λ_ext(τ)=m [a^λ β^μ -a^μ β^λ]_τ-τ0 β_μ(τ) where β^λ=v^λ/c. Expansion of the bracketed quantity reacquires the conventional equation, apart from higher-order terms in τ_0. It can be demonstrated that F=0 unambiguously implies a=0. Moreover the occurrence of the retarded time variable τ-τ0 precludes any solutions with preacceleration. A more detailed derivation is given in a forthcoming paper [S. M. Blinder, ``Classical electrodynamics with vacuum polarization: electron self-energy and radiation reaction," Repts. Math. Phys., in press].
Calculation of the Dirac equation in curved spacetimes with possible torsion using MAPLE and REDUCE
Vulcanov, Dumitru N.
2003-08-01
The article presents computer algebra procedures and routines applied to the study of the Dirac field on curved spacetimes. The main part of the procedures is devoted to the construction of Pauli and Dirac matrices algebra on an anholonomic orthonormal reference frame. Then these procedures are used to compute the Dirac equation on curved spacetimes in a sequence of special dedicated routines. A comparative review of such procedures obtained for two computer algebra platforms (REDUCE+EXCALC and MAPLE+GRTensorII) is carried out. Applications for the calculus of Dirac equation on specific examples of spacetimes with or without torsion are pointed out.
Staggered grid leap-frog scheme for the (2+1)D Dirac equation
Hammer, René
2013-01-01
A numerical scheme utilizing a grid which is staggered in both space and time is proposed for the numerical solution of the (2+1)D Dirac equation in presence of an external electromagnetic potential. It preserves the linear dispersion relation of the free Weyl equation for wave vectors aligned with the grid and facilitates the implementation of open (absorbing) boundary conditions via an imaginary potential term. This explicit scheme has second order accuracy in space and time. A functional for the norm is derived and shown to be conserved. Stability conditions are derived. Several numerical examples, ranging from generic to specific to textured topological insulator surfaces, demonstrate the properties of the scheme which can handle general electromagnetic potential landscapes.
Relativistic (Dirac equation) effects in microscopic elastic scattering calculations
Hynes, M. V.; Picklesimer, A.; Tandy, P. C.; Thaler, R. M.
1985-04-01
A simple relativistic extension of the first-order multiple scattering mechanism for the optical potential is employed within the context of a Dirac equation description of elastic nucleon-nucleus scattering. A formulation of this problem in terms of a momentum-space integral equation displaying an identifiable nonrelativistic sector is described and applied. Extensive calculations are presented for proton scattering from 40Ca and 16O at energies between 100 and 500 MeV. Effects arising from the relativistic description of the propagation of the projectile are isolated and are shown to be responsible for most of the departures from typical nonrelativistic (Schrödinger) results. Off-shell and nonlocal effects are included and these, together with uncertainties in the nuclear densities, are shown not to compromise the characteristic improvement of forward angle spin observable predictions provided by the relativistic approach. The sensitivity to ambiguities in the Lorentz scalar and vector composition of the optical potential is displayed and discussed.
Relativistic (Dirac equation) effects in microscopic elastic scattering calculations
Hynes, M.V.; Picklesimer, A.; Tandy, P.C.; Thaler, R.M.
1985-04-01
A simple relativistic extension of the first-order multiple scattering mechanism for the optical potential is employed within the context of a Dirac equation description of elastic nucleon-nucleus scattering. A formulation of this problem in terms of a momentum-space integral equation displaying an identifiable nonrelativistic sector is described and applied. Extensive calculations are presented for proton scattering from /sup 40/Ca and /sup 16/O at energies between 100 and 500 MeV. Effects arising from the relativistic description of the propagation of the projectile are isolated and are shown to be responsible for most of the departures from typical nonrelativistic (Schroedinger) results. Off-shell and nonlocal effects are included and these, together with uncertainties in the nuclear densities, are shown not to compromise the characteristic improvement of forward angle spin observable predictions provided by the relativistic approach. The sensitivity to ambiguities in the Lorentz scalar and vector composition of the optical potential is displayed and discussed.
Röken, Christian
2015-01-01
The separability of the massive Dirac equation in a rotating Kerr black hole background in advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr spacetime is described in the framework of the Newman-Penrose formalism by a local Carter tetrad, and the Dirac wave functions are given on a spin bundle in a chiral Newman-Penrose dyad representation. Applying mode analysis techniques, the Dirac equation is separated into coupled systems of radial and angular ordinary differential equations. Asymptotic radial solutions at infinity and the event and Cauchy horizons are explicitly derived and, by means of error estimates, the decay properties are analyzed. Solutions of the angular ordinary differential equations matching the Chandrasekhar-Page equation are discussed. These solutions are used in order to study the scattering of Dirac waves by the gravitational field of a Kerr black hole for an observer described by a frame without coordinate singularities at the inner horizon boundaries such t...
Schr\\"odinger-Pauli Equation for the Standard Model Extension CPT-Violating Dirac Equation
Gutierrez, Thomas D
2015-01-01
It is instructive to investigate the non-relativistic limit of the simplest Standard Model Extension (SME) CPT-violating Dirac-like equation but with minimal coupling to the electromagnetic fields. In this limit, it becomes an intuitive Schr\\"odinger-Pauli-like equation. This is comparable to the free particle treatment as explored by Kostelecky and Lane, but this exercise only considers the $a$ and $b$ CPT-violating terms and $\\vec{p}/m$ terms to first order. Several toy systems are discussed.
Inferences about interactions: Fermions and the Dirac equation
Knuth, Kevin H.
2013-08-01
At a fundamental level every measurement process relies on an interaction where one entity influences another. The boundary of an interaction is given by a pair of events, which can be ordered by virtue of the interaction. This results in a partially ordered set (poset) of events often referred to as a causal set. In this framework, an observer can be represented by a chain of events. Quantification of events and pairs of events, referred to as intervals, can be performed by projecting them onto an observer chain, or even a pair of observer chains, which in specific situations leads to a Minkowski metric replete with Lorentz transformations. We illustrate how this framework of interaction events gives rise to some of the well-known properties of the Fermions, such as Zitterbewegung. We then take this further by making inferences about events, which is performed by employing the process calculus, which coincides with the Feynman path integral formulation of quantum mechanics. We show that in the 1+1 dimensional case this results in the Feynman checkerboard model of the Dirac equation describing a Fermion at rest.
The Dirac equation applied to graphene in the presence of topological defects
Cunha, Marcio de Moura; Ribeiro, Carlos Alberto de Lima [Universidade Estadual de Feira de Santana, BA (Brazil)
2011-07-01
Full text: The Dirac equation was proposed by Paul Dirac in 1928, in an attempt to get a relativistic wave equation for particles of spin 1/2, because the Schroedinger equation does not remain invariant under Lorentz transformations and the Klein-Gordon only serves for spin 0 particles . Since then, it has been able to describe various systems, in several areas of physics, such as Field Theory, Condensed Matter, among others. Recently, some researchers have use this equation to study the graphene, a very promising material, that consist essentially in a monolayer of carbon atoms, with interesting electronic and transport properties and several possibilities of applications in Material Science and Engineering, for instance. In this work, we study the application of the Dirac equation in graphene, more specifically in the presence of topological defects, that change the physical properties of the material. This is possible because in the formalism of the Dirac equation, we can replace the derivative usual term by a term of covariant derivative, capable of describing the geometry of the space considered. From the job of Vozmediano {sup a} and others found in the literature, we write the dirac equation for graphene in presence of a defect, making a modification in the usual Dirac equation. (author)
A Matter of Principle: The Principles of Quantum Theory, Dirac's Equation, and Quantum Information
Plotnitsky, Arkady
2015-01-01
This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be be addressed as well in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac's work, which, in particular Dirac's derivation of his relativistic equation for the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall, however, also consider Heisenberg's derivation of quantum mechanics, which inspired Dirac. I argue that Heisenberg's and Dirac's work alike was guided by their adherence to and confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by G. M. D' Ariano and his coworkers on the principles of quantum information theory, which extends quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac's equation from these principles alone...
张学骜; 陈柯; 段正路
2005-01-01
Solving the Klein-Gordon equation and Dirac equation with ring-shaped non-spherical oscillator gives the exact bound state wavefunction and energy equation, and the relations between non-relativistic Schrodinger equation, KleinGordon equation and Dirac equation with equal scalar and vector potentials.
On Charge Conjugation, Chirality and Helicity of the Dirac and Majorana Equation for Massive Leptons
Eckart Marsch
2015-04-01
Full Text Available We revisit the charge-conjugation operation for the Dirac equation in its chiral representation. A new decomposition of the Dirac spinor field is suggested and achieved by means of projection operators based on charge conjugation, which is discussed here in a non-standard way. Thus, two separate two-component Majorana-type field equations for the eigenfields of the charge-conjugation operator are obtained. The corresponding free fields are entirely separated without a gauge field, but remain mixed and coupled together through an electromagnetic field term. For fermions that are charged and, thus, subjected to the gauge field of electrodynamics, these two Majorana fields can be reassembled into a doublet, which is equivalent to a standard four-component Dirac spinor field. In this way, the Dirac equation is retained in a new guise, which is fully equivalent to that equation in its chiral form.
Exact solution to the one-dimensional Dirac equation of linear potential
Long Chao-Yun; Qin Shui-Jie
2007-01-01
In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.
Solution of the Dirac equation in a curved space with static metric
Alhaidari, A D
2015-01-01
Compatibility of symmetric quantization of the Dirac equation in a curved space with general covariance gives a special representation of the spin connections in which their dot product with the Dirac gamma matrices becomes equal to the "covariant divergence" of the latter. Requiring that the square of the equation gives the conventional Klein-Gordon equation in a curved space results in an operator algebra for the Dirac gamma matrices that involves the "covariant derivative" connections and the Riemann-Christoffel connections. In 1+1 space-time with static metric, we obtain exact solutions of this Dirac equation model for some examples. We also formulate the interacting theory of the model with various coupling modes and solve it in the same space for a given potential configuration.
A detailed study of nonperturbative solutions of two-body Dirac equations
Crater, H.W.; Becker, R.L.; Wong, C.Y.; Van Alstine, P.
1992-12-01
In quark model calculations of the meson spectrums fully covariant two-body Dirac equations dictated by Dirac's relativistic constraint mechanics gave a good fit to the entire meson mass spectrum for light quark mesons as well as heavy quark mesons with constituent world scalar and vector potentials depending on just one or two parameters. In this paper, we investigate the properties of these equations that made them work so well by solving them numerically for quantum electrodynamics (QED) and related field theories. The constraint formalism generates a relativistic quantum mechanics defined by two coupled Dirac equations on a sixteen component wave function which contain Lorentz covariant constituent potentials that are initially undetermined. An exact Pauli reduction leads to a second order relativistic Schroedinger-like equation for a reduced eight component wave function determined by an effective interaction -- the quasipotential. We first determine perturbatively to lowest order the relativistic quasipotential for the Schroedinger-like equation by comparing that form with one derived from the Bethe-Salpeter equation. Insertion of this perturbative information into the minimal interaction structures of the two-body Dirac equations then completely determines their interaction structures. Then we give a procedure for constructing the full sixteen component solution to our coupled first-order Dirac equations from a solution of the second order equation for the reduced wave function. Next, we show that a perturbative treatment of these equations yields the standard spectral results for QED and related interactions.
Supersymmetry and the Dirac Equation with Vector and Scalar Coupling Potentials
无
2000-01-01
This paper shows that one type of first-order Dirac equation with vector coupling and scalar coupling potentials can be brought into the framework of non-relativistic supersymmetric quantum mechanics. The conclusion is independent of the concrete forms of the vector and scalar coupling potentials because of the nilpotent matrix realization of supersymmetric quantum mechanical algebra. The supersymmetry of this kind of Dirac equation requires that a spin-orbit coupling term be introduced into the associated supersymmetric Hamiltonian.
Dirac equation in gauge and affine-metric gravitation theories
Giachetta, G
1995-01-01
We show that the covariant derivative of Dirac fermion fields in the presence of a general linear connection on a world manifold is universal for Einstein's, gauge and affine-metric gravitation theories.
P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation
Duran, Antonio J [Departamento de Analisis Matematico, Universidad de Sevilla, Apdo (PO BOX) 1160, 41080 Sevilla (Spain); Gruenbaum, F Alberto [Department of Mathematics, University of California, Berkeley, CA 94720 (United States)
2006-04-07
The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.
Prastyaningrum, I.; Cari, C.; Suparmi, A.
2016-11-01
The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part.
Rauscher, Elizabeth A
2011-01-01
The Maxwell, Einstein, Schrödinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal
Basic quantum mechanics for three Dirac equations in a curved spacetime
Arminjon, Mayeul
2008-01-01
We consider three versions of the Dirac equation in a curved spacetime: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed tensor representation of the Dirac field (TRD). These three equations differ in the covariant derivative D_mu. A common tool is the hermitizing matrix A. Having the current conservation for any solution of the Dirac equation is equivalent to D_mu (A gamma^mu)=0, where gamma^mu is the field of Dirac matrices. This condition is always verified for DFW with its restricted choice for the field gamma^mu. It similarly restricts the choice of the field gamma^mu for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. For the Dirac Hamiltonian, a positive definite scalar product is defined for all reference frames, and a hermiticity condition is derived in a general curved spacetime with minor restrictions on the coordinate system. For DFW, the hermiticity of t...
Exact Solution of the Curved Dirac Equation in Polar Coordinates: Master Function Approach
H. Panahi
2015-01-01
Full Text Available We show that the (2+1 curved Dirac equation in polar coordinates can be transformed into Schrodinger-like differential equation for upper spinor component. We compare this equation with the Schrodinger equation derived from shape invariance property of second order differential equations of mathematical physics. This formalism enables us to determine the electrostatic potential and relativistic energy in terms of master function and corresponding weight function. We also obtain the spinor wave function in terms of orthogonal polynomials.
Fukushima, Kimichika
2015-01-01
This paper presents analytical eigenenergies for a pair of confined fundamental fermion and antifermion under a linear potential derived from the Wilson loop for the non-Abelian Yang-Mills field. We use basis functions localized in spacetime, and the Hamiltonian matrix of the Dirac equation is analytically diagonalized. The squared system eigenenergies are proportional to the string tension and the absolute value of the Dirac's relativistic quantum number related to the total angular momentum, consistent with the expectation.
Suparmi, A., E-mail: suparmiuns@gmail.com; Cari, C., E-mail: suparmiuns@gmail.com [Physics Department, Post Graduate Study, Sebelas Maret University (Indonesia); Angraini, L. M. [Physics Department, Mataram University (Indonesia)
2014-09-30
The bound state solutions of Dirac equation for Hulthen and trigonometric Rosen Morse non-central potential are obtained using finite Romanovski polynomials. The approximate relativistic energy spectrum and the radial wave functions which are given in terms of Romanovski polynomials are obtained from solution of radial Dirac equation. The angular wave functions and the orbital quantum number are found from angular Dirac equation solution. In non-relativistic limit, the relativistic energy spectrum reduces into non-relativistic energy.
Extended Wronskian Determinant Approach and Iterative Solutions of One-Dimensional Dirac Equation
XU Ying; LU Meng; SU Ru-Keng
2004-01-01
An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation, which can be solved by iterative procedure to find the wave functions, is established. We employ this approach to study the one-dimensional Dirac equation with one-well potential,and give the energy levels and wave functions up to the first order iterative approximation. For double-well potential,the energy levels up to the first order approximation are given.
Solutions for the Klein-Gordon and Dirac equations on the lattice based on Chebyshev polynomials
Faustino, Nelson
2016-01-01
The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein-Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein-Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.
A Tale of Three Equations Breit, Eddington-Guant, and Two-Body Dirac
Van Alstine, P; Alstine, Peter Van; Crater, Horace W.
1997-01-01
G.Breit's original paper of 1929 postulates the Breit equation as a correction to an earlier defective equation due to Eddington and Gaunt, containing a form of interaction suggested by Heisenberg and Pauli. We observe that manifestly covariant electromagnetic Two-Body Dirac equations previously obtained by us in the framework of Relativistic Constraint Mechanics reproduce the spectral results of the Breit equation but through an interaction structure that contains that of Eddington and Gaunt. By repeating for our equation the analysis that Breit used to demonstrate the superiority of his equation to that of Eddington and Gaunt, we show that the historically unfamiliar interaction structures of Two-Body Dirac equations (in Breit-like form) are just what is needed to correct the covariant Eddington Gaunt equation without resorting to Breit's version of retardation.
Barut, A. O.
1990-04-01
By exact explicit solution it is shown that the Lorentz-Dirac equation with radiation reaction and proper initial conditions does not violate causality, even if the force is nonanalytic. We also show that if the equation is correctly renormalized there are no runaway solutions.
Analytic solution for Gauged Dirac-Weyl equation in $(2+1)$-dimensions
Ardenghi, Juan Sebastián; Sourrouille, Lucas
2016-01-01
A gauged Dirac-Weyl equation in (2+1)-dimension is considered. This equation describe relativistic matter. In particular we are interested in matter interacting with a Chern-Simons gauge fields. We show that exact self-dual solutions are admitted. These solutions are the same as those supported by nonrelativistic matter interacting with a Chern-Simons gauge field.
The relation between Maxwell, Dirac, and the Seiberg-Witten equations
Waldyr A. Rodrigues
2003-01-01
Full Text Available We discuss unsuspected relations between Maxwell, Dirac, and the Seiberg-Witten equations. First, we present the Maxwell-Dirac equivalence (MDE of the first kind. Crucial to that proposed equivalence is the possibility of solving for ψ (a representative on a given spinorial frame of a Dirac-Hestenes spinor field the equation F=ψγ21ψ˜, where F is a given electromagnetic field. Such task is presented and it permits to clarify some objections to the MDE which claim that no MDE may exist because F has six (real degrees of freedom and ψ has eight (real degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth, even if there is no evidence until now for magnetic monopoles, because there are at least two faithful field equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail associated with Maxwell theory and the other is a (nonlinear equation (of the generalized Maxwell type satisfied by the 2-form field part of a Dirac-Hestenes spinor field that solves the Dirac-Hestenes equation for a free electron. This is a new result which can also be called MDE of the second kind. Finally, we use the MDE of the first kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A physical interpretation for those equations is proposed.
Dirac equation from the Hamiltonian and the case with a gravitational field
Arminjon, M
2006-01-01
Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the correspondence. This derivation applies in the same form to a free particle, to one in an electromagnetic field, and to one subjected to geodesic motion in a static metric, and leads to the same, usual form of the Dirac equation--in special coordinates. To use the equation in the static-gravitational case, we need to rewrite it in more general coordinates. This can be done only if the usual, spinor transformation of the wave function is replaced by the 4-vector transformation. We show that the latter also makes the flat-space-time Dirac equation Lorentz-covariant, although the Dirac matrices are not invariant. Because the equation itself is left unchanged in the flat case, the 4-vector transformation does not alter the main physical consequences of that equation in that case. However, the equation derived in the ...
Reduced Dirac equation and Lamb shift as off-mass-shell effect in quantum electrodynamics
Ni, Guang-Jiong; Xu, Jian-Jun; Lou, Sen-Yue
2011-02-01
Based on the accurate experimental data of energy-level differences in hydrogen-like atoms, especially the 1S—2S transitions of hydrogen and deuterium, the necessity of introducing a reduced Dirac equation with reduced mass as the substitution of original electron mass is stressed. Based on new cognition about the essence of special relativity, we provide a reasonable argument for the reduced Dirac equation to have two symmetries, the invariance under the (newly defined) space-time inversion and that under the pure space inversion, in a noninertial frame. By using the reduced Dirac equation and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state—-a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Reduced Dirac equation and Lamb shift as off-mass-shell effect in quantum electrodynamics
Ni Guang-Jiong; Xu Jian-Jun; Lou Sea-Yue
2011-01-01
Based on the accurate experimental data of energy-level differences in hydrogen-like atoms, especially the 1S-2S transitions of hydrogen and deuterium, the necessity of introducing a reduced Dirac equation with reduced mass as the substitution of original electron mass is stressed. Based on new cognition about the essence of special relativity, we provide a reasonable argument for the reduced Dirac equation to have two symmetries, the invariance under the (newly defined) space-time inversion and that under the pure space inversion, in a noninertial frame. By using the reduced Dirac equation and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state-a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Dirac constraint analysis and symplectic structure of anti-self-dual Yang–Mills equations
U Camci; Z Can; Y Nutku; Y Sucu; D Yazici
2006-12-01
We present the explicit form of the symplectic structure of anti-self-dual Yang–Mills (ASDYM) equations in Yang's - and -gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac's theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang–Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both - and -gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten–Zuckerman formalism. We show that the appropriate component of the Witten–Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac's theory. Finally, we present the Bäcklund transformation between the - and -gauges in order to apply Magri's theorem to the respective two Hamiltonian structures.
Analytical evaluation of the plasma dispersion function for a Fermi-Dirac distribution
B.A. Mamedov
2012-01-01
An efficient method for the analytic evaluation of the plasma dispersion function for the Fermi-Dirac distribution is proposed.The new method has been developed using the binomial expansion theorem and the Gamma functions.The general formulas obtained for the plasma dispersion function are utilized for the evaluation of the response function.The resulting series present better convergence rates.Several acceleration techniques are combined to further improve the efficiency.The obtained results for the plasma dispersion function are in good agreement with the known numerical data.
Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups
Breev, A. I.; Mosman, E. A.
2016-12-01
The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.
Najafizade, S. A.; Hassanabadi, H.; Zarrinkamar, S.
2017-09-01
In this study, the information-theoretic measures of (1+1)-dimensional Dirac equation in both position and momentum spaces are investigated for the trigonometric Rosen-Morse and the Morse potentials. The solutions of the corresponding Dirac equation are obtained in an exact analytical manner in the first step. Next, using the Fourier transformation, the position and momentum Shannon information entropies are obtained and some features of the probability densities are analyzed. The consistency with Bialynicki-Birula-Mycielski inequality and Heisenberg uncertainty is checked.
Transparent boundary conditions for the wave equation in one dimension and for a Dirac-like equation
M. Pilar Velasco
2015-11-01
Full Text Available We present a method to achieve transparent boundary conditions for the one-dimensional wave equation, and show its numerical implementation using a finite-difference method. We also present an alternative method for building the same transparent boundary conditions using a Dirac-like equation and a Spinor-like formalism. Finally, we extend our method to the three-dimensional wave equation with radial symmetry.
Spin eigen-states of Dirac equation for quasi-two-dimensional electrons
Eremko, Alexander, E-mail: eremko@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); Brizhik, Larissa, E-mail: brizhik@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); Loktev, Vadim, E-mail: vloktev@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); National Technical University of Ukraine “KPI”, Peremohy av., 37, Kyiv, 03056 (Ukraine)
2015-10-15
Dirac equation for electrons in a potential created by quantum well is solved and the three sets of the eigen-functions are obtained. In each set the wavefunction is at the same time the eigen-function of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigen-functions of Dirac equation describe three independent spin eigen-states. The energy spectrum of electrons confined by the rectangular quantum well is calculated for each of these spin states at the values of energies relevant for solid state physics. It is shown that the standard Rashba spin splitting takes place in one of such states only. In another one, 2D electron subbands remain spin degenerate, and for the third one the spin splitting is anisotropic for different directions of 2D wave vector.
Bound states of the Dirac equation with some physical potentials by the Nikiforov-Uvarov method
Setare, Mohammad R; Haidari, S [Department of Physics, University of Kurdistan, Pasdaran Avenue, Sanandaj (Iran, Islamic Republic of)], E-mail: rezakord@ipm.ir, E-mail: heidary.somayeh@gmail.com
2010-01-15
Exact analytical solutions for the s-wave Dirac equation with the reflectionless-type, Rosen-Morse and Manning-Rosen potentials are obtained, under the condition of spin symmetry. We obtained bound state energy eigenvalues and corresponding spinor wave function in the framework of the Nikiforov-Uvarov (NU) method.
Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass
YANG Jin; XIANG An-Ping; YU Wan-Lun
2003-01-01
We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable forthe case with time-independence mass.
Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass
YANGJin; XIANGAn-Ping; YUWan-Lun
2003-01-01
We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable for the case with time-independence mass.
The exact solution for the Dirac equation with the Cornell potential
Trevisan, L A; Andrade, F M
2013-01-01
An analytical solution of the Dirac equation with a Cornell potential, with identical scalar and vectorial parts, is presented. The solution is obtained by using the linear potential solution, related to Airy functions, multiplied by another function to be determined. The energy levels are obtained and we notice that they obey a band structure.
Supersymmetry and Solution of Dirac Equation with Vector and Scalar Potentials
ZHANG Xiao-Long; JU Guo-Xing; FENG Mang; REN Zhong-Zhou; GAO Ke-Lin
2008-01-01
The Dirac equations with vector and scalar potentials of the Coulomb types in two and three dimensions are solved using the supersymmetric quantum mechanics method. For the system of such potentials, the analytical expressions of the matrix dements for both position and momentum operators are obtained.
Dirac equation and optical scalars in the Einstein-Cartan theory
Timofeev, Vladimir
2016-03-01
The article deals with the Dirac equation in the Newman-Penrose formalism within the framework of Einstein-Cartan theory and behavior of isotropic congruence of autoparallels, i. e. a congruence of the curves along which tangent null vector transferred in parallel.
An Introduction to Relativistic Quantum Mechanics. I. From Relativity to Dirac Equation
De Sanctis, M
2007-01-01
By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.
Soylu, A [Department of Physics, Nigde University, 51350, Nigde (Turkey); Bayrak, O; Boztosun, I [Department of Physics, Erciyes University, 38039, Kayseri (Turkey)
2008-02-15
For any spin-orbit coupling term {kappa}, the analytical solutions of the Dirac equation for the Eckart potential are presented by using the asymptotic iteration method within the framework of the spin and pseudospin symmetry concept. The energy eigenvalues are obtained in the closed form by applying an approximation to the spin-orbit coupling potential.
Existence of Majorana fermion mode and Dirac equation in cavity quantum electrodynamics
Sarkar, Sujit, E-mail: sujit.tifr@gmail.com
2015-10-15
We present the results of low lying collective mode of coupled optical cavity arrays. We derive the Dirac equation for this system and explain the existence of Majorana fermion mode in the system. We present quite a few analytical relations between the Rabi frequency oscillation and the atom–photon coupling strength to explain the different physical situation of our study and also the condition for massless collective mode in the system. We present several analytical relations between the Dirac spinor field, order and disorder operators for our systems. We also show that the Luttinger liquid physics is one of the intrinsic concepts in our system.
Global Solutions of Einstein—Dirac Equation on the Conformal Space
LUQi－Keng; WANGShi－Kun; 等
2001-01-01
The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.
Global Solutions of Einstein-Dirac Equation on the Conformal Space
LU Qi-Keng; WANG Shi-Kun; WU Ke
2001-01-01
The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic. Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group, which is isomorphic to SL(2, ). Hence the results and their interpretation coming from the two theories would be different. In this short note we study only the Lorentz spinor manifold and, especially, the solutions of Einstein-Dirac equations on the conformal space, which is closely related to the AdS/CFT correspondence.
Large-j Expansion Method for Two-Body Dirac Equation
Askold Duviryak
2006-02-01
Full Text Available By using symmetry properties, the two-body Dirac equation in coordinate representation is reduced to the coupled pair of radial second-order differential equations. Then the large-j expansion technique is used to solve a bound state problem. Linear-plus-Coulomb potentials of different spin structure are examined in order to describe the asymptotic degeneracy and fine splitting of light meson spectra.
Generalized Klein-Gordon and Dirac Equations from Nonlocal Kinetic Approach
El-Nabulsi, Rami Ahmad
2016-09-01
In this note, I generalized the Klein-Gordon and the Dirac equations by using Suykens's nonlocal-in-time kinetic energy approach, which is motivated from Feynman's kinetic energy functional formalism where the position differences are shifted with respect to one another. I proved that these generalized equations are similar to those obtained in literature in the presence of minimal length based on the Quesne-Tkachuk algebra.
Highly covariant quantum lattice gas model of the Dirac equation
Yepez, Jeffrey
2011-01-01
We revisit the quantum lattice gas model of a spinor quantum field theory-the smallest scale particle dynamics is partitioned into unitary collide and stream operations. The construction is covariant (on all scales down to a small length {\\ell} and small time {\\tau} = c {\\ell}) with respect to Lorentz transformations. The mass m and momentum p of the modeled Dirac particle depend on {\\ell} according to newfound relations m = mo cos (2{\\pi}{\\ell}/{\\lambda}) and p = (h/2{\\pi}{\\ell}) sin(2{\\pi}{\\ell}/{\\lambda}), respectively, where {\\lambda} is the Compton wavelength of the modeled particle. These relations represent departures from a relativistically invariant mass and the de Broglie relation-when taken as quantifying numerical errors the model is physically accurate when {\\ell} {\\ll} {\\lambda}. Calculating the vacuum energy in the special case of a massless spinor field, we find that it vanishes (or can have a small positive value) for a sufficiently large wave number cutoff. This is a marked departure from th...
Relativistic integro-differential form of the Lorentz-Dirac equation in 3D without runaways
Ibison, Michael; Puthoff, Harold E.
2001-04-01
It is well known that the third-order Lorentz-Dirac equation admits runaway solutions wherein the energy of the particle grows without limit, even when there is no external force. These solutions can be denied simply on physical grounds, and on the basis of careful analysis of the correspondence between classical and quantum theory. Nonetheless, one would prefer an equation that did not admit unphysical behavior at the outset. Such an equation - an integro-differential version of the Lorentz-Dirac equation - is currently available either in 1 dimension only, or in 3 dimensions only in the non-relativistic limit. It is shown herein how the Lorentz-Dirac equation may be integrated without approximation, and is thereby converted to a second-order integro-differential equation in 3D satisfying the above requirement. I.E., as a result, no additional constraints on the solutions are required because runaway solutions are intrinsically absent. The derivation is placed within the historical context established by standard works on classical electrodynamics by Rohrlich, and by Jackson.
On the Existence of Additional (Hydrino) states in the Dirac equation
Khelashvili, Anzor
2016-01-01
In case of spinless particles there appear additional (singular) solutions in the framework of relativistic Klein-Gordon equation for Coulomb potential. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states (hydrino, small hydrogen) should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle (electron) in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions.
Dirac-Kahler equation in curved space-time, relation between spinor and tensor formulations
Red'kov, V M
2011-01-01
A common view is that generalization of a wave equation on Riemannian space-time is substantially determined by what a particle is - boson or fermion. As a rule, they say that tensor equations for bosons are extended in a simpler way then spinor equations for fermions. In that context, a very interesting problem is of extension a wave equation for Dirac--K\\"{a}hler field (Ivanenko--Landau field was historically first term, also the term a vector field of general type was used). The article relates a generally covariant tensor formalism to a spinor one when these both are applied to description of the Dirac-K\\"ahler field in a Rimannian space-time. Both methods are taken to be equivalent and the tensor equations are derived from spinor ones. It is shown that, for characterization of Dirac-K\\"ahler's tensor components, two alternative approaches are suitable: these are whether a tetrad-based pseudo tensor classification or a generally coordinate pseudo tensor one. By imposing definite restrictions on the the Di...
Fermion unification model based on the intrinsic SU(8 symmetry of a generalized Dirac equation
Eckart eMarsch
2015-10-01
Full Text Available A natural generalization of the original Dirac spinor into a multi-component spinor is achieved, which corresponds to the single lepton and the three quarks of the first family of the standard model of elementary particle physics. Different fermions result from similarity transformations of the Dirac equation, but apparently there can be no more fermions according to the maximal multiplicity revealed in this study. Rotations in the fermion state space are achieved by the unitary generators of the U(1 and the SU(3 groups, corresponding to quantum electrodynamics (QED based on electric charge and chromodynamics (QCD based on colour charge. In addition to hypercharge the dual degree of freedom of hyperspin emerges, which occurs due to the duplicity implied by the two related (Weyl and Dirac representations of the Dirac equation. This yields the SU(2 symmetry of the weak interaction, which can be married to U(1 to generate the unified electroweak interaction as in the standard model. Therefore, the symmetry group encompassing all the three groups mentioned above is SU(8, which can accommodate and unify the observed eight basic stable fermions.
QED2+1 in Graphene: Symmetries of Dirac Equation in 2+1 Dimensions
Kosiński, P.; Maślanka, P.; S´nska, J.; Zasada, I.
2012-10-01
It is well-known that the tight-binding Hamiltonian of graphene describes the low-energy excitations that appear to be massless chiral Dirac fermions. Thus, in the continuum limit one can analyze the crystal properties using the formalism of quantum electrodynamics in 2+1 dimensions (QED_{2+1}) which provides the opportunity to verify the high energy physics phenomena in the condensed matter system. We study the symmetry properties of 2+1-dimensional Dirac equation, both in the noninteracting case and in the case with constant uniform magnetic field included in the model. The maximal symmetry group of the massless Dirac equation is considered by putting it in the Jordan block form and determining the algebra of operators leaving invariant the subspace of solutions. It is shown that the resulting symmetry operators expressed in terms of Dirac matrices cannot be described exclusively in terms of γ matrices (and their products) entering the corresponding Dirac equation. It is a consequence of the reducibility of the considered representation in contrast to the 3+1-dimensional case. Symmetry algebra is demonstrated to be a direct sum of two gL(2,C) algebras plus an eight-dimensional abelian ideal. Since the matrix structure which determines the rotational symmetry has all required properties of the spin algebra, the pseudospin related to the sublattices (M. Mecklenburg and B. C. Regan, Phys. Rev. Lett. 106 (2011), 116803) gains the character of the real angular momentum, although the degrees of freedom connected with the electron's spin are not included in the model. This seems to be graphene's analogue of the phenomenon called ``spin from isospin'' in high energy physics.
Wells, J C; Eichler, J
1999-01-01
We discuss the two-center, time-dependent Dirac equation describing the dynamics of an electron during a peripheral, relativistic heavy-ion collision at extreme energies. We derive a factored form, which is exact in the high-energy limit, for the asymptotic channel solutions of the Dirac equation, and elucidate their close connection with gauge transformations which transform the dynamics into a representation in which the interaction between the electron and a distant ion is of short range. We describe the implications of this relationship for solving the time-dependent Dirac equation for extremely relativistic collisions.
Dirac equation of spin particles and tunneling radiation from a Kinnersly black hole
Li, Guo-Ping; Zu, Xiao-Tao [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Feng, Zhong-Wen [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); China West Normal University, College of Physics and Space Science, Nanchong (China); Li, Hui-Ling [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Shenyang Normal University, College of Physics Science and Technology, Shenyang (China)
2017-04-15
In curved space-time, the Hamilton-Jacobi equation is a semi-classical particle equation of motion, which plays an important role in the research of black hole physics. In this paper, starting from the Dirac equation of spin 1/2 fermions and the Rarita-Schwinger equation of spin 3/2 fermions, respectively, we derive a Hamilton-Jacobi equation for the non-stationary spherically symmetric gravitational field background. Furthermore, the quantum tunneling of a charged spherically symmetric Kinnersly black hole is investigated by using the Hamilton-Jacobi equation. The result shows that the Hamilton-Jacobi equation is helpful to understand the thermodynamic properties and the radiation characteristics of a black hole. (orig.)
Cari, C., E-mail: cari@staff.uns.ac.id; Suparmi, A., E-mail: soeparmi@staff.uns.ac.id; Yunianto, M., E-mail: muhtaryunianto@staff.uns.ac.id; Husein, A. S. [Physics Department, Faculty of Mathematics and Science, SebelasMaret University, Jl. Ir. Sutami 36A Kentingan Surakarta 57126 (Indonesia)
2016-02-08
The analytical solution of Ddimensional Dirac equation for Coulombic potential is investigated using Nikiforov-Uvarov method. The D dimensional relativistic energy spectra are obtained from relativistic energy eigenvalue equation by using Mat Lab software.The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi and Laguerre Polynomials. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy which will be applied to determine some thermodynamical properties of the system. The thermodynamical properties of the system are expressed in terms of error function and imaginary error function.
Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays
Koke, C; Angelakis, D G
2016-01-01
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. Gravitational fields can be incorporated as background spacetime if the back-action of matter on it can be neglected, yielding modified Dirac or Klein-Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many scenarios including expanding universe, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (the Unruh effect). In this work, we give a pedagogical introduction to the Dirac equation in a general 2D spacetime and show examples of spinor wave packet dynamics in some background spacetimes. In particular, we cover the phenomenon of particle creation in a time-dependent metric. Photonic analogs of these effects are then proposed, where classical light propagating in an array of coupled waveguides provides a visualis...
A Vlasov equation with Dirac potential used in fusion plasmas
Bardos, Claude [Universite Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 Place Jussieu, 75252 Paris Cedex 05 (France); Nouri, Anne [Laboratoire d' Analyse, Topologie et Probabilites (UMR 6632), Aix-Marseille Universite, 39 Rue Joliot-Curie, 13453 Marseille Cedex 13 (France)
2012-11-15
Well-posedness of the Cauchy problem is analyzed for a singular Vlasov equation governing the evolution of the ionic distribution function of a quasineutral fusion plasma. The Penrose criterium is adapted to the linearized problem around a time and space homogeneous distribution function showing (due to the singularity) more drastic differences between stable and unstable situations. This pathology appears on the full nonlinear problem, well-posed locally in time with analytic initial data, but generally ill-posed in the Hadamard sense. Eventually with a very different class of solutions, mono-kinetic, which constrains the structure of the density distribution, the problem becomes locally in time well-posed.
On exact solutions of the Dirac equation in a homogeneous magnetic field in the Lobachevsky space
Ovsiyuk, E M; Red'kov, V M
2010-01-01
There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Lobachevsky space geometry, has been obtained.
The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization
Reifler, Frank; Morris, Randall
1992-01-01
Currently, there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors.
Scattering states of Dirac particle equation with position dependent mass under the cusp potential
Chabab, M; Hassanabadi, H; Oulne, M; Zare, S
2016-01-01
We solved the one-dimensional position-dependent mass Dirac equation in the presence of the cusp potential and reported the solutions in terms of the Whittaker functions. We have derived the reflection and transmission coefficients by making use of the matching conditions on the wave functions. The effect of position dependent mass on the reflection and transmission coefficients of the system is duly investigated.
Scattering states of Dirac particle equation with position-dependent mass under the cusp potential
Chabab, M.; El Batoul, A.; Hassanabadi, H.; Oulne, M.; Zare, S.
2016-11-01
We solved the one-dimensional position-dependent mass Dirac equation in the presence of the cusp potential and reported the solutions in terms of the Whittaker functions. We have derived the reflection and transmission coefficients by making use of the matching conditions on the wave functions. The effect of the position-dependent mass on the reflection and transmission coefficients of the system is duly investigated.
Analytic Continuation in the Coupling Constant Method for the Dirac Equation
张时声; 郭建友; 张双全; 孟杰
2004-01-01
On the basis of the Dirac equation, the analytic continuation in the coupling constant method is employed to investigate the energies and widths of single-particle resonant in square-well, harmonic-oscillator, and Woodsconvergent energies and widths of single-particle resonant states can be obtained, which makes the application of the analytic continuation in the coupling constant for the relativistic mean field theory possible.
Single-site Green function of the Dirac equation for full-potential electron scattering
Kordt, Pascal
2012-05-30
I present an elaborated analytical examination of the Green function of an electron scattered at a single-site potential, for both the Schroedinger and the Dirac equation, followed by an efficient numerical solution, in both cases for potentials of arbitrary shape without an atomic sphere approximation. A numerically stable way to calculate the corresponding regular and irregular wave functions and the Green function is via the angular Lippmann-Schwinger integral equations. These are solved based on an expansion in Chebyshev polynomials and their recursion relations, allowing to rewrite the Lippmann-Schwinger equations into a system of algebraic linear equations. Gonzales et al. developed this method for the Schroedinger equation, where it gives a much higher accuracy compared to previous perturbation methods, with only modest increase in computational effort. In order to apply it to the Dirac equation, I developed relativistic Lippmann-Schwinger equations, based on a decomposition of the potential matrix into spin spherical harmonics, exploiting certain properties of this matrix. The resulting method was embedded into a Korringa-Kohn-Rostoker code for density functional calculations. As an example, the method is applied by calculating phase shifts and the Mott scattering of a tungsten impurity. (orig.)
M. R. Setare; O. Hatami
2008-01-01
Based on the shape invariance property we obtain exact solutions of the Dirac equation for an electron moving in the presence of a certain varying magnetic field, then we also show its non-relativistic limit.
Energy Loss by Radiation in Many-Particle Numerical Simulation With Lorentz-Dirac Equation
Žáček, Martin
2006-01-01
We studied the possibilities for numerical integration of Lorentz-Dirac equation that is the equation describing the motion of a charged point particle when radiation reaction is taken into account. In numerical modelling based on particle models usually the equations of motion without radiation force are used and the corrections for radiation are used consequently, expressed by laws given by averaged particle parameters as the temperature or particle density. If the complete equation of motion concluding the radiation would be used, the corrections for radiation reaction force could be used for every charged particle individually from more fundamental laws. Thus the model could be able to describe more physical phenomena. However from theory of Lorentz-Dirac equation there are known various problems with non-physical solutions and nonuniqueness that are often solved and tested by various methods. One way to eliminate the non-physical solutions is to use integro-differential equation, which is used here. The leap-frog method is used for numerical integrating and accuracy is verified for electron in magnetic field. This approach is proposed to be used for PIC (particle-in-cell) integration method, which is often used as an effective method of simulation in plasma physics for many charged particles interactinge with electromagnetic field.
Numeric Solutions of Dirac-Gursey Spinor Field Equation Under External Gaussian White Noise
Aydogmus, Fatma
2016-06-01
In this paper, we consider the Dirac-Gursey spinor field equation that has particle-like solutions derived classical field equations so-called instantons, formed by using Heisenberg ansatz, under the effect of an additional Gaussian white noise term. Our purpose is to understand how the behavior of spinor-type excited instantons in four dimensions can be affected by noise. Thus, we simulate the phase portraits and Poincaré sections of the obtained system numerically both with and without noise. Recurrence plots are also given for more detailed information regarding the system.
Solution of Dirac equation in Reissner-Nordström de Sitter space
Lyu, Yan; Cui, Song
2009-02-01
The radial parts of the Dirac equation between the outer black hole horizon and the cosmological horizon are solved in Reissner-Nordström de Sitter (RNdS) space numerically. An accurate approximation, the polynomial approximation, is used to approximate the modified tortoise coordinate \\hat r_* , which leads to the inverse function r = r(\\hat r_* ) and the potential V(\\hat r_* ). The potential V(\\hat r_* ) is replaced by a collection of step functions in sequence. Then the solution of the wave equation as well as the reflection and transmission coefficients is computed by a quantum mechanical method.
Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays
Koke, Christian, E-mail: christian.koke@stud.uni-heidelberg.de [Institut für theoretische Physik, Philosophenweg 16, D-69120 Heidelberg (Germany); Noh, Changsuk, E-mail: changsuk@kias.re.kr [Korea Institute for Advanced Study, 85 Hoegiro, Seoul 130-722 (Korea, Republic of); Angelakis, Dimitris G., E-mail: dimitris.angelakis@gmail.com [Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117542 (Singapore); School of Electronic and Computer Engineering, Technical University of Crete, Chania, Crete, 73100 (Greece)
2016-11-15
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. A gravitational field can be incorporated as a background spacetime if the back-action of matter on the field can be neglected, resulting in modifications of the Dirac or Klein–Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many situations, including the expanding-universe scenario, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (the Unruh effect). In this work, we give a pedagogical introduction to the Dirac equation in a general 2D spacetime and show examples of spinor wave packet dynamics in flat and curved background spacetimes. In particular, we cover the phenomenon of particle creation in a time-dependent metric. Photonic analogs of these effects are then proposed, where classical light propagating in an array of coupled waveguides provides a visualisation of the Dirac spinor propagating in a curved 2D spacetime background. The extent to which such a single-particle description can be said to mimic particle creation is discussed.
Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays
Koke, Christian; Noh, Changsuk; Angelakis, Dimitris G.
2016-11-01
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. A gravitational field can be incorporated as a background spacetime if the back-action of matter on the field can be neglected, resulting in modifications of the Dirac or Klein-Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many situations, including the expanding-universe scenario, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (the Unruh effect). In this work, we give a pedagogical introduction to the Dirac equation in a general 2D spacetime and show examples of spinor wave packet dynamics in flat and curved background spacetimes. In particular, we cover the phenomenon of particle creation in a time-dependent metric. Photonic analogs of these effects are then proposed, where classical light propagating in an array of coupled waveguides provides a visualisation of the Dirac spinor propagating in a curved 2D spacetime background. The extent to which such a single-particle description can be said to mimic particle creation is discussed.
A Conceptual Shift to Rectify a Defect in the Lorentz-Dirac Equation
Oliver, M A
2013-01-01
In his analysis of the Classical Theory of Radiating Electrons, Dirac (1938) draws attention to the characteristic instability of solutions to the third order equation of motion. He remarks that changing the sign of the self-force eliminates the runaway solutions and gives `reasonable behaviour'. Dirac rejects such a change and proceeds with an ad hoc modification to the solutions of the initial value problem that is not consistent with the principle of causality. We argue that his reasons for rejecting the change of sign are invalid on both physical and mathematical grounds. The conceptual shift is to treat the physical particle as a composite of the source particle and the energy-momentum that is reversibly generated in its self-field by its motion. The reversibly generated energy in the self-field is interpreted as kinetic energy, and the changes that follow result in Dirac's change of sign. Several exact solutions to the new equation of motion and its linearisation are given. For a particle in orbital mot...
Improvement of the basis for the solution of the Dirac equation in Cassini coordinates
Hahn, W.; Artemyev, A. N.; Surzhykov, A.
2017-08-01
We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in our earlier article [J. Phys. B 43, 235207 (2010)]. For the calculations in the above article, we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-splines only. Therefore, we include basis functions which are defined using functions with step-like behavior instead of B-splines. Thereby, we achieve a significant increase of accuracy of results.
Improvement of the Basis for the Solution of the Dirac Equation in Cassini Coordinates
Hahn, Walter; Surzhykov, Andrey
2016-01-01
We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in [1]. For the calculations in [1], we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-spines only. Therefore, we include basis functions which are defined using functions with step-like behaviour instead of B-splines. Thereby, we achieve a significant increase of accuracy of results as compared to [1].
Bound States of the Klein-Gordon and Dirac equations for potential V0 tanh2(r/d)
Qiang Wen-Chao
2004-01-01
The exact bound state wavefunctions and energy equations of Klein-Gordon and Dirac equations are given with equal scalar and vector potential s(r) = v(r) = V(r)/2 = V0 tanh2(r/d). The relation between the energy equation and that of relativistic harmonic is discussed.
Chicurel-Uziel, Enrique
2007-08-01
A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.
Feshbach P -Q partitioning technique and the two-component Dirac equation
Luo, Da-Wei; Pyshkin, P. V.; Yu, Ting; Lin, Hai-Qing; You, J. Q.; Wu, Lian-Ao
2016-09-01
We provide an alternative approach to relativistic dynamics based on the Feshbach projection technique. Instead of directly studying the Dirac equation, we derive a two-component equation for the upper spinor. This approach allows one to investigate the underlying physics in a different perspective. For particles with small mass such as the neutrino, the leading-order equation has a Hermitian effective Hamiltonian, implying there is no leakage between the upper and lower spinors. In the weak relativistic regime, the leading order corresponds to a non-Hermitian correction to the Pauli equation, which takes into account the nonzero possibility of finding the lower-spinor state and offers a more precise description.
The Lorentz-Dirac equation and the physical meaning of the Maxwell's fields
De Souza, M M
1995-01-01
Classical Electrodynamics is not a consistent theory because of the inadequate behaviour of its fields in the vicinity of their sources. Its problems with the electron equation of motion and with non-integrable singularity of the electron self field and of its stress tensor are well known. These inconsistencies are eliminated if the discrete and localized (classical photons) character of the electromagnetic interaction is anticipatively recognized already in a classical context. This is possible, in a manifestly covariant way, with a new model of spacetime structure, shown in a previous paper \\cite{hep-th/9505169}, that invalidates the Lorentz-Dirac equation. For the new equation of motion of a point classical electron there is no field singularity, no causality violation and no conflict with energy conservation in the electron equation of motion. The electromagnetic field must be re-interpreted in terms of average flux of classical photons. Implications of a singularity-free formalism to field theory are dis...
Buchdahl's glass dispersion coefficients calculated from Schott equation constants.
Reardon, P J; Chipman, R A
1989-08-15
A method for the rapid evaluation of Buchdahl's dispersion coefficients at an arbitrary base wavelength given the Schott equation constants is presented. Buchdahl's chromatic coordinate transformation is performed on the Schott equation for the refractive index. A Taylor series expansion of the transformed Schott equation is then equated to the square of the Buchdahl dispersion model. Equations for Buchdahl's dispersion coefficients are then obtained for any base wavelength by matching the power of Buchdahl's chromatic coordinate omega out to fourth order.
Batic, D
2005-01-01
In this paper we compute the square root of the generalized squared total angular momentum operator $J$ for a Dirac particle in the Kerr-Newman metric. The separation constant $\\lambda$ arising from the Chandrasekahr separation ansatz turns out to be the eigenvalue of $J$. After proving that $J$ is a symmetry operator, we show the completeness of Chandrasekhar Ansatz for the Dirac equation in oblate spheroidal coordinates and derive an explicit formula for the propagator $e^{-itH}$.
A new CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
Casana, R; Silva, E O; Passos, E; Santos, F E P dos
2013-01-01
In this work, we propose a CPT-even and Lorentz-violating dimension-five nonminimal coupling between fermionic and gauge fields, involving the CTP-even and Lorentz-violating gauge tensor of the SME. This nonminimal coupling modifies the Dirac equation, whose nonrelativistic regime is governed by a Hamiltonian which induces new effects, such as an electric-Zeeman-like spectrum splitting and an anomalous-like contribution to the electron magnetic moment, between others. Some of these new effects allows to constrain the magnitude of this nonminimal coupling in 1 part in 10^16.
New CPT-even and Lorentz-violating nonminimal coupling in the Dirac equation
Casana, R.; Ferreira, M. M., Jr.; Passos, E.; dos Santos, F. E. P.; Silva, E. O.
2013-02-01
In this work, we propose a CPT-even and Lorentz-violating dimension-five nonminimal coupling between fermionic and gauge fields, involving the CPT-even and Lorentz-violating gauge tensor of the Standard Model extension. This nonminimal coupling modifies the Dirac equation, whose nonrelativistic regime is governed by a Hamiltonian which induces new effects, such as an electric-Zeeman-like spectrum splitting and an anomalouslike contribution to the electron magnetic moment, among others. Some of these new effects allow one to constrain the magnitude of this nonminimal coupling in 1 part in 1016(eV)-1.
Solutions of the Dirac Equation in a Magnetic Field and Intertwining Operators
Alonso Contreras-Astorga
2012-10-01
Full Text Available The intertwining technique has been widely used to study the Schrödinger equation and to generate new Hamiltonians with known spectra. This technique can be adapted to find the bound states of certain Dirac Hamiltonians. In this paper the system to be solved is a relativistic particle placed in a magnetic field with cylindrical symmetry whose intensity decreases as the distance to the symmetry axis grows and its field lines are parallel to the x−y plane. It will be shown that the Hamiltonian under study turns out to be shape invariant.
M Hamzavi; S M Ikhdair
2014-07-01
The Hellmann potential is simply a superposition of an attractive Coulomb potential $−a/r$ plus a Yukawa potential e${}^{−δr} /r$. The generalized parametric Nikiforov–Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number in the presence of exact spin and pseudospin (p-spin) symmetries. As a particular case, we obtain the energy eigenvalues of the pure Coulomb potential in the non-relativistic limit.
Goncalves, Bruno; Dias Junior, Mario Marcio [Instituto Federal de Educacacao, Ciencia e Tecnologia Sudeste de Minas Gerais, Juiz de Fora, MG (Brazil)
2013-07-01
Full text: The discussion of experimental manifestations of torsion at low energies is mainly related to the torsion-spin interaction. In this respect the behavior of Dirac field and the spinning particle in an external torsion field deserves and received very special attention. In this work, we consider the combined action of torsion and magnetic field on the massive spinor field. In this case, the Dirac equation is not straightforward solved. We suppose that the spinor has two components. The equations have mixed terms between the two components. The electromagnetic field is introduced in the action by the usual gauge transformation. The torsion field is described by the field S{sub μ}. The main purpose of the work is to get an explicit form to the equation of motion that shows the possible interactions between the external fields and the spinor in a Hamiltonian that is independent to each component. We consider that S{sub 0} is constant and is the unique non-vanishing term of S{sub μ}. This simplification is taken just to simplify the algebra, as our main point is not to describe the torsion field itself. In order to get physical analysis of the problem, we consider the non-relativistic approximation. The final result is a Hamiltonian that describes a half spin field in the presence of electromagnetic and torsion external fields. (author)
Dirac equation for the Hulthén potential within the Yukawa-type tensor interaction
Oktay Aydo(g)du; Elham Maghsoodi; Hassan Hassanabadi
2013-01-01
Using the Nikiforov-Uvarov (NU) method,pseudospin and spin symmetric solutions of the Dirac equation for the scalar and vector Hulthén potentials with the Yukawa-type tensor potential are obtained for an arbitrary spin-orbit coupling quantum number κ.We deduce the energy eigenvalue equations and corresponding upper-and lower-spinor wave functions in both the pseudospin and spin symmetry cases.Numerical results of the energy eigenvalue equations and the upper-and lower-spinor wave functions are presented to show the effects of the external potential and particle mass parameters as well as pseudospin and spin symmetric constants on the bound-state energies and wave functions in the absence and presence of the tensor interaction.
Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity
无
2006-01-01
Based on unified theory of electromagnetic interactions and gravitational interactions, the non-relativistic limit of the equation of motion of a charged Dirac particle in gravitational field is studied. From the Schrodinger equation obtained from this non-relativistic limit, we can see that the classical Newtonian gravitational potential appears as a part of the potential in the Schrodinger equation, which can explain the gravitational phase effects found in COW experiments.And because of this Newtonian gravitational potential, a quantum particle in the earth's gravitational field may form a gravitationally bound quantized state, which has already been detected in experiments. Three different kinds of phase effects related to gravitational interactions are studied in this paper, and these phase effects should be observable in some astrophysical processes. Besides, there exists direct coupling between gravitomagnetic field and quantum spin, and radiation caused by this coupling can be used to directly determine the gravitomagnetic field on the surface of a star.
Invariance properties of the Dirac equation with external electro-magnetic ﬁeld
N D Sen Gupta
2003-01-01
In this paper, we attempt to obtain the nature of the external ﬁeld such that the Dirac equation with external electro-magnetic ﬁeld is invariant. The Poincar´e group, which is the maximal symmetry group for ﬁeld free case, is constrained by the presence of the external ﬁeld. Introducing inﬁnitesimal transformation of x andψ, we apply Lie’s extended group method to obtain the class of external ﬁeld which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic ﬁeld, though only potentials explicity appears in the equation.
Chen Gang; Chen Zi-Dong; Lou Zhi-Mei
2004-01-01
The exact bound state solutions of the Klein-Gordon equation and Dirac equation with scalar and vector pseudoharmonic oscillator potentials are obtained in this paper. Furthermore, we have used the supersymmetric quantum mechanics, shape invariance and alternative method to obtain the required results.
The Solution of Dirac Equation in Quasi-Extreme REISSNER-NORDSTRÖM de Sitter Space
Lyu, Yan; Cui, Song; Liu, Ling
The radial parts of Dirac equation between the outer black hole horizon and the cosmological horizon in quasi-extreme Reissner-Nordström de Sitter (RNdS) geometry is solved numerically. We use an accurate polynomial approximation to mimic the modified tortoise coordinate hat r*(r), for obtaining the inverse function r=r(hat r*) and V=V(hat r*). We then use a quantum mechanical method to solve the wave equation and give the reflection and transmission coefficients. We concentrate on two limiting cases. The first case is when the two horizons are close to each other, and the second case is when the horizons are far apart.
Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism
Steane, Andrew M
2014-01-01
It is widely believed that classical electromagnetism is either unphysical or inconsistent, owing to pathological behaviour when self-force and radiation reaction are non-negligible. We argue that there is no inconsistency as long as it is recognized that certain types of charge distribution are simply impossible, such as, for example, a point particle with finite charge and finite inertia. This is owing to the fact that negative inertial mass is an unphysical concept in classical physics. It remains useful to obtain an equation of motion for small charged objects that describes their motion to good approximation without requiring knowledge of the charge distribution within the object. We give a simple method to achieve this, leading to a reduced-order form of the Abraham-Lorentz-Dirac equation, essentially as proposed by Eliezer, Landau and Lifshitz.
E.Maghsoodi; H.Hassanabadi; S.Zarrinkamar
2013-01-01
Exact analytical solutions of the Dirac equation are reported for the P(o)schl-Teller double-ring-shaped Coulomb potential.The radial,polar,and azimuthal parts of the Dirac equation are solved using the Nikiforov-Uvarov method,and the exact bound-state energy eigenvalues and corresponding two-component spinor wavefunctions are reported.
Suparmi, A., E-mail: soeparmi@staff.uns.ac.id; Cari, C., E-mail: cari@staff.uns.ac.id; Pratiwi, B. N., E-mail: namakubetanurpratiwi@gmail.com [Physics Department, Faculty of Mathematics and Science, Sebelas Maret University, Jl. Ir. Sutami 36A Kentingan Surakarta 57126 (Indonesia); Deta, U. A. [Physics Department, Faculty of Science and Mathematics Education and Teacher Training, Surabaya State University, Surabaya (Indonesia)
2016-02-08
The analytical solution of D-dimensional Dirac equation for hyperbolic tangent potential is investigated using Nikiforov-Uvarov method. In the case of spin symmetry the D dimensional Dirac equation reduces to the D dimensional Schrodinger equation. The D dimensional relativistic energy spectra are obtained from D dimensional relativistic energy eigen value equation by using Mat Lab software. The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi polynomials. The thermodynamically properties of materials are generated from the non-relativistic energy eigen-values in the classical limit. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy. The thermal quantities of the system, partition function and specific heat, are expressed in terms of error function and imaginary error function which are numerically calculated using Mat Lab software.
Kazinski, P. O.; Shipulya, M. A.
2011-06-01
We present a study of planar physical solutions to the Lorentz-Dirac equation in a constant electromagnetic field. In this case, we reduced the Lorentz-Dirac equation to one second-order differential equation. We obtained the asymptotics of physical solutions to this equation at large proper times. It turns out that, in a crossed constant uniform electromagnetic field with vanishing invariants, a charged particle enters a universal regime at large times. We found that the ratios of momentum components that tend to constants are determined only by the external field. This effect is essentially due to a radiation reaction. There is no such effect for the Lorentz equation in this field.
Kazinski, P O
2010-01-01
We present a study of planar physical solutions to the Lorentz-Dirac equation in a constant electromagnetic field. In this case, we reduced the Lorentz-Dirac equation to the one second order differential equation. We found the asymptotics of physical solutions to this equation at large proper times. It turns out that, in the crossed constant uniform electromagnetic field with vanishing invariants, a charged particle goes to a universal regime at large times. We found the ratio of momentum components which tends to a constant determined only by the external field. This effect is essentially due to a radiation reaction. There is no such an effect for the Lorentz equation in this field.
Reduced Dirac Equation and Lamb Shift as an Off-mass-shell effect in Quantum Electrodynamics
Ni, G; Xu, J; Lou, Senyue; Ni, Guang-jiong; Xu, Jianjun
2005-01-01
Based on the precision experimental data of energy-level differences in hydrogenlike atoms, especially the 1S-2S transition of hydrogen and deuterium, the necessity of establishing a reduced Dirac equation (RDE) with reduced mass as the substitution of original electron mass is stressed. The theoretical basis of RDE lies on two symmetries, the invariance under the space-time inversion and that under the pure space inversion. Based on RDE and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state--a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.
Global existence for an L^2 critical Nonlinear Dirac equation in one dimension
Candy, Timothy
2011-01-01
We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the Thirring model. Local existence in $H^s$ for $s>0$, and global existence for $s>1/2$, has recently been proven by Selberg and Tesfahun by using $X^{s, b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, we first prove local existence in $L^2$ by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of $L^2$ norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all $s>0$.
Analytical solutions of the two-dimensional Dirac equation for a topological channel intersection
Anglin, J. R.; Schulz, A.
2017-01-01
Numerical simulations in a tight-binding model have shown that an intersection of topologically protected one-dimensional chiral channels can function as a beam splitter for noninteracting fermions on a two-dimensional lattice [Qiao, Jung, and MacDonald, Nano Lett. 11, 3453 (2011), 10.1021/nl201941f; Qiao et al., Phys. Rev. Lett. 112, 206601 (2014), 10.1103/PhysRevLett.112.206601]. Here we confirm this result analytically in the corresponding continuum k .p model, by solving the associated two-dimensional Dirac equation, in the presence of a "checkerboard" potential that provides a right-angled intersection between two zero-line modes. The method by which we obtain our analytical solutions is systematic and potentially generalizable to similar problems involving intersections of one-dimensional systems.
Bagci, A
2016-01-01
The author in his previous works were presented a numerical integration method, namely, global-adaptive with the Gauss-Kronrod numerical integration extension in order to accurate calculation of molecular auxiliary functions integrals involve power functions with non-integer exponents. They are constitute elements of molecular integrals arising in Dirac equation when Slater-type orbitals with non-integer principal quantum numbers are used. Binomial series representation of power functions method, so far, is used for analytical evaluation of the molecular auxiliary function integrals however, intervals of integration cover areas beyond the condition of convergence. In the present study, analytical evaluation of these integrals is re-examined. They are expressed via prolate spheroidal coordinates. An alternative analytical approximation are derived. Slowly convergent binomial series representation formulae for power functions is investigated through nonlinear sequence transformations for the acceleration of con...
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).
A CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
Ferreira Junior, Manoel; Casana, M.R.; Santos, Frederico E.P. dos; Silva, E.O. [UFMA, Sao Luis (Brazil); Passos, E. [UFCG, Campina Grande, PB (Brazil)
2013-07-01
Full text: The Standard Model Extension (SME) has been the usual framework for investigating signals of Lorentz violation in physical systems. It is the natural framework for studying properties of physical systems with Lorentz-violation since it includes Lorentz-violating terms in all sectors of the minimal standard model. The Lorentz-violating (LV) terms are generated as vacuum expectation values of tensors defined in a high energy scale. This framework has inspired a great deal of investigation in recent years. Such works encompass several distinct aspects involving fermion systems and radiative corrections, CPT- probing experiments, the electromagnetic CPT- and Lorentz-odd term, the 19 electromagnetic CPT-even coefficients. Recently, some studies involving higher dimensional operators have also been reported with great interest, including nonminimal interactions. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients. In the present work, we propose a new CPT-even, dimension-five, nonminimal coupling linking the fermionic and gauge fields in the context of the Dirac equation, involving the CPT-even tensor of the gauge term of the SME. By considering the nonrelativistic limit of the modified Dirac equation, we explicitly evaluate the new contributions to the nonrelativistic Hamiltonian. These new terms imply a direct correction on the anomalous magnetic moment, a kind of electrical Zeeman-like effect on the atomic spectrum, and a Rashba-like coupling term. These effects are then used to impose upper bounds on the magnitude of the non minimally coupled LV coefficients at the level of 1 part in 10{sub 16}. (author)
FFT-split-operator code for solving the Dirac equation in 2+1 dimensions
Mocken, Guido R.; Keitel, Christoph H.
2008-06-01
provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 474 937 No. of bytes in distributed program, including test data, etc.: 4 128 347 Distribution format: tar.gz Programming language: C++ Computer: Any, but SMP systems are preferred Operating system: Linux and MacOS X are actively supported by the current version. Earlier versions were also tested successfully on IRIX and AIX Number of processors used: Generally unlimited, but best scaling with 2-4 processors for typical problems RAM: 160 Megabytes minimum for the examples given here Classification: 2.7 External routines: FFTW Library [3,4], Gnu Scientific Library [5], bzip2, bunzip2 Nature of problem: The relativistic time evolution of wave functions according to the Dirac equation is a challenging numerical task. Especially for an electron in the presence of high intensity laser beams and/or highly charged ions, this type of problem is of considerable interest to atomic physicists. Solution method: The code employs the split-operator method [1,2], combined with fast Fourier transforms (FFT) for calculating any occurring spatial derivatives, to solve the given problem. An autocorrelation spectral method [6] is provided to generate a bound state for use as the initial wave function of further dynamical studies. Restrictions: The code in its current form is restricted to problems in two spatial dimensions. Otherwise it is only limited by CPU time and memory that one can afford to spend on a particular problem. Unusual features: The code features dynamically adapting position and momentum space grids to keep execution time and memory requirements as small as possible. It employs an object-oriented approach, and it relies on a Clifford algebra class library to represent the mathematical objects of the Dirac formalism which we employ. Besides that it includes a feature (typically called "checkpointing") which allows the resumption of an
Leviatan, A
2004-01-01
We show that the Dirac equation in 3+1 dimensions gives rise to supersymmetric patterns when the scalar and vector potentials are (i) Coulombic with arbitrary strengths or (ii) when their sum or difference is a constant, leading to relativistic pseudospin and spin symmetries. The conserved quantities and the common intertwining relation responsible for such patterns are discussed.
Leviatan, A
2004-05-21
We show that the Dirac equation in (3+1) dimensions gives rise to supersymmetric patterns when the scalar and vector potentials are (i). Coulombic with arbitrary strengths or (ii). when their sum or difference is a constant, leading to relativistic pseudospin and spin symmetries. The conserved quantities and the common intertwining relation responsible for such patterns are discussed.
Stochastic differential equations and turbulent dispersion
Durbin, P. A.
1983-01-01
Aspects of the theory of continuous stochastic processes that seem to contribute to an understanding of turbulent dispersion are introduced and the theory and philosophy of modelling turbulent transport is emphasized. Examples of eddy diffusion examined include shear dispersion, the surface layer, and channel flow. Modeling dispersion with finite-time scale is considered including the Langevin model for homogeneous turbulence, dispersion in nonhomogeneous turbulence, and the asymptotic behavior of the Langevin model for nonhomogeneous turbulence.
Koga, James
2004-10-01
Usually the motion of an electron under the influence of electromagnetic fields is influenced to a small extent by radiation damping. With the advent of high power high irradiance lasers it has become possible to generate focused laser irradiances where electrons interacting with the laser become highly relativistic over very short time and spatial scales. By focusing petawatt class lasers to very small spot sizes the amount of radiation emitted by electrons can become very large. Resultingly, the damping of the electron motion by the emission of this radiation can become large. In order to study this problem a code is written to solve a set of equations describing the evolution of a strong electromagnetic wave interacting with a single electron. Usually the equation of motion of an electron including radiation damping under the influence of electromagnetic fields is derived from the Lorentz-Dirac equation treating the damping as a perturbation. We use this equation to integrate forward in time and use the Lorentz-Dirac equation to integrate backward in time. We show that for very short wavelength electromagnetic radiation deep in the quantum regime at high irradiances differences between the perturbation equation and Lorentz-Dirac can be seen. However, for electron motion in the classical regime the differences are negligible. For electron motion in the classical regime the first order damping equation is found to be very adequate.
Nakhleh, Charles
2012-01-01
In this pedagogical note, I revisit the problem of the equation of motion of a relativistic classical electron coupled to the electromagnetic field, a problem that is not so much addressed in the education of the typical physics student as put aside en route to the more difficult problems that arise in quantum field theory. The equations governing the motion of a classical electron under the influence of its electromagnetic field have been discussed for a century, and continue to be actively investigated in the current literature, but it appears that a consistent approach to the problem from the point of view of modern renormalization theory has not previously been reported. I show that the methods of modern renormalization theory applied to the full Maxwell-Lorentz system provide a natural and intuitive derivation of the Lorentz-Dirac equation as an effective description of the electron motion valid for distances large compared to the classical electron radius. Moreover, a consistent treatment of the Lorentz...
Integrated optical Dirac physics via inversion symmetry breaking
Collins, Matthew J.; Zhang, Fan; Bojko, Richard; Chrostowski, Lukas; Rechtsman, Mikael C.
2016-12-01
Graphene and boron nitride are two-dimensional materials whose atoms are arranged in a honeycomb lattice. Their unique properties arise because their electrons behave like relativistic particles (without and with mass, respectively)—namely, they obey the Dirac equation. Here, we use a photonic analog of boron nitride to observe Dirac physics in a silicon integrated optical platform. This will allow for photonic applications of Dirac dispersions (gapped and ungapped) to be realized in an on-chip, integrated nanophotonic platform.
Fillion-Gourdeau, F; Bandrauk, A D
2015-01-01
A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron-molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.
Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis
Fillion-Gourdeau, F., E-mail: filliong@CRM.UMontreal.ca [Université du Québec, INRS – Énergie, Matériaux et Télécommunications, Varennes, J3X 1S2 (Canada); Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4 (Canada); Lorin, E., E-mail: elorin@math.carleton.ca [School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6 (Canada); Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4 (Canada); Bandrauk, A.D., E-mail: andre.bandrauk@usherbrooke.ca [Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, J1K 2R1 (Canada); Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4 (Canada)
2016-02-15
A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron–molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.
Construction of Solution for the Third Order Dispersion Evolution Equation
ZHANG Li-xun; LIU Yong-zhi; WANG Kang-ning
2004-01-01
The Golstein's strong solution formula of the second order evolution equation expands to that of the third dispersion equation by the analogy method. The semigroup expressions of its generating operator of the third order dispersion equation are obtained, and the expression to satisfy the semigroup conditions in the three orthogonal Hilbert space of the construction is also proved. Furthermore, the necessary and sufficient conditions of the generating operator's unitary semigroup are given.
A Proof Of Existence Of Particle-like Solutions Of Einstein Dirac Equations
Bird, E J
2005-01-01
We prove existence of a ground state particle-like solution to the Einstein-Dirac equations: a b ′= 1&parl0;r Ar&parr0; -wTr +m&parl0;A r&parr0; wT r- m&parl0; Ar&parr0; -1&parl0;rA r&parr0; ˙a b A′r= 1- Ar r- 16pwr T2r a2r+b 2r T′r=T rA r-1 2rAr -16pwT3r &parl0;a2r +b2r&parr0; 2rAr +32pT2r ab2r2 Ar +16pmT2r a2r -b2r 2rAr By a ground state particle-like solution we mean a solution of the above equations that satisfies the constraints: limr→∞ r1-Ar <∞lim r- ∞Tr <∞0∞ &parl0;a2r +b2r&parr0; TAdr<∞ .
The inverse problem based on a full dispersive wave equation
Gegentana Bao; Naranmandula Bao
2012-01-01
The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.
The many faces of Maxwell, Dirac and Einstein equations. A Clifford bundle approach
Rodrigues, W.A. Jr.; Oliveira, E.C. de
2007-07-01
This book is a thoughtful exposition of the algebra and calculus of differential forms, the Clifford and Spin-Clifford bundles formalisms with emphasis in calculation procedures, and vistas to a formulation of some important concepts of differential geometry necessary for a deep understanding of spacetime physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields, which were originally considered objects of a very different mathematical nature, are shown to have representatives as objects of the same mathematical nature, i.e. as sections of an appropriate Clifford bundle. This approach reveals unity in the diversity and also the many faces of the equations satisfied by those fields. Moreover, it suggests relationships which are hidden in the standard formalisms and new paths for research. Some foundational issues of relativistic field theories, in particular the one concerning the conditions for the existence of the conservation laws of energy-momentum and angular momentum in spacetime theories and many misconceptions concerning this issue is analyzed in details. The book will be useful as reference book for researchers and advanced students of theoretical physics and mathematics. Calculation procedures are illustrated by many exercises solved in detail, using the ''tricks of the trade''. Furthermore the readers will appreciate the comprehensive list of mathematical symbols as well as a list of acronyms and abbreviations. (orig.)
General solution of the Dirac equation for quasi-two-dimensional electrons
Eremko, Alexander, E-mail: eremko@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); Brizhik, Larissa, E-mail: brizhik@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); Loktev, Vadim, E-mail: vloktev@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); National Technical University of Ukraine “KPI”, Peremohy av., 37, Kyiv, 03056 (Ukraine)
2016-06-15
The general solution of the Dirac equation for quasi-two-dimensional electrons confined in an asymmetric quantum well, is found. The energy spectrum of such a system is exactly calculated using special unitary operator and is shown to depend on the electron spin polarization. This solution contains free parameters, whose variation continuously transforms one known particular solution into another. As an example, two different cases are considered in detail: electron in a deep and in a strongly asymmetric shallow quantum well. The effective mass renormalized by relativistic corrections and Bychkov–Rashba coefficients are analytically obtained for both cases. It is demonstrated that the general solution transforms to the particular solutions, found previously (Eremko et al., 2015) with the use of spin invariants. The general solution allows to establish conditions at which a specific (accompanied or non-accompanied by Rashba splitting) spin state can be realized. These results can prompt the ways to control the spin degree of freedom via the synthesis of spintronic heterostructures with the required properties.
Skeletonized wave equation of surface wave dispersion inversion
Li, Jing
2016-09-06
We present the theory for wave equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. Similar to wave-equation travel-time inversion, the complicated surface-wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the (kx,ω) domain. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2D or 3D velocity models. This procedure, denoted as wave equation dispersion inversion (WD), does not require the assumption of a layered model and is less prone to the cycle skipping problems of full waveform inversion (FWI). The synthetic and field data examples demonstrate that WD can accurately reconstruct the S-wave velocity distribution in laterally heterogeneous media.
Rivasseau, Vincent; Fuchs, Jean-Nöel
2017-01-01
This fifteenth volume of the Poincare Seminar Series, Dirac Matter, describes the surprising resurgence, as a low-energy effective theory of conducting electrons in many condensed matter systems, including graphene and topological insulators, of the famous equation originally invented by P.A.M. Dirac for relativistic quantum mechanics. In five highly pedagogical articles, as befits their origin in lectures to a broad scientific audience, this book explains why Dirac matters. Highlights include the detailed "Graphene and Relativistic Quantum Physics", written by the experimental pioneer, Philip Kim, and devoted to graphene, a form of carbon crystallized in a two-dimensional hexagonal lattice, from its discovery in 2004-2005 by the future Nobel prize winners Kostya Novoselov and Andre Geim to the so-called relativistic quantum Hall effect; the review entitled "Dirac Fermions in Condensed Matter and Beyond", written by two prominent theoreticians, Mark Goerbig and Gilles Montambaux, who consider many other mater...
Ovsiyuk, E M; Red'kov, V M
2010-01-01
There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in the space of constant positive curvature, spherical Riemann space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Riemann space geometry, is obtained.
Debergh, N M; Samsonov, B F; Van den Bossche, B
2002-01-01
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined through the knowledge of only two eigenfunctions of the first Dirac Hamiltonian. Moreover this operator together with its adjoint and the two Hamiltonians generate a quadratic deformation of the superalgebra subtending the usual supersymmetric quantum mechanics. Our developments are illustrated on the free particle case and the generalized Coulomb interaction. In the latter case, a relativistic counterpart of shape-invariance is observed.
Altuğ Arda
2017-01-01
Full Text Available We find the exact bound state solutions and normalization constant for the Dirac equation with scalar-vector-pseudoscalar interaction terms for the generalized Hulthén potential in the case where we have a particular mass function m(x. We also search the solutions for the constant mass where the obtained results correspond to the ones when the Dirac equation has spin and pseudospin symmetry, respectively. After giving the obtained results for the nonrelativistic case, we search then the energy spectra and corresponding upper and lower components of Dirac spinor for the case of PT-symmetric forms of the present potential.
DISSIPATION AND DISPERSION APPROXIMATION TO HYDRODYNAMICAL EQUATIONS AND ASYMPTOTIC LIMIT
Hsiao Ling; Li Hailiang
2008-01-01
The compressible Euler equations with dissipation and/or dispersion correction are widely used in the area of applied sciences, for instance, plasma physics,charge transport in semiconductor devices, astrophysics, geophysics, etc. We consider the compressible Euler equation with density-dependent (degenerate) viscosities and capillarity, and investigate the global existence of weak solutions and asymptotic limit.
Amplitude-phase calculations of Regge poles obtained from coupled radial Dirac equations
Thylwe, K-E [KTH-Mechanics, Royal lnstitute of Technology, S-100 44 Stockholm (Sweden); McCabe, P, E-mail: ket@mech.kth.se [CCDC, 12 Union Road, CB2 1EZ, Cambridge (United Kingdom)
2011-07-08
A recently developed amplitude-phase method for spinor-wave solutions is applied to the calculations of Regge pole positions and residues of Dirac particles. At a given energy the Dirac spin causes two sets of Regge poles that tend to coalesce in the non-relativistic limit. For the particular case of equal Lorentz-type vector and scalar potentials there is only one pole string, located very close to the non-relativistic pole string.
Hu, Huayu
2015-01-01
Nonperturbative calculation of QED processes participated by a strong electromagnetic field, especially provided by strong laser facilities at present and in the near future, generally resorts to the Furry picture with the usage of analytical solutions of the particle dynamical equation, such as the Klein-Gordon equation and Dirac equation. However only for limited field configurations such as a plane-wave field could the equations be solved analytically. Studies have shown significant interests in QED processes in a strong field composed of two counter-propagating laser waves, but the exact solutions in such a field is out of reach. In this paper, inspired by the observation of the structure of the solutions in a plane-wave field, we develop a new method and obtain the analytical solution for the Klein-Gordon equation and equivalently the action function of the solution for the Dirac equation in this field, under a largest dynamical parameter condition that there exists an inertial frame in which the particl...
Precise dispersion equations of absorbing filter glasses
Reichel, S.; Biertümpfel, Ralf
2014-05-01
The refractive indices versus wavelength of optical transparent glasses are measured at a few wavelengths only. In order to calculate the refractive index at any wavelength, a so-called Sellmeier series is used as an approximation of the wavelength dependent refractive index. Such a Sellmeier representation assumes an absorbing free (= loss less) material. In optical transparent glasses this assumption is valid since the absorption of such transparent glasses is very low. However, optical filter glasses have often a rather high absorbance in certain regions of the spectrum. The exact description of the wavelength dependent function of the refractive index is essential for an optimized design for sophisticated optical applications. Digital cameras use an IR cut filter to ensure good color rendition and image quality. In order to reduce ghost images by reflections and to be nearly angle independent absorbing filter glass is used, e.g. blue glass BG60 from SCHOTT. Nowadays digital cameras improve their performance and so the IR cut filter needs to be improved and thus the accurate knowledge of the refractive index (dispersion) of the used glasses must be known. But absorbing filter glass is not loss less as needed for a Sellmeier representation. In addition it is very difficult to measure it in the absorption region of the filter glass. We have focused a lot of effort on measuring the refractive index at specific wavelength for absorbing filter glass - even in the absorption region. It will be described how to do such a measurement. In addition we estimate the use of a Sellmeier representation for filter glasses. It turns out that in most cases a Sellmeier representation can be used even for absorbing filter glasses. Finally Sellmeier coefficients for the approximation of the refractive index will be given for different filter glasses.
Universal properties of materials with the Dirac dispersion relation of low-energy excitations
Protogenov, A. P., E-mail: alprot@appl.sci-nnov.ru [Russian Academy of Sciences, Institute for Applied Physics (Russian Federation); Chulkov, E. V. [Universidad del Pais Vasco, Departamento de Fisica de Materiales (Spain)
2015-12-15
The N-terminal scheme is considered for studying the contribution of edge states to the response of a two-dimensional topological insulator. A universal distribution of the nonlocal resistance between terminals is determined in the ballistic transport approach. The calculated responses are identical to experimentally observed values. The spectral properties of surface electronic states in Weyl semimetals are also studied. The density of surface states is accurately determined. The universal behavior of these characteristics is a distinctive feature of the considered Dirac materials which can be used in practical applications.
Spin-orbit splittings in heavy-light mesons and Dirac equation
Riazuddin, [Quaid-i-Azam University Campus, National Centre for Physics, Islamabad (Pakistan); Shafiq, Sidra [National University of Science and Technology, Centre for Advance Mathematics and Physics, Islamabad (Pakistan)
2012-03-15
The spin-orbit splitting in heavy-light mesons is seen to be suppressed experimentally, which may be due to a relativistic dynamical symmetry for the Dirac Hamiltonian. An alternative derivation of such a symmetry is given. Furthermore, the dynamics necessary for a qualitative understanding of the spin-orbit splitting seen experimentally is discussed. (orig.)
A test of Lorentz-Dirac and Lienard-Wiechert equations
Comay, E.
1987-12-01
Gedanken experiments of two charges moving uniformly along a circle are used for testing both the Lorentz-Dirac radiation reaction force and the Lienard-Wiechert formulas of retarted potentials. It is shown that if some additional postulates hold then these expressions are acceptable only as low order approximations.
Dirac equation with spin symmetry for the modiﬁed Pöschl–Teller potential in dimensions
D Agboola
2011-06-01
We present solutions of the Dirac equation with spin symmetry for vector and scalar modiﬁed Pöschl–Teller potentials within the framework of an approximation of the centrifugal term. The relativistic energy spectrum is obtained using the Nikiforov–Uvarov method and the two-component spinor wave functions obtained are in terms of the Jacobi polynomials. It is found that there exist only positive energy states for bound states under spin symmetry, and the energy of a level with ﬁxed value of , increases with increase in dimension of space time and the potential range parameter .
Gromes, Dieter
2015-01-01
We calculate the energy radiated coherently by a system of $N$ charged non relativistic particles. It disagrees with the energy loss which is obtained if one employs the Lorentz Abraham Dirac (LAD) equation for each particle, and sums up the contributions. This fact was already clearly stated in the classical literature long ago. The reason for the discrepancy is the omission of the mixing terms in the Poynting vector. For some simple systems we present a generalized equation for the radiation reaction force which cures this defect. The counter examples show that the LAD equation cannot be generally valid and that all "proofs" must fail somewhere. We demonstrate this failure for some popular examples in the literature.
THE CAUCHY PROBLEM FOR SOME DISPERSIVE WAVE EQUATIONS
Zhang Wenling
2004-01-01
In this paper, we consider the Cauchy problem for some dispersive equations. By means of nonlinear estimate in Besov spaces and fixed point theory, we prove the global well-posedness of the above problem. What's more, we improve the scattering result obtained in [1].
Pseudo Dirac dispersion in Mn-intercalated graphene on SiC
Kahaly, M. Upadhyay
2013-07-01
The atomic and electronic structures of bulk C6Mn, bulk C 8Mn, and Mn-intercalated graphene on SiC(0 0 0 1) and SiC(0001̄) are investigated by density functional theory. We find for both configurations of Mn-intercalated graphene a nonmagnetic state, in agreement with the experimental situation for SiC(0 0 0 1), and explain this property. The electronic structures around the Fermi energy are dominated by Dirac-like cones at energies consistent with data from angular resolved photoelectron spectroscopy [Gao et al., ACS Nano. 6 (2012) 6562]. However, our results demonstrate that the corresponding states trace back to hybridized Mn d orbitals, and not to the graphene. © 2013 Elsevier B.V. All rights reserved.
The many faces of Maxwell, Dirac and Einstein equations a Clifford bundle approach
Rodrigues, Jr, Waldyr A
2016-01-01
This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research. This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields. The exercises with solut...
Consistency of multi-time Dirac equations with general interaction potentials
Deckert, Dirk-André; Nickel, Lukas
2016-07-01
In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac's multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.
Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations
Wan, Renhui; Chen, Jiecheng
2016-08-01
In this paper, we obtain global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Our works are consistent with the corresponding works by Elgindi-Widmayer (SIAM J Math Anal 47:4672-4684, 2015) in the special case {A=κ=1}. In addition, our result concerning the SQG equation can be regarded as the borderline case of the work by Cannone et al. (Proc Lond Math Soc 106:650-674, 2013).
BETA NUR PRATIWI; A SUPARMI; C CARI; ANDRI SOFYAN HUSEIN
2017-02-01
Analytical solution of the Dirac equation for the modified Pöschl–Teller potential and trigonometric Scarf II non-central potential for spin symmetry is studied using asymptotic iteration method. One-dimensional Dirac equation consisting of the radial and angular parts can be obtained by the separation of variables. By usingasymptotic iteration method, the relativistic energy equation and orbital quantum number (l) equation can be obtained, where both are interrelated. Relativistic energy equation is calculated numerically by the Matlab software. The increase in the radial quantum number $n_r$ causes a decrease in the energy value, and the wave functions of the radial and the angular parts are expressed in terms of hypergeometric functions. Some thermodynamical properties of the system can be determined by reducing the relativistic energy equation to the non-relativisticenergy equation. Thermodynamical properties such as vibrational partition function, vibrational specific heat function and vibrational mean energy function are expressed in terms of error function.
Pratiwi, Beta Nur; Suparmi, A.; Cari, C.; Husein, Andri Sofyan
2017-02-01
Analytical solution of the Dirac equation for the modified Pöschl-Teller potential and trigonometric Scarf II non-central potential for spin symmetry is studied using asymptotic iteration method. One-dimensional Dirac equation consisting of the radial and angular parts can be obtained by the separation of variables. By using asymptotic iteration method, the relativistic energy equation and orbital quantum number ( l) equation can be obtained, where both are interrelated. Relativistic energy equation is calculated numerically by the Matlab software. The increase in the radial quantum number n r causes a decrease in the energy value, and the wave functions of the radial and the angular parts are expressed in terms of hypergeometric functions. Some thermodynamical properties of the system can be determined by reducing the relativistic energy equation to the non-relativistic energy equation. Thermodynamical properties such as vibrational partition function, vibrational specific heat function and vibrational mean energy function are expressed in terms of error function.
A Partially-ordered-set Based Approach to the Dirac Equation in 3+1 space-time
Earle, Keith; Knuth, Kevin
2012-02-01
Recent work by Knuth and co-workers has shown how insights into Einstein's Theory of Special Relativity may be obtained by careful reasoning about consistent quantification of a poset. The Feynman Chessboard problem in 1+1 spacetime can be treated from this perspective, for example. Alternative methods of solution based on techniques borrowed from statistical mechanics have also been developed over the years to solve the Feynman Chessboard model in 1+1 spacetime. One particularly intriguing solution is based on a master-equation approach developed by McKeon and Ord for 1+1 spacetime. We will show how this model may be extended to 3+1 spacetime using techniques developed by Bialynicki-Birula, thus providing an alternative derivation of the Dirac equation. An external electromagnetic field can be accommodated very naturally in the formalism from which a pleasing pictorial representation of electromagnetic interactions in the lattice picture emerges.
Suparmi
2014-12-01
Full Text Available The bound state solution of the Dirac equation for generalized PöschlTeller and trigonometric Pöschl-Teller non-central potentials was obtained using SUSY quantum mechanics and the idea of shape invariance potential. The approximate relativistic energy spectrum was expressed in the closed form. The radial and polar wave functions were obtained using raising and lowering of radial and polar operators. The orbital quantum numbers were found from the polar Dirac equation, which was solved using SUSY quantum mechanics and the idea of shape invariance.
Langevin equation model of dispersion in the convective boundary layer
Nasstrom, J S
1998-08-01
This dissertation presents the development and evaluation of a Lagrangian stochastic model of vertical dispersion of trace material in the convective boundary layer (CBL). This model is based on a Langevin equation of motion for a fluid particle, and assumes the fluid vertical velocity probability distribution is skewed and spatially homogeneous. This approach can account for the effect of large-scale, long-lived turbulent structures and skewed vertical velocity distributions found in the CBL. The form of the Langevin equation used has a linear (in velocity) deterministic acceleration and a skewed randomacceleration. For the case of homogeneous fluid velocity statistics, this ""linear-skewed" Langevin equation can be integrated explicitly, resulting in a relatively efficient numerical simulation method. It is shown that this approach is more efficient than an alternative using a "nonlinear-Gaussian" Langevin equation (with a nonlinear deterministic acceleration and a Gaussian random acceleration) assuming homogeneous turbulence, and much more efficient than alternative approaches using Langevin equation models assuming inhomogeneous turbulence. "Reflection" boundary conditions for selecting a new velocity for a particle that encounters a boundary at the top or bottom of the CBL were investigated. These include one method using the standard assumption that the magnitudes of the particle incident and reflected velocities are positively correlated, and two alternatives in which the magnitudes of these velocities are negatively correlated and uncorrelated. The constraint that spatial and velocity distributions of a well-mixed tracer must be the same as those of the fluid, was used to develop the Langevin equation models and the reflection boundary conditions. The two Langevin equation models and three reflection methods were successfully tested using cases for which exact, analytic statistical properties of particle velocity and position are known, including well
Dispersion Equation of the Coaxial-Radial Line
WANG Wenxiang; YUE Lingna; YU Guofen; GONG Yubing; HUANG Minzhi
2004-01-01
An all-metal slow-wave structure,coaxial-radial line,which is suitable for application in broadband high power traveling wave tube (TWT) and relativistic TWT as a RF system is introduced.Making use of the field matching method and variational method together with the orthogonality of the Bessel function and the Floquet Theroem for the periodic system,the dispersion characteristic expression is derived.This equation is more rigorous than that of precious reports.
Zhang, Zhendong
2016-07-26
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh wave dispersion curve using a difference approximation to the gradient of the misfit function. We call this wave equation inversion of skeletonized surface waves because the skeletonized dispersion curve for the fundamental-mode Rayleigh wave is inverted using finite-difference solutions to the multi-dimensional elastic wave equation. The best match between the predicted and observed dispersion curves provides the optimal S-wave velocity model. Our method can invert for lateral velocity variations and also can mitigate the local minimum problem in full waveform inversion with a reasonable computation cost for simple models. Results with synthetic and field data illustrate the benefits and limitations of this method. © 2016 Elsevier B.V.
Crater, Horace; Yang, Dujiu
1991-09-01
A semirelativistic expansion in powers of 1/c2 is canonically matched through order (1/c4) of the two-particle total Hamiltonian of Wheeler-Feynman vector and scalar electrodynamics to a similar expansion of the center of momentum (c.m.) total energy of two interacting particles obtained from covariant generalized mass shell constraints derived with the use of the classical Todorov equation and Dirac's Hamiltonian constraint mechanics. This determines through order 1/c4 the direct interaction used in the covariant Todorov constraint equation. We show that these interactions are momentum independent in spite of the extensive and complicated momentum dependence of the potential energy terms in the Wheeler-Feynman Hamiltonian. The invariant expressions for the relativistic reduced mass and energy of the fictitious particle of relative motion used in the Todorov equation are also dynamically determined through this order by this same procedure. The resultant covariant Todorov equation then not only reproduces the noncovariant Wheeler-Feynman dynamics through order 1/c4 but also implicitly provides a rather simple covariant extrapolation of it to all orders of 1/c2.
Fang, Zhi; Shi, Min; Guo, Jian-You; Niu, Zhong-Ming; Liang, Haozhao; Zhang, Shi-Sheng
2017-02-01
Resonances play critical roles in the formation of many physical phenomena, and many techniques have been developed for the exploration of resonances. In a recent letter [Phys. Rev. Lett. 117, 062502 (2016), 10.1103/PhysRevLett.117.062502], we proposed a new method for probing single-particle resonances by solving the Dirac equation in complex momentum representation for spherical nuclei. Here, we develop the theoretical formalism of this method for deformed nuclei. We elaborate numerical details and calculate the bound and resonant states in 37Mg. The results are compared with those from the coordinate representation calculations with a satisfactory agreement. In particular, the present method can expose clearly the resonant states in a complex momentum plane and determine precisely the resonance parameters for not only narrow resonances but also broad resonances that were difficult to obtain before.
Hall, Richard L.; Zorin, Petr
2016-04-01
The classic comparison theorem of quantum mechanics states that if two potentials are ordered then the corresponding energy eigenvalues are similarly ordered, that is to say if Va≤ Vb, then Ea≤ Eb. Such theorems have recently been established for relativistic problems even though the discrete spectra are not easily characterized variationally. In this paper we improve on the basic comparison theorem for the Dirac equation with spin and pseudo-spin symmetry in d≥ 1 dimensions. The graphs of two comparison potentials may now cross each other in a prescribed manner implying that the energy values are still ordered. The refined comparison theorems are valid for the ground state in one dimension and for the bottom of an angular momentum subspace in d>1 dimensions. For instance in a simplest case in one dimension, the condition Va≤ Vb is replaced by Ua≤ Ub, where Ui(x)=int0x Vi(t)dt, xin[0,∞), and i=a or b.
Mapping curved spacetimes into Dirac spinors
Sabín, Carlos
2016-01-01
We show how to transform a Dirac equation in curved spacetime into a Dirac equation in flat spacetime. In particular, we show that any solution of the free massless Dirac equation in a 1+1 dimensional flat spacetime can be transformed via a local phase transformation into a solution of the corresponding Dirac equation in a curved background, where the spacetime metric is encoded into the phase. In this way, the existing quantum simulators of the Dirac equation can naturally incorporate curved spacetimes. As a first example we use our technique to obtain solutions of the Dirac equation in a particular family of interesting spacetimes in 1+1 dimensions.
Benchmark values for molecular three-center integrals arising in Dirac equation
Bagci, A
2015-01-01
The authors in their previous papers obtained compact, arbitrarily accurate expressions for two center one and two electron Dirac integrals expressed over Slater orbitals with noninteger principal quantum numbers. In this present study, the accuracy limits of given expressions is examined for three-center nuclear attraction integrals, which are the first integral set do not have analytically closed form relations. They are expressed through new molecular auxiliary functions obtained via Neumann expansion of Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrarily values of orbital parameters, principal quantum numbers in Slater orbitals. Several methods, such as Laplace expansion of Coulomb interaction, single center expansions, Fourier transformation methods, have been developed for evaluation of three-center integrals when integer principal quantum number is used. This is the first attempts to study the three-center integrals with noninteger Slater orbita...
Quiney, H. M.; Glushkov, V. N.; Wilson, S.; Sabin,; Brandas, E
2001-01-01
A comparison is made of the accuracy achieved in finite difference and finite basis set approximations to the Dirac equation for the ground state of the hydrogen molecular ion. The finite basis set calculations are carried out using a distributed basis set of Gaussian functions the exponents and pos
Quiney, H. M.; Glushkov, V. N.; Wilson, S.; Sabin,; Brandas, E
2001-01-01
A comparison is made of the accuracy achieved in finite difference and finite basis set approximations to the Dirac equation for the ground state of the hydrogen molecular ion. The finite basis set calculations are carried out using a distributed basis set of Gaussian functions the exponents and
Bruning, J.; Dobrokhotov, S.Y.; Katsnelson, M.I.; Minenkov, D.S.
2016-01-01
We consider the two-dimensional stationary Schrodinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of st
Quantum-classical correspondence of the Dirac equation with a scalar-like potential
Mai-Lin Liang; Shun-Lin Shu; Bing Yuan
2009-05-01
Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein–Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.
TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS
无
2002-01-01
The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.
Photoconductivity in Dirac materials
J. M. Shao
2015-11-01
Full Text Available Two-dimensional (2D Dirac materials including graphene and the surface of a three-dimensional (3D topological insulator, and 3D Dirac materials including 3D Dirac semimetal and Weyl semimetal have attracted great attention due to their linear Dirac nodes and exotic properties. Here, we use the Fermi’s golden rule and Boltzmann equation within the relaxation time approximation to study and compare the photoconductivity of Dirac materials under different far- or mid-infrared irradiation. Theoretical results show that the photoconductivity exhibits the anisotropic property under the polarized irradiation, but the anisotropic strength is different between 2D and 3D Dirac materials. The photoconductivity depends strongly on the relaxation time for different scattering mechanism, just like the dark conductivity.
Photoconductivity in Dirac materials
Shao, J. M.; Yang, G. W., E-mail: stsygw@mail.sysu.edu.cn [State Key Laboratory of Optoelectronic Materials and Technologies, Nanotechnology Research Center, School of Materials & Engineering, School of Physics & Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong (China)
2015-11-15
Two-dimensional (2D) Dirac materials including graphene and the surface of a three-dimensional (3D) topological insulator, and 3D Dirac materials including 3D Dirac semimetal and Weyl semimetal have attracted great attention due to their linear Dirac nodes and exotic properties. Here, we use the Fermi’s golden rule and Boltzmann equation within the relaxation time approximation to study and compare the photoconductivity of Dirac materials under different far- or mid-infrared irradiation. Theoretical results show that the photoconductivity exhibits the anisotropic property under the polarized irradiation, but the anisotropic strength is different between 2D and 3D Dirac materials. The photoconductivity depends strongly on the relaxation time for different scattering mechanism, just like the dark conductivity.
Point-particle effective field theory III: relativistic fermions and the Dirac equation
Burgess, C. P.; Hayman, Peter; Rummel, Markus; Zalavári, László
2017-09-01
We formulate point-particle effective field theory (PPEFT) for relativistic spin-half fermions interacting with a massive, charged finite-sized source using a first-quantized effective field theory for the heavy compact object and a second-quantized language for the lighter fermion with which it interacts. This description shows how to determine the near-source boundary condition for the Dirac field in terms of the relevant physical properties of the source, and reduces to the standard choices in the limit of a point source. Using a first-quantized effective description is appropriate when the compact object is sufficiently heavy, and is simpler than (though equivalent to) the effective theory that treats the compact source in a second-quantized way. As an application we use the PPEFT to parameterize the leading energy shift for the bound energy levels due to finite-sized source effects in a model-independent way, allowing these effects to be fit in precision measurements. Besides capturing finite-source-size effects, the PPEFT treatment also efficiently captures how other short-distance source interactions can shift bound-state energy levels, such as due to vacuum polarization (through the Uehling potential) or strong interactions for Coulomb bound states of hadrons, or any hypothetical new short-range forces sourced by nuclei.
Sharma, Anushrut
2014-01-01
It is well-known that in the Newman-Penrose formalism the Riemann tensor can be expressed as a set of eighteen complex first-order equations, in terms of the twelve spin coefficients, known as Ricci identities. The Ricci tensor herein is determined via the Einstein equations. It is also known that the Dirac equation in a curved spacetime can be written in the Newman-Penrose formalism as a set of four first-order coupled equations for the spinor components of the wave-function. In the present article we suggest that it might be possible to think of the Dirac equations in the N-P formalism as a special case of the Ricci identities, after an appropriate identification of the four Dirac spinor components with four of the spin coefficients, provided torsion is included in the connection, and after a suitable generalization of the energy-momentum tensor. We briefly comment on similarities with the Einstein-Cartan-Sciama-Kibble theory. The motivation for this study is to take some very preliminary steps towards deve...
Kononets, Yu. V.
2016-12-01
The presented enhanced version of Eriksen's theorem defines an universal transform of the Foldy-Wouthuysen type and in any external static electromagnetic field (ESEMF) reveals a discrete symmetry of Dirac's equation (DE), responsible for existence of a highly influential conserved quantum number—the charge index distinguishing two branches of DE spectrum. It launches the charge-index formalism (CIF) obeying the charge-index conservation law (CICL). Via its unique ability to manipulate each spectrum branch independently, the CIF creates a perfect charge-symmetric architecture of Dirac's quantum mechanics (DQM), which resolves all the riddles of the standard DE theory (SDET). Besides the abstract CIF algebra, the paper discusses: (1) the novel accurate charge-symmetric definition of the electric-current density; (2) DE in the true-particle representation, where electrons and positrons coexist on equal footing; (3) flawless "natural" scheme of second quantization; and (4) new physical grounds for the Fermi-Dirac statistics. As a fundamental quantum law, the CICL originates from the kinetic-energy sign conservation and leads to a novel single-particle physics in strong-field situations. Prohibiting Klein's tunneling (KT) in Klein's zone via the CICL, the precise CIF algebra defines a new class of weakly singular DE solutions, strictly confined in the coordinate space and experiencing the total reflection from the potential barrier.
Yuan Hongjun; Jin Yang
2005-01-01
The aim of this paper is to discuss the existence and uniqueness of solutions for the porous medium equation ut - (um)xx = μ(x) in (x,t) ∈ R × (0, +∞) with initial condition u(x, 0) = uo(x) x ∈ (-∞, +∞),whereμ(x) is a nonnegative finite Radon measure, u0 ∈ L1 (R)∩L∞ (R) is a nonnegative function, and m ＞ 1, and R ≡ (-∞, +∞).
Derivation of the Convective Dispersion Equation with Adsorption by Markov Random Ways
WU Jing-Chun; QIN Sheng-Gao; WANG Yang
2009-01-01
The convective dispersion equation with adsorption is derived on the basis of the Chapman-Kolmogroff equation which expresses the statistical properties of the Markov transition probability. The acquired equation has the same expression as the one derived on the basis of the combination of both the mass balance equation and the particles retention kinetics equation. The probability variables that describe the random movement of solute particles have a definite physical significance associated with the parameters in the convective dispersion equation. The derivation confirms the validity of the Markov process to describe the particles movement in the process of convective dispersion.
Pratiwi, B. N.; Suparmi, A.; Cari, C.; Husein, A. S.; Yunianto, M.
2016-08-01
We apllied asymptotic iteration method (AIM) to obtain the analytical solution of the Dirac equation in case exact pseudospin symmetry in the presence of modified Pcischl- Teller potential and trigonometric Scarf II non-central potential. The Dirac equation was solved by variables separation into one dimensional Dirac equation, the radial part and angular part equation. The radial and angular part equation can be reduced into hypergeometric type equation by variable substitution and wavefunction substitution and then transform it into AIM type equation to obtain relativistic energy eigenvalue and wavefunctions. Relativistic energy was calculated numerically by Matlab software. And then relativistic energy spectrum and wavefunctions were visualized by Matlab software. The results show that the increase in the radial quantum number nr causes decrease in the relativistic energy spectrum. The negative value of energy is taken due to the pseudospin symmetry limit. Several quantum wavefunctions were presented in terms of the hypergeometric functions.
Anderson's absolute objects and constant timelike vector hidden in Dirac matrices
2001-01-01
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute objects of the Dirac equation. There are two ways of the Dirac matrices transformation: (1) Dirac matrices form a 4-vector and wave function is a scalar, (2) Dirac matrices are scalars and the wave function is a spinor. In the first case the Dirac equation ...
New multi-soliton solutions and travelling wave solutions of the dispersive long-wave equations
张解放; 郭冠平; 吴锋民
2002-01-01
Using the extended homogeneous balance method, the (1+1)-dimensional dispersive Iong-wave equations have been solved. Starting from the homogeneous balance method, we have obtained a nonlinear transformation for simplifying a dispersive long-wave equation into a linear partial differential equation. Usually, we can obtain only a type of soliton-like solution. In this paper, we have further found some new multi-soliton solutions and exact travelling solutions of the dispersive long-wave equations from the linear partial equation.
量子与经典对应：Dirac方程中的速度算符%Quantum and classical correspondence for velocity operator in Dirac equation
张治国; 封文江; 郑伟; 陈皓; 崔崧
2016-01-01
Due to Bohr correspondence principle in the quantum mechanics,quantum mechanics go to classical mechanics in the case of large quantum number. Based upon Heisenberg correspondence principle, quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Using Heisenberg correspondence principle, quantum-classical correspondence of the relativistic free particle and the 1/2 spin charged particles in a constant magnetic field are studied.Applying Heisenberg correspondence principle to Dirac equation in relativistic realm,the operator of free particle in the Dirac theory and quantum-classical correspondence are obtained,and 1/2 spin charged particle in a constant magnetic field in Dirac equation is also studied.For the relativistic free particle or the charged particle in a constant magnetic field,the velocity operator in the Dirac theory will reduce to the classical velocity.%由量子力学中的Bohr对应原理可知,在大量子数情形下,量子力学应过渡到经典力学。在经典极限下,由 Heisenberg 对应原理可知,厄密算符的量子矩阵元对应经典物理量的Fourier展开系数。利用 Heisenberg对应原理研究相对论效应的自由粒子和在匀磁场中的带电粒子的量子经典对应问题。将 Heisenberg对应原理应用到相对论领域的 Dirac 方程,计算出自由粒子的Dirac方程中的α算符及其经典近似,并且研究自旋1/2的带电粒子在匀磁场中的Dirac 方程情形。对于相对论效应的自由粒子和在匀磁场中的带电粒子,Dirac理论中的α算符将对应经典的速度。
Barik, N.; Das, M.
1983-12-01
The effect of confinement on the magnetic moment of a quark has been studied in a simple independent-quark model based on the Dirac equation with a power-law potential. The magnetic moments so obtained for the constituent quarks, which are found to be significantly different from their corresponding Dirac moments, are used in predicting the magnetic moments of baryons in the nucleon octet as well as those in the charmed and b-flavored sectors. We not only get an improved result for the proton magnetic moment, but the calculation for the rest of the nucleon octet also turns out to be in reasonable agreement with experiment. The overall predictions for the charmed and b-flavored baryons are also comparable with other model predictions.
WANG Qi; CHEN Yong; LI Biao; ZHANG Hong-Qing
2004-01-01
Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e.,the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.
Menculini, L; Roy, P
2013-01-01
We study the (2+1) dimensional Dirac equation in an homogeneous magnetic field (relativistic Landau problem) within a minimal length, or generalized uncertainty principle -GUP-, scenario. We derive exact solutions for a given explicit representation of the GUP and provide expressions of the wave functions in the momentum representation. We find that in the minimal length case the degeneracy of the states is modified and that there are states that do not exist in the ordinary quantum mechanics limit (\\beta -->0). We also discuss the mass-less case which may find application in describing the behavior of charged fermions in new materials like Graphene.
Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
Eldeberky, Y.; Madsen, Per A.
1999-01-01
This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary c...
Sochichiu, Corneliu
2011-01-01
We study the emergence of Dirac fermionic field in the low energy description of non-relativistic dynamical network models. The Dirac fermionic field appears as the effective field describing the excitations above point-like Fermi levels. Together with the Dirac fermionic field an effective space-time metric is also emerging. We analyze the conditions for such Fermi points to appear in general, paying special attention to the case of two and three spacial dimensions.
Das, Amiya; Ganguly, Asish
2017-07-01
The paper deals with Kadomtsev-Petviashvili (KP) equation in presence of a small dispersion effect. The nature of solutions are examined under the dispersion effect by using Lyapunov function and dynamical system theory. We prove that when dispersion is added to the KP equation, in certain regions, yet there exist bounded traveling wave solutions in the form of solitary waves, periodic and elliptic functions. The general solution of the equation with or without the dispersion effect are obtained in terms of Weirstrass ℘ functions and Jacobi elliptic functions. New form of kink-type solutions are established by exploring a new technique based on factorization method, use of functional transformation and the Abel's first order nonlinear equation. Furthermore, the stability analysis of the dispersive solutions are examined which shows that the traveling wave velocity is a bifurcation parameter which governs between different classes of waves. We use the phase plane analysis and show that at a critical velocity, the solution has a transcritical bifurcation.
Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics
Ratliff, Daniel J.; Bridges, Thomas J.
2016-10-01
Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1 + 1 and 2 + 1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein-Gordon equation is used to illustrate the theory.
Maxwell and Dirac theories as an already unified theory
1995-01-01
In this paper we formulate Maxwell and Dirac theories as an already unified theory (in the sense of Misner and Wheeler). We introduce Dirac spinors as "Dirac square root" of the Faraday bivector, and use this in order to find a spinorial representation of Maxwell equations. Then we show that under certain circunstances this spinor equation reduces to an equation formally identical to Dirac equation. Finally we discuss certain conditions under which this equation can be really interpreted as D...
Hyperbolic Mild Slope Equations with Inclusion of Amplitude Dispersion Effect: Regular Waves
JIN Hong; ZOU Zhi-li
2008-01-01
A new form of hyperbolic mild slope equations is derived with the inclusion of the amplitude dispersion of nonlinear waves. The effects of including the amplitude dispersion effect on the wave propagation are discussed. Wave breaking mechanism is incorporated into the present model to apply the new equations to surf zone. The equations are solved numerically for regular wave propagation over a shoal and in surf zone, and a comparison is made against measurements. It is found that the inclusion of the amplitude dispersion can also improve model's performance on prediction of wave heights around breaking point for the wave motions in surf zone.
Moduli Space of Integrable Dirac Structures
Milani, Vida
2009-01-01
In this paper we introduce the notion of integrable Dirac structures on Hermitian modules. The moduli space of the space of integrable Dirac structures is studied. Then a necessary and sufficient condition for the integrability of a Dirac structure is obtained as the solution of a certain partial differential equation.
Hnizdo, V.
1988-06-01
Refutation is given of a recent claim that both the Lorentz-Dirac radiation reaction force and Lienard-Wiechert retarded potentials satisfy energy conservation only to a low order of approximation in a system of two charges which move uniformly along a circle. When correctly calculated, the power radiated by such a system equals exactly the rate at which work is done on the system by external forces.
QingdongCAI
1999-01-01
A new technique in the formulation of numerical scheme for hyperbolic equation is developed.It is different from the classical FD and FE methods.We begin with the algebraic equations with some undefined parameters,and get the difference equation through Taylor-series expansion.When the parameters in the partial difference equation are defined,the equation is what the scheme will simulated.The numerical example of the viscous BUrgers equation shows the validity of the scheme.This method deals with the numerical viscosity and dispersion exactly ,giving a preliminary explanation to some problems that the CFD face now.
Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension
Bisio, Alessandro; D’Ariano, Giacomo Mauro; Tosini, Alessandro, E-mail: alessandro.tosini@unipv.it
2015-03-15
We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be efficiently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing. In this framework, the discretization is a central ingredient and not only a tool for performing non-perturbative calculation as in lattice gauge theory. The automaton model, endowed with a precise notion of local observables and a full probabilistic interpretation, could lead to a coherent unification of a hypothetical discrete Planck scale with the usual Fermi scale of high-energy physics. - Highlights: • The free Dirac field in one space dimension as a quantum cellular automaton. • Large scale limit of the automaton and the emergence of the Dirac equation. • Dispersive differential equation for the evolution of smooth states on the automaton. • Optimal discrimination between the
Precise analysis of pion-pion scattering data from Roy equations and forward dispersion relations
Peláez, J R; Kaminski, R; Ynduráin, F J
2008-01-01
We review our recent analysis of pion-pion scattering data in terms of Roy equations and Forward Dispersion Relations, and present some preliminary results in terms of a new set of once-subtracted coupled equations for partial waves. The first analysis consists of independent fits to the different pion-pion channels that satisfies rather well the dispersive representation. In the second analysis we constrain the fit with the dispersion relations. The latter provides a very precise and model independent description of data using just analyticity, causality and crossing.
Two-phase flow equations for a dilute dispersion of gas bubbles in liquid
Biesheuvel, A.; Wijngaarden, van L.
1984-01-01
Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non
Shi Jing
2014-01-01
Full Text Available The solving processes of the homogeneous balance method, Jacobi elliptic function expansion method, fixed point method, and modified mapping method are introduced in this paper. By using four different methods, the exact solutions of nonlinear wave equation of a finite deformation elastic circular rod, Boussinesq equations and dispersive long wave equations are studied. In the discussion, the more physical specifications of these nonlinear equations, have been identified and the results indicated that these methods (especially the fixed point method can be used to solve other similar nonlinear wave equations.
Analytical solutions of time–space fractional, advection–dispersion andWhitham–Broer–Kaup equations
M D Khan; I Naeem; M Imran
2014-12-01
In this article, we study time–space fractional advection–dispersion (FADE) equation and time–space fractional Whitham–Broer–Kaup (FWBK) equation that have significant roles in hydrology. We introduce suitable transformations to convert fractional-order derivatives to integerorder derivatives and as a result these equations transform into partial differential equations (PDEs). Then the Lie symmetries and the corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations.
The zero dispersion limit for the Korteweg-deVries KdV equation.
Lax, P D; Levermore, C D
1979-08-01
We use the inverse scattering method to determine the weak limit of solutions of the Korteweg-deVries equation as dispersion tends to zero. The limit, valid for all time, is characterized in terms of a quadratic programming problem which can be solved with the aid of function theoretic methods. For large t, the solutions satisfy Whitham's averaged equations at some times and the equations found by Flaschka et al. at other times.
Boundary value problemfor multidimensional fractional advection-dispersion equation
Khasambiev Mokhammad Vakhaevich
2015-05-01
Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the
A simple and accurate model for Love wave based sensors: Dispersion equation and mass sensitivity
Jiansheng Liu
2014-07-01
Full Text Available Dispersion equation is an important tool for analyzing propagation properties of acoustic waves in layered structures. For Love wave (LW sensors, the dispersion equation with an isotropic-considered substrate is too rough to get accurate solutions; the full dispersion equation with a piezoelectric-considered substrate is too complicated to get simple and practical expressions for optimizing LW-based sensors. In this work, a dispersion equation is introduced for Love waves in a layered structure with an anisotropic-considered substrate and an isotropic guiding layer; an intuitive expression for mass sensitivity is also derived based on the dispersion equation. The new equations are in simple forms similar to the previously reported simplified model with an isotropic substrate. By introducing the Maxwell-Weichert model, these equations are also applicable to the LW device incorporating a viscoelastic guiding layer; the mass velocity sensitivity and the mass propagation loss sensitivity are obtained from the real part and the imaginary part of the complex mass sensitivity, respectively. With Love waves in an elastic SiO2 layer on an ST-90°X quartz structure, for example, comparisons are carried out between the velocities and normalized sensitivities calculated by using different dispersion equations and corresponding mass sensitivities. Numerical results of the method presented in this work are very close to those of the method with a piezoelectric-considered substrate. Another numerical calculation is carried out for the case of a LW sensor with a viscoelastic guiding layer. If the viscosity of the layer is not too big, the effect on the real part of the velocity and the mass velocity sensitivity is relatively small; the propagation loss and the mass loss sensitivity are proportional to the viscosity of the guiding layer.
Kong Cuicui [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)], E-mail: cuicuikong@yahoo.com.cn; Wang Dan; Song Lina; Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2009-01-30
In this paper, with the aid of symbolic computation and a general ansaetz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansaetz. The method can also be applied to other nonlinear partial differential equations.
A Short Biography of Paul A. M. Dirac and Historical Development of Dirac Delta Function
Debnath, Lokenath
2013-01-01
This paper deals with a short biography of Paul Dirac, his first celebrated work on quantum mechanics, his first formal systematic use of the Dirac delta function and his famous work on quantum electrodynamics and quantum statistics. Included are his first discovery of the Dirac relativistic wave equation, existence of positron and the intrinsic…
Cari, C.; Suparmi, A.; Yunianto, M.; Pratiwi, B. N.
2016-08-01
The Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.
Mohammadi, Vahid; Chenaghlou, Alireza
2017-09-01
The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum E3‧, anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang-Coulomb monopole (YCM) superintegrable system (a SU(2) non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.
A new dispersion-relation preserving method for integrating the classical Boussinesq equation
Jang, T. S.
2017-02-01
In this paper, a dispersion-relation preserving method is proposed for nonlinear dispersive waves, starting from the oldest weakly nonlinear dispersive wave mathematical model in shallow water waves, i.e., the classical Boussinesq equation. It is a semi-analytic procedure, however, which preserves, as a distinctive feature, the dispersion-relation imbedded in the model equation without adding (unwelcome) numerical effects, i.e., the proposed method has the same dispersion-relation as the original classical Boussinesq equation. This remarkable (dispersion-relation) preserving property is proved mathematically for small wave motion in present study. The property is also numerically examined by observing both the local wave number and the local frequency of a slowly varying water-wave group. The dispersion-relation preserving method proposed here is powerful as well for observing nonlinear wave phenomena such as solitary waves and their collision. In fact, the main features of nonlinear wave characteristics are clearly seen through not only a single propagating solitary wave but counter-propagating (head-on) solitary wave collisions. They are compared with known (exact) nonlinear solutions, the results of which represent a major improvement over existing solution formulations in the literature.
Complexiton solutions of the (2+1)-dimensional dispersive long wave equation
Chen Yong; Fan En-Gui
2007-01-01
In this pager a pure algebraic method implemented in a computer algebraic system, named multiple Riccati equations rational expansion method, is presented to construct a novel class of complexiton solutions to integrable equations and nonintegrable equations. By solving the (2+1)-dimensional dispersive long wave equation, it obtains many new types of complexiton solutions such as various combination of trigonometric periodic and hyperbolic function solutions,various combination of trigonometric periodic and rational function solutions, various combination of hyperbolic and rational function solutions, etc.
Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations
Liu, Hailiang
2009-01-01
This paper is concerned with a class of nonlocal dispersive models -- the $\\theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {\\it Acta Math. Appl. Sin.} Engl. Ser. {\\bf 24}(3)(2008)423--440]: cted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {\\it J. reine angew. Math.}, {\\bf 624} (2008)51--80.
Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics
Clamond, Didier; Mitsotakis, Dimitrios
2015-01-01
For surface gravity waves propagating in shallow water, we propose a variant of the fully nonlinear Serre-Green-Naghdi equations involving a free parameter that can be chosen to improve the dispersion properties. The novelty here consists in the fact that the new model conserves the energy, contrary to other modified Serre's equations found in the literature. Numerical comparisons with the Euler equations show that the new model is substantially more accurate than the classical Serre equations, specially for long time simulations and for large amplitudes.
Canonical structure of evolution equations with non-linear dispersive terms
B Talukdar; J Shamanna; S Ghosh
2003-07-01
The inverse problem of the variational calculus for evolution equations characterized by non-linear dispersive terms is analysed with a view to clarify why such a system does not follow from Lagrangians. Conditions are derived under which one could construct similar equations which admit a Lagrangian representation. It is shown that the system of equations thus obtained can be Hamiltonized by making use of the Dirac’s theory of constraints. The speciﬁc results presented refer to the third- and ﬁfth-order equations of the so-called distinguished subclass.
Convergence rates for dispersive approximation schemes to nonlinear Schr\\"odinger equations
Ignat, Liviu I
2011-01-01
This article is devoted to the analysis of the convergence rates of several numerical approximation schemes for linear and nonlinear Schr\\"odinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schr\\"odinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L2(R), a fact that can not be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for H^s(R) data if 0 < s < 1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic decay rates.
Casajús Ramo, A
2006-01-01
DIRAC is the LHCb Workload and Data Management System. Based on a service-oriented architecture, it enables generic distributed computing with lightweight Agents and Clients for job execution and data transfers. DIRAC implements a client-server architecture exposing server methods through XML Remote Procedure Call (XML-RPC) protocol. DIRAC is mostly coded in python. DIRAC security infrastructure has been designed to be a completely generic XML-RPC transport over a SSL tunnel. This new security layer is able to handle standard X509 certificates as well as grid-proxies to authenticate both sides of the connection. Serve and client authentication relies over OpenSSL and py-Open SSL, but to be able to handle grid proxies some modifications have been added to those libraries. DIRAC security infrastructure handles authorization and authorization as well as provides extended capabilities like secure connection tunneling and file transfer. Using this new security infrastructure all LHCb users can safely make use o...
Solving Time-Fractional Advection-Dispersion Equation by Variable Weights Particle Tracking Method
Cao, Shaohua; Jiang, Jianguo; Wu, Jichun
2017-09-01
Particle tracking method is an efficient and reliable method to solve time-fractional advection-dispersion equation, which can describe anomalous transport in heterogeneous porous media. However, this method will lead to severe fluctuation or disappearance of solutions if the concentration value is small. A variable weights method is developed to conquer the shortcoming of particle tracking method. Then, one-dimensional and two-dimensional time-fractional advection-dispersion equations are solved by the variable weights particle tracking method. Compared to traditional particle tracking method, the variable weights version may eliminate the fluctuation and improve the accuracy by orders of magnitude without more computational cost.
Wave-equation dispersion inversion of surface waves recorded on irregular topography
Li, Jing
2017-08-17
Significant topographic variations will strongly influence the amplitudes and phases of propagating surface waves. Such effects should be taken into account, otherwise the S-velocity model inverted from the Rayleigh dispersion curves will contain significant inaccuracies. We now show that the recently developed wave-equation dispersion inversion (WD) method naturally takes into account the effects of topography to give accurate S-velocity tomograms. Application of topographic WD to demonstrates that WD can accurately invert dispersion curves from seismic data recorded over variable topography. We also apply this method to field data recorded on the crest of mountainous terrain and find with higher resolution than the standard WD tomogram.
New Complexiton Solutions of (1+1)-Dimensional Dispersive Long Wave Equation
无
2006-01-01
By means of two different Riccati equations with different parameters as subequation in the components of finite rational expansion method, new complexiton solutions for the (1+1 )-dimensional dispersive long wave equation are successfully constructed, which include various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, and various combination of hyperbolic and rational function solutions.
Xia Ji; Wei Cai; Pingwen Zhang
2008-01-01
In this paper, we analyze the transmission and reflection properties of a high order discontinuous Galerkin method for dispersive Maxwell's equations, originally proposed by Lu et al. [J. Comput. Phys. 200 (2004), pp. 549-580]. We study the reflection and transmission properties of the numerical method for up to second-order polynomial elements for one-and two-dimensional Maxwell's equations with rectangular meshes. High order accuracy has been shown for reflection and transmission coefficients near material interfaces.
Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method
Jerome L.V. Lewandowski
2005-01-25
A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
Dirac oscillators and quasi-exactly solvable operators
Brihaye, Y
2005-01-01
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a quite general spherically symmetric form for these potentials and we analyse some exactly and quasi exactly solvable properties of the underlying matricial linear operators.
Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations
Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping
2016-10-01
Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.
Héctor Torres-Silva
2008-11-01
Full Text Available In the present article we propose a simple equality involving the Dirac operator and the Maxwell operators from a chiral approach. This equality establishes a direct connection between solutions of the two systems. Moreover, we show that the connection is valid when a fairly natural relationship between the frequency of the electromagnetic wave and the energy of the Dirac particle is fulfilled, if the electric field is parallel to the magnetic field . Our analysis is based on the quaternionic form of the Dirac equation and on the quaternionic form of the Maxwell equations. In both cases the quaternionic reformulations are completely equivalent to the traditional form of the Dirac and Maxwell systems. This theory is a new quantum mechanics (QM interpretation. The research below shows that the QM represents the electrodynamics of the curvilinear closed chiral waves. This concords entirely with the modern interpretation and results of the quantum field theory.En el presente artículo se propone un simple igualdad que considera el operador de Dirac y los operadores de Maxwell bajo un enfoque quiral. Esta igualdad establece una conexión directa entre las soluciones de los dos sistemas. Además, se muestra que es válida cuando una relación muy natural se cumple entre la frecuencia de la onda electromagnética y la energía de la partícula Dirac, si el campo eléctrico es paralelo al campo magnético . Este análisis se basa en la forma cuaterniónica de la ecuación de Dirac y la forma cuaterniónica de las ecuaciones de Maxwell. En ambos casos las reformulaciones con cuaterniones son completamente equivalentes a la forma tradicional de los sistemas de Dirac y Maxwell. Esta teoría es una nueva interpretación de la mecánica cuántica. Este trabajo prueba que la mecánica cuántica representa la electrodinámica de ondas quirales curvilíneas cerradas. Esto está enteramente de acuerdo con la moderna interpretación y resultados de la teoría cu
Boundary value problem for one-dimensional fractional differential advection-dispersion equation
Khasambiev Mokhammad Vakhaevich
2014-07-01
Full Text Available An equation commonly used to describe solute transport in aquifers has attracted more attention in recent years. After a formal study of some aspects of the advection-diffusion equation, basically from the mathematical point of view with the solution of a differential equation with fractional derivative, the main interest to this problem shifted onto physical aspects of the dynamical system, such as the total energy and the dynamical response. In this regard it should be pointed out that the interaction with environment is expressed in terms of stochastic arrow of time. This allows one also to reach a progress in one more issue. Formerly the equation of advection-diffusion was not obtained from any physical principles. However, mainly the success concerns linear fractional systems. In fact, there are many cases in which linear treatments are not sufficient. The more general systems described by nonlinear fractional differential equations have not been studied enough. The ordinary calculus brings out clearly that essentially new phenomena occur in nonlinear systems, which generally cannot occur in linear systems. Due to vast range of application of the fractional advection-dispersion equation, a lot of work has been done to find numerical solution and fundamental solution of this equation. The research on the analytical solution of initial-boundary problem for space-fractional advection-dispersion equation is relatively new and is still at an early stage of development. In this paper, we will take use of the method of variable separation to solve space-fractional advection-dispersion equation with initial boundary data.
Wigner function for the Dirac oscillator in spinor space
MA Kai; WANG Jian-Hua; YUAN Yi
2011-01-01
The Wigner function for the Dirac oscillator in spinor space is studied in this paper.Firstly,since the Dirac equation is described as a matrix equation in phase space,it is necessary to define the Wigner function as a matrix function in spinor space.Secondly,the matrix form of the Wigner function is proven to support the Dirac equation.Thirdly,by solving the Dirac equation,energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.
The Dirac operator and gamma matrices for quantum Minkowski spaces
1997-01-01
Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.
Dispersion Interactions between Rare Gas Atoms: Testing the London Equation Using ab Initio Methods
Halpern, Arthur M.
2011-01-01
A computational chemistry experiment is described in which students can use advanced ab initio quantum mechanical methods to test the ability of the London equation to account quantitatively for the attractive (dispersion) interactions between rare gas atoms. Using readily available electronic structure applications, students can calculate the…
New Optical Solitons in High-Order Dispersive Cubic-Quintic Nonlinear Schrodinger Equation
LI Hua-Mei; XU You-Shen; LIN Ji
2004-01-01
By using the generalized tanh-function method, we find bright and dark solitary wave solutions to an extended nonlinear Schrodinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. At the same time, we also obtained other types of exact solutions.
Shen, S.; Liu, F.; Anh, V.; Turner, I.
2008-12-01
In this paper, we consider a Riesz fractional advection-dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order{alpha} [isin] (0, 1) and {beta} [isin] (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection-dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection-dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.
New explicit exact solutions to a nonlinear dispersive-dissipative equation
Naranmandula; Wang Ke-Xie
2004-01-01
Using the first-integral method, we obtain a series of new explicit exact solutions such as exponential function solutions, triangular function solutions, singular solitary wave solution and kink solitary wave solution of a nonlinear dispersive-dissipative equation, which describes weak nonlinear ion-acoustic waves in plasma consisting of cold ions and warm electrons.
Compact and noncompact structures of the nonlinearly dispersive GNLS(m,n,k,l equation
Bülent Kılıç
2015-02-01
Full Text Available In this paper, we establish exact-special solutions of the generalized nonlinear dispersion GNLS(m,n,k,l equation. We use the ansatz method for acquiring the compactons, solitary patterns, solitons and other types of solutions.
Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation
M. B. A. Mansour
2009-01-01
In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations (ODEs). Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.
Lim, C. W.; Wu, B. S.; He, L. H.
2001-12-01
A novel approach is presented for obtaining approximate analytical expressions for the dispersion relation of periodic wavetrains in the nonlinear Klein-Gordon equation with even potential function. By coupling linearization of the governing equation with the method of harmonic balance, we establish two general analytical approximate formulas for the dispersion relation, which depends on the amplitude of the periodic wavetrain. These formulas are valid for small as well as large amplitude of the wavetrain. They are also applicable to the large amplitude regime, which the conventional perturbation method fails to provide any solution, of the nonlinear system under study. Three examples are demonstrated to illustrate the excellent approximate solutions of the proposed formulas with respect to the exact solutions of the dispersion relation. (c) 2001 American Institute of Physics.
On the Schrodinger equations with isotropic and anisotropic fourth-order dispersion
Elder J. Villamizar-Roa
2016-01-01
Full Text Available This article concerns the Cauchy problem associated with the nonlinear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $$ i\\partial_tu+\\epsilon \\Delta u+\\delta A u+\\lambda|u|^\\alpha u=0,\\quad x\\in\\mathbb{R}^{n},\\; t\\in \\mathbb{R}, $$ where A is either the operator $\\Delta^2$ (isotropic dispersion or $\\sum_{i=1}^d\\partial_{x_ix_ix_ix_i}$, $1\\leq d
Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation
Abdon Atangana
2013-01-01
Full Text Available Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in porous medium. A space-time fractional advection-dispersion equation (FADE is a generalization of the classical ADE in which the first-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order , and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order . We derive the solution of the new equation in terms of Mittag-Leffler functions using Laplace transfrom. Some examples are given. The results from comparison let no doubt that the FADE is better in prediction than ADE.
Hyperbolic Mild Slope Equations with Inclusion of Amplitude Dispersion Effect: Random Waves
无
2008-01-01
New hyperbolic mild slope equations for random waves are developed with the inclusion of amplitude dispersion. The frequency perturbation around the peak frequency of random waves is adopted to extend the equations for regular waves to random waves. The nonlinear effect of amplitude dispersion is incorporated approximately into the model by only considering the nonlinear effect on the carrier waves of random waves, which is done by introducing a representative wave amplitude for the carrier waves. The computation time is greatly saved by the introduction of the representative wave amplitude. The extension of the present model to breaking waves is also considered in order to apply the new equations to surf zone. The model is validated for random waves propagate over a shoal and in surf zone against measurements.
On the Klein-Gordon equation using the dispersion relation of Doubly Special Relativity
Felipe, Yese J.
2017-01-01
The theory of Doubly Special Relativity or Deformed Special Relativity (DSR), proposes that there is a maximum energy scale and a minimum length scale that is invariant for all observers. These maximum energy and minimum length correspond to the Planck energy and the Planck length, respectively. As a consequence, the dispersion relation is modified to be E2 =p2c2 +m2c4 + λE3 + ... Previous work has been done to express Quantum Mechanics using the dispersion relation of DSR. Solutions of the free particle, the harmonic oscillator, and the Hydrogen atom have been obtained from the DSR Schrodinger equation. We explore how the DSR Klein-Gordon equation can be consistently approximated in the non-relativistic limit in order to derive the DSR Schrodinger equation.
Kambe, Takahide; Saito, Koichi
2016-01-01
As the interior density of a neutron star can become very high, it has been expected and discussed that quark matter may exist inside it. To describe the transition from hadron to quark phases (and vice versa), there are mainly two methods; one is the first-order phase transition, and the other is the crossover phenomenon. In the present study, using the flavor-SU (3) NJL model with the vector coupling interaction, we have calculated the equation of state for the quark phase at high density. Furthermore, for the hadron phase at low density, we have used two kinds of the equations of state; one is a relatively soft one by the QHD model, and the other is a stiff one calculated with relativistic Brueckner-Hartree-Fock approximation. Using those equations of state for the two phases, we have investigated the influence of various choices of parameters concerning the crossover region on the mass and radius of a neutron star.
Gómez, F; Afanasev, L; Benayoun, M; Brekhovskikh, V; Caragheorgheopol, G; Cechák, T; Chiba, M; Constantinescu, S; Doudarev, A; Dreossi, D; Drijard, Daniel; Ferro-Luzzi, M; Gallas, M V; Gerndt, J; Giacomich, R; Gianotti, P; Goldin, D; Gorin, A; Gortchakov, O; Guaraldo, C; Hansroul, M; Hosek, R; Iliescu, M; Jabitski, M; Kalinina, N; Karpoukhine, V; Kluson, J; Kobayshi, M; Kokkas, P; Komarov, V; Koulikov, A; Kouptsov, A; Krouglov, V; Krouglova, L; Kuroda, K I; Lanaro, A; Lapshine, B; Lednicky, R; Leruste, P; Levisandri, P; López-Aguera, A; Lucherini, V; Mäki, T; Manuilov, I; Montanet, L; Narjoux, J L; Nemenov, L; Nikitin, M; Nunez Pardo, T; Okada, K; Olchevskii, V; Pazos, A; Pentia, M; Penzo, Aldo L; Perreau, J M; Petrascu, C; Pló, M; Ponta, T; Pop, D; Riazantsev, A; Rodríguez, J M; Rodriguez Fernandez, A; Rykaline, V; Santamarina, C; Saborido, J; Schacher, J; Sidorov, A; Smolik, J; Takeutchi, F; Tarasov, A; Tauscher, L; Tobar, M J; Trusov, S; Vasquez, P; Vlachos, S; Yazkov, V; Yoshimura, Y; Zrelov, P
2001-01-01
The main objective of DIRAC experiment is the measurement of the lifetime tau of the exotic hadronic atom consisting of pi /sup +/ and pi /sup -/ mesons. The lifetime of this atom is determined by the decay mode pi /sup +/ pi /sup -/ to pi /sup 0/ pi /sup 0/ due to the strong interaction. Through the precise relationship between the lifetime and the S-wave pion-pion scattering length difference a/sub 0/-a/sub 2/ for isospin 0 and 2 (respectively), a measurement of tau with an accuracy of 10% will allow a determination of a/sub 0/-a/sub 2/at a 5% precision level. Pion-pion scattering lengths have been calculated in the framework of chiral perturbation theory with an accuracy below 5%. In this way DIRAC experiment will provide a crucial test of the chiral symmetry breaking scheme in QCD effective theories at low energies. (19 refs).
A novel method for analytically solving a radial advection-dispersion equation
Lai, Keng-Hsin; Liu, Chen-Wuing; Liang, Ching-Ping; Chen, Jui-Sheng; Sie, Bing-Ruei
2016-11-01
An analytical solution for solute transport in a radial flow field has a variety of practical applications in the study of the transport in push-pull/divergent/convergent flow tracer tests, aquifer remediation by pumping and aquifer storage and recovery. However, an analytical solution for radial advective-dispersive transport has been proven very difficult to develop and relatively few in subsurface hydrology have made efforts to do so, because variable coefficients in the governing partial differential equations. Most of the solutions for radial advective-dispersive transport presented in the literature have generally been solved semi-analytically with the final concentration values being obtained with the help of a numerical Laplace inversion. This study presents a novel solution strategy for analytically solving the radial advective-dispersive transport problem. A Laplace transform with respect to the time variable and a generalized integral transform technique with respect to the spatial variable are first performed to convert the transient governing partial differential equations into an algebraic equation. Subsequently, the algebraic equation is solved using simple algebraic manipulations, easily yielding the solution in the transformed domain. The solution in the original domain is ultimately obtained by successive applications of the Laplace and corresponding generalized integral transform inversions. A convergent flow tracer test is used to demonstrate the robustness of the proposed method for deriving an exact analytical solution to the radial advective-dispersive transport problem. The developed analytical solution is verified against a semi-analytical solution taken from the literature. The results show perfect agreement between our exact analytical solution and the semi-analytical solution. The solution method presented in this study can be applied to create more comprehensive analytical models for a great variety of radial advective-dispersive
Pérez Guerrero, J. S.; Skaggs, T. H.
2010-08-01
SummaryMathematical models describing contaminant transport in heterogeneous porous media are often formulated as an advection-dispersion transport equation with distance-dependent transport coefficients. In this work, a general analytical solution is presented for the linear, one-dimensional advection-dispersion equation with distance-dependent coefficients. An integrating factor is employed to obtain a transport equation that has a self-adjoint differential operator, and a solution is found using the generalized integral transform technique (GITT). It is demonstrated that an analytical expression for the integrating factor exists for several transport equation formulations of practical importance in groundwater transport modeling. Unlike nearly all solutions available in the literature, the current solution is developed for a finite spatial domain. As an illustration, solutions for the particular case of a linearly increasing dispersivity are developed in detail and results are compared with solutions from the literature. Among other applications, the current analytical solution will be particularly useful for testing or benchmarking numerical transport codes because of the incorporation of a finite spatial domain.
A KdV-like advection-dispersion equation with some remarkable properties
Sen, Abhijit; Thyagaraja, Anantanarayanan; Krishnaswami, Govind S
2011-01-01
We discuss a new non-linear advection-dispersion equation u_t + (2 u_{xx}/u) u_x = epsilon u_{xxx}, invariant under scaling of dependent variable and referred to here as SIdV. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the Korteweg-de Vries (KdV) solitary wave solution. Indeed, there is a one-parameter family of first order advection equations with cubic dispersion sharing the KdV solitary wave, that interpolate between SIdV and KdV. SIdV is one of the two simplest such translation and space-time reflection-symmetric equations invariant under rescaling of wave amplitude u. The scale-invariant advection in SIdV is reminiscent of the (E x B)/B^2 velocity of plasma physics. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, SIdV possesses solitary and periodic travelling waves and recurrence properties usually associated with integrabl...
Lijun Song; Lu Li; Guosheng Zhou
2005-01-01
The effect of third-order dispersion on breathing localized solutions in the quintic complex GinzburgLandau (CGL) equation is investigated. It is found that even small third-order dispersion can cause dramatic changes in the behavior of the solutions, such as breathing solution asymmetrically and travelling slowly towards the right for the positive third-order dispersion.
Ali S Wadi; Mourad F Dimian; Fayez N Ibrahim
2014-08-01
We present simple analytical solutions for the unsteady advection–dispersion equations describing the pollutant concentration (, ) in one dimension. The solutions are obtained by using Laplace transformation technique. In this study we divided the river into two regions ≤ 0 and ≥0 and the origin at = 0. The variation of (, ) with the time from = 0 up to → ∞ (the steady state case) is taken into account in our study. The special case for which the dispersion coefficient = 0 is studied in detail. The parameters controlling the pollutant concentration along the river are determined.
Quiney, HM; Glushkov, VN; Wilson, S
2002-01-01
Using basis sets of distributed s-type Gaussian functions with positions and exponents optimized so as to support Hartree-Fock total energies with an accuracy approaching the sub-muHartree level, Dirac-Hartree-Fock-Coulomb calculations are reported for the ground states of the H-2, LiH, and BH molec
无
2005-01-01
The dispersion of a solid particle in a liquid may lead to the formation of solvation film onthe particle surface, which can strongly increase the repulsive force between particles and thus strongly affect the stability of dispersions. The solvation film thickness, which varies with the variation of the property of suspension particles and solutions, is one of the most important parameters of the solvation film, and is also one of the most difficult parameters that can be measured accurately. In this paper, a method, based on the Einstein viscosity equation of dispersions, for determining the solvation film thickness of particles is developed. This method was tested on two kinds of silica spherical powders (namely M1 and M2) dispersed in ethyl alcohol, in water, and in a water-ethyl alcohol mixture (1:1 by volume) through measuring the relative viscosity of dispersions of the particles as a function of the volume fraction of the dry particles in the dispersion, and of the specific surface area and the density of the particles. The calculated solvation film thicknesses on M1 are 7.48, 18.65 and 23.74 nm in alcohol, water and the water-ethyl alcohol mixture, 12.41, 12.71 and 13.13 nm on M2 in alcohol, water and the water-ethyl alcohol mixture, respectively.
The determination of an unknown source for a space fractional advection dispersion equation
Aldoghaither, Abeer
2014-09-01
In this paper, we are interested in the estimation of the source term for a space fractional advection dispersion equation using concentration and flux measurements at final time. An example of application is the identification of contamination source in groundwater transport. We propose to use the socalled modulating functions method which has been introduced for parameters estimation. This method allows to transfer the estimation problem into solving a system of algebraic equations. Numerical examples are given to illustrate the effectiveness and the robustness of the proposed method. Finally, a comparison between a Tikhonov-based optimization method and the modulating functions approach is presented.
A numerical scheme for space-time fractional advection-dispersion equation
Javadi, S; Jani, M
2015-01-01
In this paper, we develop a numerical resolution of the space-time fractional advection-dispersion equation. After time discretization, we utilize collocation technique and implement a product integration method in order to simplify the evaluation of the terms involving spatial fractional order derivatives. Then utilizing Bernstein polynomials as basis, the problem is transformed into a linear system of algebraic equations. Error analysis and order of convergence for the proposed method are also discussed. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method and to confirm the analytic results.
Arqub, Omar Abu; El-Ajou, Ahmad; Momani, Shaher
2015-07-01
Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.
Magnetotransport properties near the Dirac point of Dirac semimetal Cd3As2 nanowires
Wang, Li-Xian; Wang, Shuo; Li, Jin-Guang; Li, Cai-Zhen; Xu, Jun; Yu, Dapeng; Liao, Zhi-Min
2017-02-01
Three-dimensional (3D) Dirac semimetals are featured by 3D linear energy-momentum dispersion relation, which have been proposed to be a desirable system to study Dirac fermions in 3D space and Weyl fermions in solid-state materials. Significantly, to reveal exotic transport properties of Dirac semimetals, the Fermi level should be close to the Dirac point, around which the linear dispersion is retained. Here we report the magnetotransport properties near the Dirac point in Cd3As2 nanowires, manifesting the evolution of band structure under magnetic field. Ambipolar field effect is observed with the Dirac point at V g = 3.9 V. Under high magnetic field, there is a resistivity dip in transfer curve at the Dirac point, which is caused by the Zeeman splitting enhanced density of state around the Dirac point. Furthermore, the low carrier density in the nanowires makes it feasible to enter into the quantum limit regime, resulting in the quantum linear magnetoresistance being observed even at room temperature. Besides, the dramatic reduction of bulk conductivity due to the low carrier density, together with a large surface to volume ratio of the nanowire, collectively help to reveal the Shubnikov-de Haas oscillations from the surface states. Our studies on transport properties around the Dirac point therefore provide deep insights into the emerging exotic physics of Dirac and Weyl fermions.
Dirac-Kahler Theory and Massless Fields
Pletyukhov, V A
2010-01-01
Three massless limits of the Dirac-Kahler theory are considered. It is shown that the Dirac-Kahler equation for massive particles can be represented as a result of the gauge-invariant mixture (topological interaction) of the above massless fields.
On localization of Dirac fermions by disorder
Medvedyeva, Mariya Vyacheslavivna
2011-01-01
This thesis is devoted to the effects of disorder on two-dimensional systems of Dirac fermions. Disorder localizes the usual electron system governed by the Schroedinger equation. The influence of disorder on Dirac fermions is qualitevely different. We concentrate on a random mass term in the Dira
Dirac and Weyl semimetals with holographic interactions
Jacobs, V.P.J.
2015-01-01
Dirac and Weyl semimetals are states of matter exhibiting the relativistic physics of, respectively, the Dirac and Weyl equation in a three-dimensional bulk material. These three-dimensional semimetals have recently been realized experimentally in various crystals. Theoretically, especially the noni
Birol İbiş
2014-01-01
Full Text Available This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE involving Jumarie’s modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM. FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs.
Direct and inverse source problems for a space fractional advection dispersion equation
Aldoghaither, Abeer
2016-05-15
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.
Non-completely elastic interactions in a (2+1)-dimensional dispersive long wave equation
Chen Wei-Lu; Zhang Wen-Ting; Zhang Li-Pu; Dai Chao-Qing
2012-01-01
With the help of a modified mapping method,we obtain two kinds of variable separation solutions with two arbitrary functions for the (2+1)-dimensional dispersive long wave equation.When selecting appropriate multi-valued functions in the variable separation solution,we investigate the interactions among special multi-dromions,dromion-like multi-peakons,and dromion-like multi-semifoldons,which all demonstrate non-completely elastic properties.
Chen Guowang; Xue Hongxia
2008-01-01
In this article, the existence, uniqueness and regularities of the global gener-alized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation utt -uxx-auxxtt+bux4 - duxxt= f(u)xx are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
HUANG; Guanhua; HUANG; Quanzhong; ZHAN; Hongbin
2005-01-01
The newly developed Fractional Advection-Dispersion Equation (FADE), which is FADE was extended and used in this paper for modelling adsorbing contaminant transport by adding an adsorbing term. A parameter estimation method and its corresponding FORTRAN based program named FADEMain were developed on the basis of Nonlinear Least Square Algorithm and the analytical solution for one-dimensional FADE under the conditions of step input and steady state flow. Data sets of adsorbing contaminants Cd and NH4+-N transport in short homogeneous soil columns and conservative solute NaCI transport in a long homogeneous soil column, respectively were used to estimate the transport parameters both by FADEMain and the advection-dispersion equation (ADE) based program CXTFIT2.1. Results indicated that the concentration simulated by FADE agreed well with the measured data. Compared to the ADE model, FADE can provide better simulation for the concentration in the initial lower concentration part and the late higher concentration part of the breakthrough curves for both adsorbing contaminants. The dispersion coefficients for ADE were from 0.13 to 7.06 cm2/min, while the dispersion coefficients for FADE ranged from 0.119 to 3.05 cm1.856/min for NaCI transport in the long homogeneous soil column. We found that the dispersion coefficient of FADE increased with the transport distance, and the relationship between them can be quantified with an exponential function. Less scale-dependent was also found for the dispersion coefficient of FADE with respect to ADE.
Zhang, Da; Zhang, Zhaoyang; Ahmed, Noor; Zhang, Yanpeng; Li, Fuli; Belić, Milivoj R; Xiao, Min
2016-01-01
We establish a link between the fractional Schr\\"odinger equation (FSE) and light propagation in the honeycomb lattice (HCL) - the Dirac-Weyl equation (DWE). The fractional Laplacian in FSE causes a modulation of the dispersion relation of the system, which in the limiting case becomes linear. In the HCL, the dispersion relation is already linear around the Dirac point, suggesting a possible connection with the FSE. Here, we demonstrate this connection by describing light propagation in both FSE and HCL, using DWE. Thus, we propagate Gaussian beams according to FSE, HCL around the Dirac point, and DWE, to discover very similar behavior - the conical diffraction. However, if an additional potential is brought into the system, the link between FSE and HCL is broken, because the added potential serves as a perturbation, which breaks the translational periodicity of HCL and destroys Dirac cones in the dispersion relation.
First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals
Mei, Jun
2012-07-24
By using the k•p method, we propose a first-principles theory to study the linear dispersions in phononic and photonic crystals. The theory reveals that only those linear dispersions created by doubly degenerate states can be described by a reduced Hamiltonian that can be mapped into the Dirac Hamiltonian and possess a Berry phase of -π. Linear dispersions created by triply degenerate states cannot be mapped into the Dirac Hamiltonian and carry no Berry phase, and, therefore should be called Dirac-like cones. Our theory is capable of predicting accurately the linear slopes of Dirac and Dirac-like cones at various symmetry points in a Brillouin zone, independent of frequency and lattice structure. © 2012 American Physical Society.
Van der Waals equation of state revisited: importance of the dispersion correction.
de Visser, Sam P
2011-04-28
One of the most basic equations of state describing nonideal gases and liquids is the van der Waals equation of state, and as a consequence, it is generally taught in most first year undergraduate chemistry courses. In this work, we show that the constants a and b in the van der Waals equation of state are linearly proportional to the polarizability volume of the molecules in a gas or liquid. Using this information, a new thermodynamic one-parameter equation of state is derived that contains experimentally measurable variables and physics constants only. This is the first equation of state apart from the Ideal Gas Law that contains experimentally measurable variables and physics constants only, and as such, it may be a very useful and practical equation for the description of dilute gases and liquids. The modified van der Waals equation of state describes pV as the sum of repulsive and attractive intermolecular interaction energies that are represented by an exponential repulsion function between the electron clouds of the molecules and a London dispersion component, respectively. The newly derived equation of state is tested against experimental data for several gas and liquid examples, and the agreement is satisfactory. The description of the equation of state as a one-parameter function also has implications on other thermodynamic functions, such as critical parameters, virial coefficients, and isothermal compressibilities. Using our modified van der Waals equation of state, we show that all of these properties are a function of the molecular polarizability volume. Correlations of experimental data confirm the derived proportionalities.
New multisoliton solutions of the （2＋1） —dimensional dispersive long wave equations
JiefangZHANG; PingHAN
2001-01-01
Using the extension homogeneous balance method,we have obtained a nonlinear transformation to simplify the (2+1) dimensional dispersive long wave equations into a linear partial differential equation,and found some new special types of the multisoliton solutions and exact solutions of the linear partial differential equation.
Sadykov, N. R.
2011-03-01
It is suggested to extend the results obtained for Maxwell's equations in Majorana form (spin-1 particles) for spin particles with a half-integer spin and a nonzero mass. It is shown that in an unbounded "chiral medium" (twisted media) the degeneration existing between particles of different helicities is removed. For ultrarelativistic particles, an analog to the inverse optical Magnus effect follows where the effect is determined by the chirality of the medium. From the inverse scattering problem for the transforms under consideration it follows that the amplitude of the wave function of a particle in a chiral medium can vary with time according to a linear law (for example, the process of neutrino (antineutrino) production or annihilation), and the parameters of the medium satisfy the evolution equation.
Anderson's absolute objects and constant timelike vector hidden in Dirac matrices
Rylov, Yu A
2001-01-01
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute objects of the Dirac equation. There are two ways of the Dirac matrices transformation: (1) Dirac matrices form a 4-vector and wave function is a scalar, (2) Dirac matrices are scalars and the wave function is a spinor. In the first case the Dirac equation is nonrelativistic, in the second one it is relativistic. Transforming Dirac equation to another scalar-vector variables, one shows that the first way of transformation is valid, and the Dirac equation is not relativistic
Liu Cheng-Shi
2007-01-01
Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation (ODE), whose general solution can be obtained using the trick of one-parameter group.Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.
Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations
Brandolese, Lorenzo
2014-08-01
We unify a few of the best known results on wave breaking for the Camassa-Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that is strictly negative in at least one point . Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean's necessary and sufficient condition for blowup, our approach applies to other equations that are not integrable: we illustrate this fact by establishing new local-in-space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods.
Bound states in a model of interaction of Dirac field with material plane
Pismak Yu. M.
2016-01-01
Full Text Available In the framework of the Symanzik approach model of the interaction of the Dirac spinor field with the material plane in the 3 + 1-dimensional space is constructed. The model contains eight real parameters characterizing the properties of the material plane. The general solution of the Euler-Lagrange equations of the model and dispersion equations for bound states are analyzed. It is shown that there is a choice of parameters of the model in which the connected states are characterized by dispersion law of a mass-less particle moving along the material plane with the dimensionless Fermi velocity not exceeding one.
Ghoumaid, A.; Benamira, F.; Guechi, L. [Laboratoire de Physique Théorique, Département de Physique, Faculté des Sciences Exactes, Université des Frères Mentouri, Constantine, Route d’Ain El Bey, Constantine (Algeria)
2016-02-15
It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given is incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions {sub 2}F{sub 1}(a, b, c; z). The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function.
On Huygens' principle for Dirac operators associated to electromagnetic fields
CHALUB FABIO A.C.C.
2001-01-01
Full Text Available We study the behavior of massless Dirac particles, i.e., solutions of the Dirac equation with m = 0 in the presence of an electromagnetic field. Our main result (Theorem 1 is that for purely real or imaginary fields any Huygens type (in Hadamard's sense Dirac operators is equivalent to the free Dirac operator, equivalence given by changes of variables and multiplication (right and left by nonzero functions.
Dirac particle in gravitational quantum mechanics
Pedram, Pouria
2011-08-01
In this Letter, we consider the effects of the Generalized (Gravitational) Uncertainty Principle (GUP) on the eigenvalues and the eigenfunctions of the Dirac equation. This form of GUP is consistent with various candidates of quantum gravity such as string theory, loop quantum gravity, doubly special relativity and black hole physics and predicts both a minimum measurable length and a maximum measurable momentum. The modified Hamiltonian contains two additional terms proportional to a( and a( where αi are Dirac matrices and a∼1/MPlc is the GUP parameter. For the case of the Dirac free particle and the Dirac particle in a box, we solve the generalized Dirac equation and find the modified energy eigenvalues and eigenfunctions.
Torsion Gravity for Dirac Fields
Fabbri, Luca
2016-01-01
In this article we will take into account the most complete back-ground with torsion and curvature, providing the most exhaustive coupling for the Dirac field: we will discuss the integrability of the interaction of the matter field and the reduction of the matter field equations.
Dispersive Blow-Up Ⅱ.Schr(o)dinger-Type Equations,Optical and Oceanic Rogue Waves
Jerry L.BONA; Jean-Claude SAUT
2010-01-01
Addressed here is the occurrence of point singularities which owe to the fo-cusing of short or long waves,a phenomenon labeled dispersive blow-up.The context of this investigation is linear and nonlinear,strongly dispersive equations or systems of equa-tions.The present essay deals with linear and nonlinear Schr(o)dinger equations,a class of fractional order SchrSdinger equations and the linearized water wave equations,with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.
Nakatsuji, Hiroshi
2012-09-18
Just as Newtonian law governs classical physics, the Schrödinger equation (SE) and the relativistic Dirac equation (DE) rule the world of chemistry. So, if we can solve these equations accurately, we can use computation to predict chemistry precisely. However, for approximately 80 years after the discovery of these equations, chemists believed that they could not solve SE and DE for atoms and molecules that included many electrons. This Account reviews ideas developed over the past decade to further the goal of predictive quantum chemistry. Between 2000 and 2005, I discovered a general method of solving the SE and DE accurately. As a first inspiration, I formulated the structure of the exact wave function of the SE in a compact mathematical form. The explicit inclusion of the exact wave function's structure within the variational space allows for the calculation of the exact wave function as a solution of the variational method. Although this process sounds almost impossible, it is indeed possible, and I have published several formulations and applied them to solve the full configuration interaction (CI) with a very small number of variables. However, when I examined analytical solutions for atoms and molecules, the Hamiltonian integrals in their secular equations diverged. This singularity problem occurred in all atoms and molecules because it originates from the singularity of the Coulomb potential in their Hamiltonians. To overcome this problem, I first introduced the inverse SE and then the scaled SE. The latter simpler idea led to immediate and surprisingly accurate solution for the SEs of the hydrogen atom, helium atom, and hydrogen molecule. The free complement (FC) method, also called the free iterative CI (free ICI) method, was efficient for solving the SEs. In the FC method, the basis functions that span the exact wave function are produced by the Hamiltonian of the system and the zeroth-order wave function. These basis functions are called complement
Some New Exact Solutions to the Dispersive Long-Wave Equation in(2+1)-Dimensional Spaces
LI De-Sheng; ZHANG Hong-Qing
2003-01-01
In this paper, by using a further extended tanh method and symbolic computation system, some newsoliton-like and period form solutions of the dispersive long-wave equation in (2+1)-dimensional spaces are obtained.
无
2007-01-01
In this paper, we consider a Riesz space-fractional reaction-dispersion equation (RSFRDE). The RSFRDE is obtained from the classical reaction-dispersion equation by replacing the second-order space derivative with a Riesz derivative of order β∈(1, 2].We propose an implicit finite difference approximation for RSFRDE. The stability and convergence of the finite difference approximations are analyzed. Numerical results are found in good agreement with the theoretical analysis.
Effects of Higher Order Dispersion Terms in the Nonlinear Schrodinger Equation
Robert Beech
2005-01-01
Full Text Available This study presents a concise graphical analysis of solitonic solutions to a nonlinear Schrodinger equation (NLSE. A sequence of code using the standard NDSolve function has been developed in Mathematica to investigate the acceptable accuracy of the NLSE in relatively small ranges of the dispersive parameter space. An operator splitting approach was used in the numerical solutions to expand the boundaries and reduce the artifacts for a reliable solution. These numerical routines were implemented through the use with Mathematica and the results give a very clear view of this interesting and important practical phenomenon.
Exotic Localized Coherent Structures of the (2+1)-Dimensional Dispersive Long-Wave Equation
ZHANG JieFang
2002-01-01
This article is concerned with the extended homogeneous balance method for studying thc abundantlocalized solution structures in the (2-k1)-dimensional dispersive long-wave equations uty + xx + (u2)xy/2 = 0, ηt +(u + u + uxy)x = 0. Starting from the homogeneous balance method, we find that the richness of the localized coherentstructures of the model is caused by the entrance of two variable-separated arbitrary functions. For some special selectionsof the arbitrary functions, it is shown that the localized structures of the model may be dromions, lumps, breathers,instantons and ring solitons.
MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION
Li Yanling; Ma Yicheng
2005-01-01
The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region.Finally, a few examples of application are given.
Periodic folded waves for a (2+1)-dimensional modified dispersive water wave equation
Huang Wen-Hua
2009-01-01
A general solution,including three arbitrary functions,is obtained for a (2+1)-dimensional modified dispersive water-wave (MDWW) equation by means of the WTC truncation method.Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution,special types of periodic folded waves are derived.In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations.The interactions of the periodic folded waves and the degenerated single folded solitary waves axe investigated graphically and found to be completely elastic.
Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations revisited
Egorova, I. E.; Kopylova, E. A.; Marchenko, V. A.; Teschl, G.
2016-06-01
It is shown that for a one-dimensional Schrödinger operator with a potential whose first moment is integrable the elements of the scattering matrix are in the unital Wiener algebra of functions with integrable Fourier transforms. This is then used to derive dispersion estimates for solutions of the associated Schrödinger and Klein-Gordon equations. In particular, the additional decay conditions are removed in the case where a resonance is present at the edge of the continuous spectrum. Bibliography: 29 titles.
SPACE-TIME DISCONTINUOUS GALERKIN METHOD FOR MAXWELL EQUATIONS IN DISPERSIVE MEDIA
汪波; 谢资清; 张智民
2014-01-01
In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O?τr+1+hk+1/2? are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r+1 in temporal variable t.
The scattering transform for the Benjamin-Ono equation in the small-dispersion limit
Miller, Peter D.; Wetzel, Alfredo N.
2016-10-01
Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained in Miller and Wetzel [17], we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself. Our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering, and they also provide new details not previously reported in the literature.
Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow
Zhijian, Yang
2006-01-01
The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say [alpha], it proves that when [alpha]>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when [alpha][greater-or-equal, slanted]5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2
Dispersion equations for field-aligned cyclotron waves in axisymmetric magnetospheric plasmas
N. I. Grishanov
2006-03-01
Full Text Available In this paper, we derive the dispersion equations for field-aligned cyclotron waves in two-dimensional (2-D magnetospheric plasmas with anisotropic temperature. Two magnetic field configurations are considered with dipole and circular magnetic field lines. The main contribution of the trapped particles to the transverse dielectric permittivity is estimated by solving the linearized Vlasov equation for their perturbed distribution functions, accounting for the cyclotron and bounce resonances, neglecting the drift effects, and assuming the weak connection of the left-hand and right-hand polarized waves. Both the bi-Maxwellian and bi-Lorentzian distribution functions are considered to model the ring current ions and electrons in the dipole magnetosphere. A numerical code has been developed to analyze the dispersion characteristics of electromagnetic ion-cyclotron waves in an electron-proton magnetospheric plasma with circular magnetic field lines, assuming that the steady-state distribution function of the energetic protons is bi-Maxwellian. As in the uniform magnetic field case, the growth rate of the proton-cyclotron instability (PCI in the 2-D magnetospheric plasmas is defined by the contribution of the energetic ions/protons to the imaginary part of the transverse permittivity elements. We demonstrate that the PCI growth rate in the 2-D axisymmetric plasmasphere can be significantly smaller than that for the straight magnetic field case with the same macroscopic bulk parameters.
Modelling drug degradation in a spray dried polymer dispersion using a modified Arrhenius equation.
Patterson, Adele; Ferreira, Ana P; Banks, Elizabeth; Skeene, Kirsty; Clarke, Graham; Nicholson, Sarah; Rawlinson-Malone, Clare
2015-01-15
The Pharmaceutical industry is increasingly utilizing amorphous technologies to overcome solubility challenges. A common approach is the use of drug in polymer dispersions to prevent recrystallization of the amorphous drug. Understanding the factors affecting chemical and physical degradation of the drug within these complex systems, e.g., temperature and relative humidity, is an important step in the selection of a lead formulation, and development of appropriate packaging/storage control strategies. The Arrhenius equation has been used as the basis of a number of models to predict the chemical stability of formulated product. In this work, we investigate the increase in chemical degradation seen for one particular spray dried dispersion formulation using hydroxypropyl methylcellulose acetate succinate (HPMC-AS). Samples, prepared using polymers with different substitution levels, were placed on storage for 6 months under a range of different temperature and relative humidity conditions and the degradant level monitored using high-performance liquid chromatography (HPLC). While the data clearly illustrates the impact of temperature and relative humidity on the degradant levels detected, it also highlighted that these terms do not account for all the variability in the data. An extension of the Arrhenius equation to include a term for the polymer chemistry, specifically the degree of succinoyl substitution on the polymer backbone, was shown to improve the fit of the model to the data.
无
2009-01-01
Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system.A population balance equation(PBE),a non-linear hyperbolic equation of the number density function,is usually employed to describe the micro-behavior(aggregation,breakage,growth,etc.) of a disperse phase and its effect on particle size distribution.Numerical solution is the only choice in most cases.In this paper,three different numerical methods(direct discretization methods,Monte Carlo methods,and moment methods) for the solution of a PBE are evaluated with regard to their ease of implementation,computational load and numerical accuracy.Special attention is paid to the relatively new and superior moment methods including quadrature method of moments(QMOM),direct quadrature method of moments(DQMOM),modified quadrature method of moments(M-QMOM),adaptive direct quadrature method of moments(ADQMOM),fixed pivot quadrature method of moments(FPQMOM),moving particle ensemble method(MPEM) and local fixed pivot quadrature method of moments(LFPQMOM).The prospects of these methods are discussed in the final section,based on their individual merits and current state of development of the field.
SU JunWei; GU ZhaoLin; XU X.Yun
2009-01-01
Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system.A population balance equation(PBE),a non-linear hyperbolic equation of the number density function,is usually employed to describe the micro-behavior(aggregation,breakage,growth,etc.)of a disperse phase and its effect on particle size distribution.Numerical solution is the only choice in most cases.In this paper,three different numerical methods(direct discretization methods,Monte Carlo methods,and moment methods)for the solution of a PBE are evaluated with regard to their ease of implementation,computational load and numerical accuracy.Special attention is paid to the relatively new and superior moment methods including quadrature method of moments(QMOM),direct quadrature method of moments(DQMOM),modified quadrature method of moments(M-QMOM),adaptive direct quadrature method of moments(ADOMOM),fixed pivot quadrature method of moments(FPQMOM),moving particle ensemble method(MPEM)and local fixed pivot quadrature method of moments(LFPQMOM).The prospects of these methods ere discussed in the final section,based on their individual merits and current state of development of the field.
Dirac dynamical resonance states around Schwarzschild black holes
Zhou, Xiang-Nan; Yang, Ke; Liu, Yu-Xiao
2013-01-01
Recently, a novel kind of scalar wigs around Schwarzschild black holes---scalar dynamical resonance states were introduced in [Phys. Rev. D 84, 083008 (2011)] and [Phys. Rev. Lett. 109, 081102 (2012)]. In this paper, we investigate the existence and evolution of Dirac dynamical resonance states. First we look for stationary resonance states of a Dirac field around a Schwarzchild black hole by using the Schrodinger-like equations reduced from the Dirac equation in Schwarzschild spacetime. Then Dirac pseudo-stationary configurations are constructed from the stationary resonance states. We use these configurations as initial data and investigate their numerical evolutions and energy decay. These dynamical solutions are the so-called "Dirac dynamical resonance states". It is found that the energy of the Dirac dynamical resonance states shows an exponential decay. The decay rate of energy is affected by the resonant frequency, the mass of Dirac field, the total angular momentum, and the spin-orbit interaction. In ...
Nurlybek A. Ispulov
2017-01-01
Full Text Available The investigation of thermoelastic wave propagation in elastic media is bound to have much influence in the fields of material science, geophysics, seismology, and so on. The heat conduction equations and bound equations of motions differ by the difficulty level and presence of many physical and mechanical parameters in them. Therefore thermoelasticity is being extensively studied and developed. In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalent set of first-order differential equations. Moreover, for given set of equations, the structure of fundamental solutions for the general case has been derived and also dispersion relations are obtained.
无
2005-01-01
Using trial equation method, abundant exact envelope traveling wave solutions of high-order dispersive cubic-quintic nonlinear Schrodinger equation, which include envelope soliton solutions, triangular function envelope solutions, and Jacobian elliptic function envelope solutions, are obtained. To our knowledge, all of these results are new.In particular, our proposed method is very simple and can be applied to a lot of similar equations.
Optical analogue of relativistic Dirac solitons in binary waveguide arrays
Tran, Truong X., E-mail: truong.tran@mpl.mpg.de [Department of Physics, Le Quy Don University, 236 Hoang Quoc Viet str., 10000 Hanoi (Viet Nam); Max Planck Institute for the Science of Light, Günther-Scharowsky str. 1, 91058 Erlangen (Germany); Longhi, Stefano [Department of Physics, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano (Italy); Biancalana, Fabio [Max Planck Institute for the Science of Light, Günther-Scharowsky str. 1, 91058 Erlangen (Germany); School of Engineering and Physical Sciences, Heriot-Watt University, EH14 4AS Edinburgh (United Kingdom)
2014-01-15
We study analytically and numerically an optical analogue of Dirac solitons in binary waveguide arrays in the presence of Kerr nonlinearity. Pseudo-relativistic soliton solutions of the coupled-mode equations describing dynamics in the array are analytically derived. We demonstrate that with the found soliton solutions, the coupled mode equations can be converted into the nonlinear relativistic 1D Dirac equation. This paves the way for using binary waveguide arrays as a classical simulator of quantum nonlinear effects arising from the Dirac equation, something that is thought to be impossible to achieve in conventional (i.e. linear) quantum field theory. -- Highlights: •An optical analogue of Dirac solitons in nonlinear binary waveguide arrays is suggested. •Analytical solutions to pseudo-relativistic solitons are presented. •A correspondence of optical coupled-mode equations with the nonlinear relativistic Dirac equation is established.
Parabolic metamaterials and Dirac bridges
Colquitt, D. J.; Movchan, N. V.; Movchan, A. B.
2016-10-01
A new class of multi-scale structures, referred to as `parabolic metamaterials' is introduced and studied in this paper. For an elastic two-dimensional triangular lattice, we identify dynamic regimes, which corresponds to so-called `Dirac Bridges' on the dispersion surfaces. Such regimes lead to a highly localised and focussed unidirectional beam when the lattice is excited. We also show that the flexural rigidities of elastic ligaments are essential in establishing the `parabolic metamaterial' regimes.
Liu, Z. T.; Xing, X. Z.; Li, M. Y.; Zhou, W.; Sun, Y.; Fan, C. C.; Yang, H. F.; Liu, J. S.; Yao, Q.; Li, W.; Shi, Z. X.; Shen, D. W.; Wang, Z.
2016-07-01
CaFeAs2 is a parent compound of recently discovered 112-type iron-based superconductors. It is predicted to be a staggered intercalation compound that naturally integrates both quantum spin Hall insulating and superconducting layers and an ideal system for the realization of Majorana modes. We performed a systematical angle-resolved photoemission spectroscopy and first-principles calculation study of the slightly electron-doped CaFeAs2. We found that the zigzag As chain of 112-type iron-based superconductors play a considerable role in the low-energy electronic structure, resulting in the characteristic Dirac-cone like band dispersion as the prediction. Our experimental results further confirm that these Dirac cones only exist around the X but not Y points in the Brillouin zone, breaking the S4 symmetry at iron sites. Our findings present the compelling support to the theoretical prediction that the 112-type iron-based superconductors might host the topological nontrivial edge states. The slightly electron doped CaFeAs2 would provide us a unique opportunity to realize and explore Majorana fermion physics.
Wigner function for the Dirac oscillator in spinor space
马凯; 王剑华; 袁毅
2011-01-01
The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space.
The Dirac oscillator in a rotating frame of reference
Strange, P.; Ryder, L. H.
2016-10-01
The Dirac equation in a rotating frame of reference is derived from first principles within a linear approximation. This equation is employed to exhibit an equivalence between a particle in a Dirac oscillator potential and a free particle in a rotating frame of reference. A zero-point contribution to the energy of the particle, resulting from its spin, is also noted.
Dali Zhang
2012-01-01
Full Text Available This paper deals with an inverse problem for identifying multiparameters in 1D space fractional advection dispersion equation (FADE on a finite domain with final observations. The parameters to be identified are the fractional order, the diffusion coefficient, and the average velocity in the FADE. The forward problem is solved by a finite difference scheme, and then an optimal perturbation regularization algorithm is introduced to determine the three parameters simultaneously. Numerical inversions are performed both with the accurate data and noisy data, and several factors having influences on realization of the algorithm are discussed. The inversion solutions are in good approximations to the exact solutions demonstrating the efficiency of the proposed algorithm.
Forward dispersion relations and Roy equations in $\\pi\\pi$ scattering
Kaminski, R; Ynduráin, F J
2007-01-01
We review results of an analysis of pipi interactions in S, P and D waves for two-pion effective mass from threshold to about 1.4 GeV. In particular we show a recent improvement of this analysis above the K anti-K threshold using more data for phase shifts and including the S0 wave inelasticity from pipi -> K anti-K. In addition, we have improved the fit to the f2(1270) resonance and used a more flexible P wave parametrization above the K anti-K threshold and included an estimation of the D2 wave inelasticity. The better accuracy thus achieved also required a refinement of the Regge analysis above 1.42 GeV. We have checked that the pipi scattering amplitudes obtained in this approach satisfy remarkably well forward dispersion relations and Roy's equations.
Li, Wan-Tong; Wang, Jia-Bing; Zhang, Li
2016-08-01
This paper is concerned with the new types of entire solutions other than traveling wave solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats. We first establish the existence and properties of spatially periodic solutions connecting two steady states. Then new types of entire solutions are constructed by combining the rightward and leftward pulsating traveling fronts with different speeds and a spatially periodic solution. Finally, for a class of special heterogeneous reaction, we further establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters, such as wave speeds and the shifted variables. In other words, we build a five-dimensional manifold of solutions and the traveling wave solutions are on the boundary of the manifold.
Gravitationally induced inhibitions of dispersion according to the Schr\\"odinger-Newton Equation
Giulini, Domenico
2011-01-01
We re-consider the time dependent Schr\\"odinger-Newton equation as a model for the self-gravitational interaction of a quantum system. We numerically locate the onset of gravitationally induced inhibitions of dispersion of Gaussian wave packets and find them to occur at mass values more than 6 orders of magnitude higher than reported by Salzman and Carlip (2006, 2008), namely at about $10^{10}\\,\\mathrm{u}$. This fits much better to simple analytical estimates but unfortunately also questions the experimental realisability of the proposed laboratory test of quantum gravity in the foreseeable future, not just because of large masses, but also because of the need to provide sufficiently long coherence times.
Superconductivity in doped Dirac semimetals
Hashimoto, Tatsuki; Kobayashi, Shingo; Tanaka, Yukio; Sato, Masatoshi
2016-07-01
We theoretically study intrinsic superconductivity in doped Dirac semimetals. Dirac semimetals host bulk Dirac points, which are formed by doubly degenerate bands, so the Hamiltonian is described by a 4 ×4 matrix and six types of k -independent pair potentials are allowed by the Fermi-Dirac statistics. We show that the unique spin-orbit coupling leads to characteristic superconducting gap structures and d vectors on the Fermi surface and the electron-electron interaction between intra and interorbitals gives a novel phase diagram of superconductivity. It is found that when the interorbital attraction is dominant, an unconventional superconducting state with point nodes appears. To verify the experimental signature of possible superconducting states, we calculate the temperature dependence of bulk physical properties such as electronic specific heat and spin susceptibility and surface state. In the unconventional superconducting phase, either dispersive or flat Andreev bound states appear between point nodes, which leads to double peaks or a single peak in the surface density of states, respectively. As a result, possible superconducting states can be distinguished by combining bulk and surface measurements.
A general polynomial solution to convection–dispersion equation using boundary layer theory
Jiao Wang; Ming’an Shao; Laiming Huang; Xiaoxu Jia
2017-04-01
A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute transport in semi-infinite systems such as soil column. However, previous studies have only proposed low order polynomial solutions such as parabolic and cubic polynomials. This paper presents a general polynomial boundary layer solution to CDE. Comparison with exact solution suggests the prediction accuracy of the boundary layer solution varies with the order of polynomial expression and soil transport parameters. The results show that prediction accuracy increases with increasing order up to parabolic or cubic polynomial function and with no distinct relationship between accuracy and order for higher order polynomials ($n\\geqslant 3$). Comparison of two critical solute transport parameters (i.e., dispersion coefficient and retardation factor), estimated by the boundary layer solution and obtained by CXTFIT curve-fitting, shows a good agreement. The study shows that the general solution can determine the appropriate orders of polynomials for approximate CDE solutions that best describe solute concentration profiles and optimal solute transport parameters. Furthermore, the general polynomial solution to CDE provides a simple approach to solute transport problems, a criterion for choosing the right orders of polynomials for soils with different transport parameters. It is also a potential approach for estimating solute transport parameters of soils in the field.
Equations of state for crystalline zirconium iodide: The role of dispersion
Rossi, Matthew L., E-mail: mrossi@lanl.gov [Materials Science and Technology, MST-6, Los Alamos National Lab, Los Alamos, NM (United States); Taylor, Christopher D. [Materials Science and Technology, MST-6, Los Alamos National Lab, Los Alamos, NM (United States)
2013-02-15
We present the first-principle equations of state of several zirconium iodides, ZrI{sub 2}, ZrI{sub 3}, and ZrI{sub 4}, computed using density functional theory methods that apply various methods for introducing the dispersion correction. Iodides formed due to reaction of molecular or atomic iodine with zirconium and zircaloys are of particular interest due to their application to the cladding material used in the fabrication of nuclear fuel rods. Stress corrosion cracking (SCC), associated with fission product chemistry with the clad material, is a major concern in the life cycle of nuclear fuels, as many of the observed rod failures have occurred due to pellet–cladding chemical interactions (PCCI) [A. Atrens, G. Dannhäuser, G. Bäro, Stress-corrosion-cracking of zircaloy-4 cladding tubes, Journal of Nuclear Materials 126 (1984) 91–102; P. Rudling, R. Adamson, B. Cox, F. Garzarolli, A. Strasser, High burn-up fuel issues, Nuclear Engineering and Technology 40 (2008) 1–8]. A proper understanding of the physical properties of the corrosion products is, therefore, required for the development of a comprehensive SCC model. In this particular work, we emphasize that, while existing modeling techniques include methods to compute crystal structures and associated properties, it is important to capture intermolecular forces not traditionally included, such as van der Waals (dispersion) correction. Furthermore, crystal structures with stoichiometries favoring a high I:Zr ratio are found to be particularly sensitive, such that traditional density functional theory approaches that do not incorporate dispersion incorrectly predict significantly larger volumes of the lattice. This latter point is related to the diffuse nature of the iodide electron cloud.
Quantum game interpretation of Dirac spinor field
Zhi, Haizhao
2011-01-01
This paper introduced the classical prisoner dilemma with the character and structure of quantum prisoner dilemma's strategy space. Associate with the Dirac spinor field, apply the basic quantum game strategy to the translation of the dynamics of Dirac equation. Decompose the real space and time to lattice we found that the basic interaction of spinor could be translated into quantum game theory. At the same time, we gained the new dynamics of quantized spacial evolutionary game.
Klein-Gordon and Dirac gyroscopes
SadurnI, E [Instituto de Ciencias Fisicas, Universidad Nacional Autonoma de Mexico, Cuernavaca, Morelos (Mexico)], E-mail: sadurni@fis.unam.mx
2009-01-09
The formulation of a rigid body in relativistic quantum mechanics is studied. Departing from an alternate approach at the relativistic classical level, the corresponding Klein-Gordon and Dirac operators for the rigid body are obtained in covariant form. The resulting wave equations are shown to be consistent, by construction, with earlier definitions of a relativistic rigid body by Aldinger et al (1983 Phys. Rev. D 28 3020). Wavefunctions and spectra for both cases are obtained explicitly, including the Dirac gyroscope with asymmetries.
Taohua Liu
2017-01-01
Full Text Available Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K and computational cost of O(KlogK. Traditionally, the Gaussian elimination method requires storage of O(K2 and computational cost of O(K3. Finally, the accuracy and efficiency of the method are checked with a numerical example.
X. Wang
2015-01-01
Full Text Available We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE in one-dimensional (1D and two-dimensional (2D cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.
Gravitational Repulsion and Dirac Antimatter
Kowitt, Mark E.
1996-03-01
Based on an analogy with electron and hole dynamics in semiconductors, Dirac's relativistic electron equation is generalized to include a gravitational interaction using an electromagnetic-type approximation of the gravitational potential. With gravitational and inertial masses decoupled, the equation serves to extend Dirac's deduction of antimatter parameters to include the possibility of gravitational repulsion between matter and antimatter. Consequences for general relativity and related “antigravity” issues are considered, including the nature and gravitational behavior of virtual photons, virtual pairs, and negative-energy particles. Basic cosmological implications of antigravity are explored—in particular, potential contributions to inflation, expansion, and the general absence of detectable antimatter. Experimental and observational tests are noted, and new ones suggested.
Lingju Kong
2013-04-01
Full Text Available We study the existence of multiple solutions to the boundary value problem $$displaylines{ frac{d}{dt}Big(frac12{}_0D_t^{-eta}(u'(t+frac12{}_tD_T^{-eta}(u'(t Big+lambda abla F(t,u(t=0,quad tin [0,T],cr u(0=u(T=0, }$$ where $T>0$, $lambda>0$ is a parameter, $0leqeta<1$, ${}_0D_t^{-eta}$ and ${}_tD_T^{-eta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $eta$, $F: [0,T]imesmathbb{R}^Nomathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $eta=0$. Examples are provided to illustrate the applicability of our results.
Hydrodynamics of the Dirac spectrum
Liu, Yizhuang, E-mail: yizhuang.liu@stonybrook.edu [Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800 (United States); Warchoł, Piotr, E-mail: piotr.warchol@uj.edu.pl [M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krakow (Poland); Zahed, Ismail, E-mail: ismail.zahed@stonybrook.edu [Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800 (United States)
2016-02-10
We discuss a hydrodynamical description of the eigenvalues of the Dirac spectrum in even dimensions in the vacuum and in the large N (volume) limit. The linearized hydrodynamics supports sound waves. The hydrodynamical relaxation of the eigenvalues is captured by a hydrodynamical (tunneling) minimum configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of unbroken chiral symmetry to a phase of broken chiral symmetry occurs over a time set by the speed of sound.
Plasmon modes of a massive Dirac plasma, and their superlattices
Sachdeva, Rashi; Thakur, Anmol; Vignale, Giovanni; Agarwal, Amit
2015-05-01
We explore the collective density oscillations of a collection of charged massive Dirac particles, in one, two, and three dimensions, and their one-dimensional (1D) superlattice. We calculate the long-wavelength limit of the dynamical polarization function analytically, and use the random phase approximation to obtain the plasmon dispersion. The density dependence of the long-wavelength plasmon frequency in massive Dirac systems is found to be different compared to systems with parabolic and gapless Dirac dispersion. We also calculate the long-wavelength plasmon dispersion of a 1D metamaterial made from 1D and 2D massive Dirac plasma. Our analytical results will be useful for exploring the use of massive Dirac materials as electrostatically tunable plasmonic metamaterials and can be experimentally verified by infrared spectroscopy, as in the case of graphene [L. Ju et al., Nat. Nanotechnol. 6, 630 (2011), 10.1038/nnano.2011.146].
Wu, Junru; Layman, Christopher; Liu, Jun
2004-02-01
A fundamental mathematical framework for applications of Doublet Mechanics to ultrasound propagation in a discrete material is introduced. A multiscale wave equation, dispersion relation for longitudinal waves, and shear waves are derived. The van Hove singularities and corresponding highest frequency limits for the Mth-order wave equations of longitudinal and shear waves are determined for a widely used microbundle structure. Doublet Mechanics is applied to soft tissue and low-density polyethylene. The experimental dispersion data for soft tissue and low-density polyethylene are compared with results predicted by Doublet Mechanics and an attenuation model based on a Kramers-Kronig relation in classical continuum mechanics.
On the method of strained parameters for a KdV type of equation with exact dispersion property
Karjanto, N
2016-01-01
This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact dispersion property is adopted as the governing equation for unidirectional wave packet evolution. Following the idea from Zakharov's seminal paper (Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. \\textit{Journal of Applied Mechanics and Technical Physics}, {\\bf 9}, 190--194), the equation is transformed from the spatial--temporal domain to the wavenumber--temporal domain. The solution of the transformed equation is sought using the perturbation theory, for which the ansatz is expressed in the form of a regular expansion in the increasing order of a small parameter. After implementing the na\\"{i}ve perturbation method, due to nonlinear mode generation and particular combinations of wavenumbers, the third-order solution contain...
The Dirac-Electron Vacuum Wave
Daywitt W. C.
2016-07-01
Full Text Available This paper argues that the Dirac equation can be interpreted as an interaction between the electron core and the Planck vacuum state, where the positive and negative solutions represent respectively the dynamics of the electron core and a vacuum wave propagating within the vacuum state. Results show that the nonrelativistic positive solution reduces to the Schrödinger wave equation
Chen Yong; Li Biao; Zhang Hong-Qing
2004-01-01
@@ An extended Jacobi elliptic function method is proposed for constructing the exact double periodic solutions of nonlinear partial differential equations (PDEs) in a unified way. It is shown that these solutions exactly degenerate to the many types of soliton solutions in a limited condition. The Wu-Zhang equation (which describes the (2+1)-dimensional dispersive long wave) is investigated by this means and more formal double periodic solutions are obtained.
WANG Qi; CHEN Yong; ZHANG Hong-Qing
2005-01-01
In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.
Chen Yong; Zhang Hong Qin
2003-01-01
Based on the idea of homogenous balance method and with the help of Mathematica, we obtain a new auto-Baecklund transformation for modified nonlinear dispersive equation mK(m,n). Then based on the Baecklund transformation, some solitary patterns solution for mK(m,n) equation are derived. In addition, we also obtain the general solutions for mK(n,n) in higher dimensional spatial domains, even in N dimensional space.
Clobert, J.; Danchin, E.; Dhondt, A.A.; Nichols, J.D.
2001-01-01
The ability of species to migrate and disperse is a trait that has interested ecologists for many years. Now that so many species and ecosystems face major environmental threats from habitat fragmentation and global climate change, the ability of species to adapt to these changes by dispersing, migrating, or moving between patches of habitat can be crucial to ensuring their survival. This book provides a timely and wide-ranging overview of the study of dispersal and incorporates much of the latest research. The causes, mechanisms, and consequences of dispersal at the individual, population, species and community levels are considered. The potential of new techniques and models for studying dispersal, drawn from molecular biology and demography, is also explored. Perspectives and insights are offered from the fields of evolution, conservation biology and genetics. Throughout the book, theoretical approaches are combined with empirical data, and care has been taken to include examples from as wide a range of species as possible.
The Clifford algebra of physical space and Dirac theory
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Semi-Dirac points in phononic crystals
Zhang, Xiujuan
2014-01-01
A semi-Dirac cone refers to a peculiar type of dispersion relation that is linear along the symmetry line but quadratic in the perpendicular direction. It was originally discovered in electron systems, in which the associated quasi-particles are massless along one direction, like those in graphene, but effective-mass-like along the other. It was reported that a semi-Dirac point is associated with the topological phase transition between a semi-metallic phase and a band insulator. Very recently, the classical analogy of a semi-Dirac cone has been reported in an electromagnetic system. Here, we demonstrate that, by accidental degeneracy, two-dimensional phononic crystals consisting of square arrays of elliptical cylinders embedded in water are also able to produce the particular dispersion relation of a semi-Dirac cone in the center of the Brillouin zone. A perturbation method is used to evaluate the linear slope and to affirm that the dispersion relation is a semi-Dirac type. If the scatterers are made of rubber, in which the acoustic wave velocity is lower than that in water, the semi-Dirac dispersion can be characterized by an effective medium theory. The effective medium parameters link the semi-Dirac point to a topological transition in the iso-frequency surface of the phononic crystal, in which an open hyperbola is changed into a closed ellipse. This topological transition results in drastic change in wave manipulation. On the other hand, the theory also reveals that the phononic crystal is a double-zero-index material along the x-direction and photonic-band-edge material along the perpendicular direction (y-direction). If the scatterers are made of steel, in which the acoustic wave velocity is higher than that in water, the effective medium description fails, even though the semi-Dirac dispersion relation looks similar to that in the previous case. Therefore different wave transport behavior is expected. The semi-Dirac points in phononic crystals described in
Gugushvili, S.; Spreij, P.
2016-01-01
We consider the problem of non-parametric estimation of the deterministic dispersion coefficient of a linear stochastic differential equation based on discrete time observations on its solution. We take a Bayesian approach to the problem and under suitable regularity assumptions derive the posteror
WEN Xiao-Yong
2009-01-01
With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the (2+1)-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.
DIRAC distributed secure framework
Casajus, A.; Graciani, R.; LHCb DIRAC Team
2010-04-01
DIRAC, the LHCb community Grid solution, provides access to a vast amount of computing and storage resources to a large number of users. In DIRAC users are organized in groups with different needs and permissions. In order to ensure that only allowed users can access the resources and to enforce that there are no abuses, security is mandatory. All DIRAC services and clients use secure connections that are authenticated using certificates and grid proxies. Once a client has been authenticated, authorization rules are applied to the requested action based on the presented credentials. These authorization rules and the list of users and groups are centrally managed in the DIRAC Configuration Service. Users submit jobs to DIRAC using their local credentials. From then on, DIRAC has to interact with different Grid services on behalf of this user. DIRAC has a proxy management service where users upload short-lived proxies to be used when DIRAC needs to act on behalf of them. Long duration proxies are uploaded by users to a MyProxy service, and DIRAC retrieves new short delegated proxies when necessary. This contribution discusses the details of the implementation of this security infrastructure in DIRAC.
Forward dispersion relations and Roy equations in {pi}{pi} scattering
Kaminski, R. [Polish Academy of Sciences, Department of Theoretical Physics Henryk Niewodniczanski Institute of Nuclear Physics, Krakow (Poland); Pelaez, J.R. [Universidad Complutense de Madrid, Departamento de Fisica Teorica, II (Metodos Matematicos), Facultad de Ciencias Fisicas, Madrid (Spain); Yndurain, F.J. [Universidad Autonoma de Madrid, Canto Blanco, Departamento de Fisica Teorica, Madrid (Spain)
2007-03-15
We first review the results of an analysis of {pi}{pi} interactions in S, P and D waves for the two-pion effective mass from threshold to about 1.4 GeV. In particular, we show a recent improvement of this analysis above the K anti K threshold using more data for phase shifts and including the S0-wave inelasticity from {pi}{pi}{yields}K anti K. In addition, we have improved the fit to the f{sub 2}(1270)-resonance and used a more flexible P-wave parametrization above the K anti K threshold and included an estimation of the D2-wave inelasticity. The better accuracy thus achieved also required a refinement of the Regge analysis above 1.42 GeV. Finally, in this work we check that the {pi}{pi} scattering amplitudes obtained in this approach satisfy remarkably well forward dispersion relations and Roy's equations. (orig.)
Tunable multiple layered Dirac cones in optical lattices.
Lan, Z; Celi, A; Lu, W; Öhberg, P; Lewenstein, M
2011-12-16
We show that multiple layered Dirac cones can emerge in the band structure of properly addressed multicomponent cold fermionic gases in optical lattices. The layered Dirac cones contain multiple copies of massless spin-1/2 Dirac fermions at the same location in momentum space, whose different Fermi velocity can be tuned at will. On-site microwave Raman transitions can further be used to mix the different Dirac species, resulting in either splitting of or preserving the Dirac point (depending on the symmetry of the on-site term). The tunability of the multiple layered Dirac cones allows us to simulate a number of fundamental phenomena in modern physics, such as neutrino oscillations and exotic particle dispersions with E~p(N) for arbitrary integer N.
On the spring and mass of the Dirac oscillator
Crawford, James P.
1993-01-01
The Dirac oscillator is a relativistic generalization of the quantum harmonic oscillator. In particular, the square of the Hamiltonian for the Dirac oscillator yields the Klein-Gordon equation with a potential of the form: (ar(sub 2) + b(L x S)), where a and b are constants. To obtain the Dirac oscillator, a 'minimal substitution' is made in the Dirac equation, where the ordinary derivative is replaced with a covariant derivative. However, an unusual feature of the covariant derivative in this case is that the potential is a non-trivial element of the Clifford algebra. A theory which naturally gives rise to gage potentials which are non-trivial elements of the Clifford algebra is that based on local automorphism invariance. An exact solution of the automorphism gage field equations which reproduces both the potential term and the mass term of the Dirac oscillator is presented.
The right inverse of Dirac operator in octonionic space
Wang, Haiyan; Bian, Xiaoli
2017-09-01
The octonion Dirac equation also called wave equation is an important equation which formulates the localization spaces for subluminal and superluminal particles. The purpose of this paper is to look for the right inverse operator of octonion Dirac operator in Hölder space. However, some difficulties will arise in noncommutative and nonassociative setting. We note that the associator is available to overcome the difficulties.
Dirac spinor in a nonstationary Godel-type cosmological Universe
Villalba, Victor M
2015-01-01
In the present article we solve, via separation of variables, the massless Dirac equation in a nonstationary rotating, causal G\\"odel-type cosmological universe, having a constant rotational speed in all the points of the space. We compute the frequency spectrum. We show that the spectrum of massless Dirac particles is discrete and unbounded.
Tunneling of Dirac Particles from Kaluza-Klein Black Hole
ZENG Xiao-Xiong; LI Qiang
2009-01-01
Applying the fermions tunneling method, proposed by Kerner and Mann recently, we discuss the tunneling characteristics of Dirac particles from the stationary Kaluza-Klein black hole. To choose Gamma matrix conveniently and avoid the ergosphere dragging effect, we perform it in the dragging coordinate frame. The result shows that Hawking temperature in this case also can be reproduced by the general Dirac equation.
Methods and Algorithms for Solving Inverse Problems for Fractional Advection-Dispersion Equations
Aldoghaither, Abeer
2015-11-12
Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final
Deconstructing non-dissipative non-Dirac-hermitian relativistic quantum systems
Ghosh, Pijush K
2011-01-01
A method to construct non-dissipative non-Dirac-hermitian relativistic quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-hermitian operators, which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry
Deconstructing non-dissipative non-Dirac-Hermitian relativistic quantum systems
Ghosh, Pijush K.
2011-08-01
A method to construct non-dissipative non-Dirac-Hermitian relativistic quantum system that is isospectral with a Dirac-Hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-Hermitian operators, which are Hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-Hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry.
田野; 陈静; 张忠飞
2012-01-01
In this paper, the separation transformation approach is extended to the （N ＋ 1）-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of SHe superfluid. This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation. Then the general solutions of the set of partial differential equations are obta/ned and the nonlinear ordinary differential equation is solved by F-expansion method. Finally, many new exact solutions of the （N ＋ 1）-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation. For the case of N 〉 2, there is an arbitrary function in the exact solutions, which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.
Scarring of Dirac fermions in chaotic billiards.
Ni, Xuan; Huang, Liang; Lai, Ying-Cheng; Grebogi, Celso
2012-07-01
Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrödinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions.
Dirac-Coulomb scattering with plane wave energy eigenspinors on de Sitter expanding universe
Cotaescu, Ion I
2007-01-01
The lowest order contribution of the amplitude of Dirac-Coulomb scattering in de Sitter spacetime is calculated assuming that the initial and final states of the Dirac field are described by exact solutions of the free Dirac equation on de Sitter spacetime with a given energy and helicity. We find that the total energy is conserved in the scattering process.
Zheng Xuedong; Chen Yong; Zhang Hongqing
2003-05-12
Making use of a new generalized ansatzes, we present the generalized extended tanh-function method for constructing the exact solutions of nonlinear partial differential equations (NPDEs) in a unified way. Applying the generalized method, with the aid of MAPLE, we consider the Wu-Zhang equation (which describes (1+1)-dimensional dispersive long wave). As a result, we can successfully obtain the solitary wave solutions that can be found by the extended tanh-function method and the modified extended tanh-function method. More importantly, for the equation, we also obtain other new and more general solutions at the same time. The results include kink-profile solitary wave solutions, bell-profile solitary wave solutions, periodic wave solutions, rational solutions, singular solutions and other new formal solutions. As an illustrative sample, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.
Rodrigues, R. de Lima [Universidade Federal de Campina Grande (UFCG), Cuite, PB (Brazil). Centro de Tecnologia. Unidade Academica de Educacao]. E-mail: rafael@df.ufcg.edu.br; rafaelr@cbpf.br
2007-07-01
In the present work we obtain a new representation for the Dirac oscillator based on the Clifford algebra C 7. The symmetry breaking and the energy eigenvalues for our model of the Dirac oscillator are studied in the non-relativistic limit. (author)
Remiddi, Ettore
2016-01-01
It is shown that the study of the imaginary part and of the corresponding dispersion relations of Feynman graph amplitudes within the differential equations method can provide a powerful tool for the solution of the equations, especially in the massive case. The main features of the approach are illustrated by discussing the simple cases of the 1-loop self-mass and of a particular vertex amplitude, and then used for the evaluation of the two-loop massive sunrise and the QED kite graph (the problem studied by Sabry in 1962), up to first order in the (d-4) expansion.
Remiddi, Ettore; Tancredi, Lorenzo
2016-06-01
It is shown that the study of the imaginary part and of the corresponding dispersion relations of Feynman graph amplitudes within the differential equations method can provide a powerful tool for the solution of the equations, especially in the massive case. The main features of the approach are illustrated by discussing the simple cases of the 1-loop self-mass and of a particular vertex amplitude, and then used for the evaluation of the two-loop massive sunrise and the QED kite graph (the problem studied by Sabry in 1962), up to first order in the (d - 4) expansion.
A. H. Bhrawy
2013-01-01
Full Text Available We propose Jacobi-Gauss-Lobatto collocation approximation for the numerical solution of a class of fractional-in-space advection-dispersion equation with variable coefficients based on Caputo derivative. This approach has the advantage of transforming the problem into the solution of a system of ordinary differential equations in time this system is approximated through an implicit iterative method. In addition, some of the known spectral collocation approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters and . Finally, numerical results are provided to demonstrate the effectiveness of the proposed spectral algorithms.
Andrea Prosperetti
2006-03-24
The report briefly describes the activities carried out in the course of the project. A first line of research was the development of systematic closure relations for averaged equations for disperse multiphase flow. A second line was the development of efficient numerical methods for the simulation of Navier-Stokes flows with many suspended particles. The report also lists the 21 journal articles in which this work is more fully decsribed.
Grava, T
2012-01-01
We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\\epsilon^{2}u_{xxx}=0$ for $\\epsilon\\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.
Cui, Jianbo, E-mail: jianbocui@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Hong, Jialin, E-mail: hjl@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Liu, Zhihui, E-mail: liuzhihui@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Zhou, Weien, E-mail: weienzhou@nudt.edu.cn [College of Science, National University of Defense Technology, Changsha 410073 (China)
2017-08-01
We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.
Analysis of the small dispersion limit of a non-integrable generalized Korteweg-de Vries equation
Zakeri, Gholam-Ali; Yomba, Emmanuel
2013-08-01
A generalized non-integrable Korteweg-de Vries (KdV) equation is investigated for the qualitative behavior of its solutions with a small dispersion limit. We obtained two reduced ordinary differential equations using a similarity analysis and discussed the solutions of generalized KdV (gKdV) by employing singular perturbation and asymptotic methods. We found a new closed form solution and provided various approximate solutions. We have shown that for sech-type initial value data the cumulative primitive function of the gKdV solution converges point-wise as the coefficient of the dispersive term goes to zero. Our numerical experiments provide strong evidence that for each fixed time, the solutions of gKdV are bounded by well-defined envelopes as the coefficient of the dispersion term goes to zero. We have shown that for a higher order of nonlinearity, the soliton becomes shaper, with a larger amplitude, but remains bounded. Comparatively, for a smaller coefficient of the dispersion term, its base gets smaller and the soliton becomes narrower, but the amplitude of the soliton remains the same.
Consequences of Dirac Theory of the Positron
Heisenberg, W K
1936-01-01
According to Dirac's theory of the positron, an electromagnetic field tends to create pairs of particles which leads to a change of Maxwell's equations in the vacuum. These changes are calculated in the special case that no real electrons or positrons are present and the field varies little over a Compton wavelength.
Gogoi, R; Kalita, L; Devi, N, E-mail: runmoni_gogoi@rediffmail.co, E-mail: latikakalita@rediffmail.co, E-mail: nirupama_cotton@rediffmail.co [Department of Mathematics, Cotton College, Guwahati-781001, Assam (India)
2010-02-01
Much interest was shown towards the studies on nonlinear stability in the late sixties. Plasma instabilities play an important role in plasma dynamics. More attention has been given towards stability analysis after recognizing that they are one of the principal obstacles in the way of a successful resolution of the problem of controlled thermonuclear fusion. Nonlinearity and dispersion are the two important characteristics of plasma instabilities. Instabilities and nonlinearity are the two important and interrelated terms. In our present work, the continuity and momentum equations for both ions and electrons together with the Poisson equation are considered as cold plasma model. Then we have adopted the modified reductive perturbation technique (MRPT) from Demiray [1] to derive the higher order equation of the Nonlinear Schroedinger equation (NLSE). In this work, detailed mathematical expressions and calculations are done to investigate the changing character of the modulation of ion acoustic plasma wave through our derived equation. Thus we have extended the application of MRPT to derive the higher order equation. Both progressive wave solutions as well as steady state solutions are derived and they are plotted for different plasma parameters to observe dark/bright solitons. Interesting structures are found to exist for different plasma parameters.
DIFFUSIVE-DISPERSIVE TRAVELING WAVES AND KINETIC RELATIONS IV.COMPRESSIBLE EULER EQUATIONS
无
2003-01-01
The authors consider the Euler equations for a compressible fluid in one space dimensionwhen the equation of state of the fluid does not fulfill standard convexity assumptions andviscosity and capillarity effects are taken into account. A typical example of nonconvex con-stitutive equation for fluids is Van der Waals' equation. The first order terms of these partialdifferential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class ofnonconvex equations of state, an existence theorem of traveling waves solutions with arbitrarylarge amplitude is established here. The authors distinguish between classical (compressive) andnonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequali-ties, and are characterized by the so-called kinetic relation, whose properties are investigatedin this paper.
Monti, Dalida
1996-01-01
Relativamente poco noto al gran pubblico, il premio Nobel Paul Adrien Maurice Dirac appartiene a quel gruppo di uomini di ingegno che nei primi decenni del secolo contribuirono a dare alla nostra concezione del mondo fisico la sua impronta attuale. Assolutamente cruciali, per una valutazione dell'opera di Dirac, sono gli anni compresi tra il 1925 e il 1931: un periodo in cui il fisico fornisce la prima spiegazione chiara e coerente delle proprietà di spin dell'elettrone (equazione di Dirac) e perviene, in forza della pura deduzione matematica, alla scoperta dell'esistenza dell'elettrone positivo o positrone.
张爱萍; 强稳朝
2007-01-01
In this paper, the relativistic Rosen-Morse Ⅱ potential is investigated by solving the Klein-Gordon and the Dirac equations with equal attractive scalar s(r) and repulsive vector v(r) potentials. The exact energy equations of the bound state are obtained by the method of supersymmetric and shape invariance. Finally, a kind of special potential about Rosen-Morse Ⅱ potential is discussed.%在标量势和矢量势相等的情形,研究了Rosen-MorseⅡ势的相对论效应,应用超对称和形状不变势方法通过求解Klein-Gordon方程和Dirac方程得到了束缚态能量本征值,最后,讨论了Rosen-MorseⅡ势的一种特殊情况.
On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension
Pelinovsky, D. E.; Stefanov, A.
2008-11-01
Based on the recent work [Komech et al., "Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations," Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrödinger operator, Hϕ =(-Δ+V)ϕ=-(ϕn +1+ϕn -1-2ϕn)+Vnϕn. We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates ||eitHPa.c.(H)||lσ2→l-σ2≲t-3/2 for any fixed σ >5/2 and any t >0, where Pa.c.(H) denotes the spectral projection to the absolutely continuous spectrum of H. In addition, based on the scattering theory for the discrete Jost solutions and the previous results by Stefanov and Kevrekidis ["Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations," Nonlinearity 18, 1841 (2005)], we find new dispersive estimates ||eitH Pa.c.(H)||l1→l∞≲t-1/3, which are sharp for the discrete Schrödinger operators even for V =0.
Three dimensional Dirac semimetals
Zaheer, Saad
We extend the physics of graphene to three dimensional systems by showing that Dirac points can exist on the Fermi surface of realistic materials in three dimensions. Many of the exotic electronic properties of graphene can be ascribed to the pseudorelativistic behavior of its charge carriers due to two dimensional Dirac points on the Fermi surface. We show that certain nonsymmorphic spacegroups exhibit Dirac points among the irreducible representations of the appropriate little group at high symmetry points on the surface of the Brillouin zone. We provide a list of all Brillouin zone momenta in the 230 spacegroups that can host Dirac points. We describe microscopic considerations necessary to design materials in one of the candidate spacegroups such that the Dirac point appears at the Fermi energy without any additional non-Dirac-like Fermi pockets. We use density functional theory based methods to propose six new Dirac semimetals: BiO 2 and SbO2 in the beta-cristobalite lattice (spacegroup 227), and BiCaSiO4, BiMgSiO4, BiAlInO 4, and BiZnSiO4 in the distorted spinels lattice (spacegroup 74). Additionally we derive effective Dirac Hamiltonians given group representative operators as well as tight binding models incorporating spin-orbit coupling. Finally we study the Fermi surface of zincblende (spacegroup 216) HgTe which is effectively point-like at Gamma in the Brillouin zone and exhibits accidental degeneracies along a threefold rotation axis. Whereas compressive strain gaps the band structure into a topological insulator, tensile strain shifts the accidental degeneracies away from Gamma and enlarges the Fermi surface. States on the Fermi surface exhibit nontrivial spin texture marked by winding of spins around the threefold rotation axis and by spin vortices indicating a change in the winding number. This is confirmed by microscopic calculations performed in tensile strained HgTe and Hg0.5Zn 0.5 Te as well as k.p theory. We conclude with a summary of recent
Two Qubits in the Dirac Representation
Rajagopal, A K
2000-01-01
A general two qubit system expressed in terms of the complete set of unit and fifteen traceless, Hermitian Dirac matrices, is shown to exhibit novel features of this system. The well-known physical interpretations associated with the relativistic Dirac equation involving the symmetry operations of time-reversal T, charge conjugation C, parity P, and their products are reinterpreted here by examining their action on the basic Bell states. The transformation properties of the Bell basis states under these symmetry operations also reveal that C is the only operator that does not mix the Bell states whereas all others do. In a similar fashion, expressing the various logic gates introduced in the subject of quantum computers in terms of the Dirac matrices shows for example, that the NOT gate is related to the product of time-reversal and parity operators.
DIRAC Workload Management System
Paterson, S
2007-01-01
DIRAC (Distributed Infrastructure with Remote Agent Control) is the Workload and Data Management system (WMS) for the LHCb experiment. The DIRAC WMS offers a transparent way for LHCb users to submit jobs to the EGEE Grid as well as local clusters and individual PCs. This paper will describe workload management optimizations, which ensure high job efficiency and minimized job start times. The computing requirements of the LHCb experiment can only be fulfilled through the use of many distributed compute resources. DIRAC provides a robust platform to run data productions on all the resources available to LHCb including the EGEE Grid. More recently, user support was added to DIRAC that greatly simplifies the procedure of submitting, monitoring and retrieving output of Grid jobs for the LHCb user community. DIRAC submits Pilot Agents to the EGEE Grid via the gLite WMS as normal jobs. Pilot Agents then request jobs from the DIRAC Workload Management System after the local environment has been checked. Therefore DIR...
Second-order envelope equation of graphene electrons
Luo, Ji
2014-10-01
A treatment of graphene's electronic states based on the tight-binding method is presented. Like Dirac equation, this treatment uses envelope functions to eliminate crystal potential. Besides, a density-functional-theory Kohn-Sham (KS) orbital of an isolated carbon atom is employed. By locally expanding envelope functions into second-order polynomials and by involving up to third-nearest atoms in calculating orbital integrals, the second-order envelope equation is obtained. This equation does not contain any experimental data except graphene's crystal structure, and its coefficients are determined through several kinds of integrals of the carbon KS orbital. As an improvement, it leads to more accurate energy dispersion than Dirac equation including the triangular warping effect and asymmetry for electrons and holes, and gives the Fermi velocity which is in good agreement with the experimental value.
Szybisz, Leszek
1990-06-01
The self-consistency of solutions obtained from a recently proposed numerical relaxation method of solving the Euler-Lagrange equations for the ground state of inhomogeneous Bose systems at zero temperature is investigated. For this kind of system at least three different dispersion relations can be formulated, all of them providing information about the same eigenstates. The quality of our optimization scheme is studied by analyzing the convergence of the low-lying eigenvalues and eigenfunctions of these dispersion relations. Numerical results for the spectrum and spatial shape of elementary excitations of a thin film of liquid 4He supported by an external potential are reported. The optimal lowest-lying eigenvalues are compared with estimations based on simple theoretical approaches and with calculations performed by other authors.
Exotic Localized Coherent Structures of the （2＋1）—Dimensional Dispersive Long—Wave Equation
ZHANGJie－Fang
2002-01-01
This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2+1)-dimensional dispersive long-wave equations uty+ηxx+(u2)xy/2=0,ηt+(uη+u+uxy)x=0.Starting from the homogeneous balance method,we find that the richness of the localized coberent structures of the model is caused by the entrance of two variable-separated arbitrary functions.for some special selections of the arbitrary functions,it is shown that the localized structures of the model may be dromions,lumps,breathers,instantons and ring solitons.
Bezák, V
2003-02-01
The Waxman-Peck theory of population genetics is discussed in regard of soil bacteria. Each bacterium is understood as a carrier of a phenotypic parameter p. The central objective is the calculation of the probability density with respect to p, Phi(p,t;p(0)), of the carriers living at time t>0, provided that initially at t(0)=0, all bacteria carried the phenotypic parameter p(0)=0. The theory involves two small parameters: the mutation probability mu and a parameter gamma involved in a function w(p) defining the fitness of the bacteria to survive the generation time tau and give birth to an offspring. The mutation from a state p to a state q is defined by a Gaussian with a dispersion sigma(2)(m). The author focuses our attention on a function phi(p,t) which determines uniquely the function Phi(p,t;p(0)) and satisfies a linear equation (Waxman's equation). The Green function of this equation is mathematically identical with the one-particle Bloch density matrix, where mu characterizes the order of magnitude of the potential energy. (In the x representation, the potential energy is proportional to the inverted Gaussian with the dispersion sigma(2)(m)). The author solves Waxman's equation in the standard style of a perturbation theory and discusses how the solution depends on the choice of the fitness function w(p). In a sense, the function c(p)=1-w(p)/w(0) is analogous to the dispersion function E(p) of fictitious quasiparticles. In contrast to Waxman's approximation, where c(p) was taken as a quadratic function, c(p) approximately gammap(2), the author exemplifies the problem with another function, c(p)=gamma[1-exp(-ap(2))], where gamma is small but a may be large. The author shows that the use of this function in the theory of the population genetics is the same as the use of a nonparabolic dispersion law E=E(p) in the density-matrix theory. With a general function c(p), the distribution function Phi(p,t;0) is composed of a delta-function component, N
Red'kov, V M
2011-01-01
Tetrad based equation for Dirac-K\\"{a}hler particle is solved in spherical coordinates in the flat Minkocski space-time. Spherical solutions of boson type (J =0,1,2,...) are constructed. After performing a special transformation over spherical boson solutions of the Dirac-K\\"{a}hler equation, 4 \\times 4-matrices U(x) \\Longrightarrow V(x), simple linear expansions of the four rows of new representativeof the Dirac--K\\"{a}hler field V(x) in terms of spherical fermion solutions \\Psi_{i}(x) of the four ordinary Dirac equations have been derived. However, this fact cannot be interpreted as the possibility not to distinguish between the Dirac-K\\"{a}hler field and the system four Dirac fermions. The main formal argument is that the special transformation (I \\otimes S(x)) involved does not belong to the group of tetrad local gauge transformation for Dirac-K\\"{a}hler field, 2-rank bispinor under the Lorentz group. Therefore, the linear expansions between boson and fermion functions are not gauge invariant under the gr...
Dark solitons in the Lugiato-Lefever equation with normal dispersion
Parra-Rivas, Pedro; Gomila, Damia; Gelens, Lendert
2016-01-01
The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.
Dark solitons in the Lugiato-Lefever equation with normal dispersion
Parra-Rivas, Pedro; Gomila, Damia; Gelens, Lendert; Knobloch, Edgar
2016-01-01
The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understa...
Dark solitons in the Lugiato-Lefever equation with normal dispersion
Parra-Rivas, P.; Knobloch, E.; Gomila, D.; Gelens, L.
2016-06-01
The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
Pedersen, Martin Erland Vestergaard; Cheng, Ji; Xu, Chris
2013-01-01
An improved version of the generalized nonlinear Schrödinger equation is derived, which takes into account the correct dispersion of the transverse field distribution. The new improved version of the generalized nonlinear Schrödinger equation is verified to give the same results as the standard...
无
2006-01-01
By making use of the generalized sine-Gordon equation expansion method, we find cnoidal periodic wave solutions and fundamental bright and dark optical solitarywave solutions for the fourth-order dispersive and the quintic nonlinear Schrodinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.
Feng, Xin; Ye, Xingyou; Park, Jun-Bom; Lu, Wenli; Morott, Joe; Beissner, Brad; Lian, Zhuoyang John; Pinto, Elanor; Bi, Vivian; Porter, Stu; Durig, Tom; Majumdar, Soumyajit; Repka, Michael A
2015-01-01
The recrystallization of an amorphous drug in a solid dispersion system could lead to a loss in the drug solubility and bioavailability. The primary objective of the current research was to use an improved kinetic model to evaluate the recrystallization kinetics of amorphous structures and to further understand the factors influencing the physical stability of amorphous solid dispersions. Amorphous solid dispersions of fenofibrate with different molecular weights of hydroxypropylcellulose, HPC (Klucel™ LF, EF, ELF) were prepared utilizing hot-melt extrusion technology. Differential scanning calorimetry was utilized to quantitatively analyze the extent of recrystallization in the samples stored at different temperatures and relative humidity (RH) conditions. The experimental data were fitted into the improved kinetics model of a modified Avrami equation to calculate the recrystallization rate constants. Klucel LF, the largest molecular weight among the HPCs used, demonstrated the greatest inhibition of fenofibrate recrystallization. Additionally, the recrystallization rate (k) decreased with increasing polymer content, however exponentially increased with higher temperature. Also k increased linearly rather than exponentially over the range of RH studied.
Aldoghaither, Abeer
2015-12-01
In this paper, a new method, based on the so-called modulating functions, is proposed to estimate average velocity, dispersion coefficient, and differentiation order in a space-fractional advection-dispersion equation, where the average velocity and the dispersion coefficient are space-varying. First, the average velocity and the dispersion coefficient are estimated by applying the modulating functions method, where the problem is transformed into a linear system of algebraic equations. Then, the modulating functions method combined with a Newton\\'s iteration algorithm is applied to estimate the coefficients and the differentiation order simultaneously. The local convergence of the proposed method is proved. Numerical results are presented with noisy measurements to show the effectiveness and robustness of the proposed method. It is worth mentioning that this method can be extended to general fractional partial differential equations.
LI Zi-Ping; LI Ai-Min; JIANG Jin-Huan; WANG Yong-Long
2005-01-01
The extended canonical Noether identities and canonical first Noether theorem derived from an extended action in phase space for a system with a singular Lagrangian are formulated. Using these canonical Noether identities,it can be shown that the constraint multipliers connected with the first-class constraints may not be independent, so a query to a conjecture of Dirac is presented. Based on the symmetry properties of the constrained Hamiltonian system in phase space, a counterexample to a conjecture of Dirac is given to show that Dirac's conjecture fails in such a system.We present here a different way rather than Cawley's examples and other's ones in that there is no linearization of constraints in the problem. This example has a feature that neither the primary first-class constraints nor secondary first-class constraints are generators of the gauge transformation.
Dirac and Weyl Materials: Fundamental Aspects and Some Spintronics Applications
Yang, Shengyuan A.
2016-09-01
Dirac and Weyl materials refer to a class of solid materials which host low-energy quasiparticle excitations that can be described by the Dirac and Weyl equations in relativistic quantum mechanics. Starting with the advent of graphene as the first prominent example, these materials have been attracting tremendous interest owing to their novel fundamental properties as well as the great potential for applications. Here we introduce the basic concepts and notions related to Dirac and Weyl materials and briefly review some recent works in this field, particularly on the conceptual development and the possible spintronics/pseudospintronics applications.
Accidental degeneracy of double Dirac cones in a phononic crystal
Chen, Ze-Guo
2014-04-09
Artificial honeycomb lattices with Dirac cone dispersion provide a macroscopic platform to study the massless Dirac quasiparticles and their novel geometric phases. In this paper, a quadruple-degenerate state is achieved at the center of the Brillouin zone in a two-dimensional honeycomb lattice phononic crystal, which is a result of accidental degeneracy of two double-degenerate states. In the vicinity of the quadruple-degenerate state, the dispersion relation is linear. Such quadruple degeneracy is analyzed by rigorous representation theory of groups. Using method, a reduced Hamiltonian is obtained to describe the linear Dirac dispersion relations of this quadruple-degenerate state, which is well consistent with the simulation results. Near such accidental degeneracy, we observe some unique properties in wave propagating, such as defect-insensitive propagating character and the Talbot effect.
Abel, Steven [Durham Univ. (United Kingdom). Inst. for Particle Physics Phenomenology; CERN, Geneva (Switzerland); Goodsell, Mark [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2011-02-15
A simple and natural model is presented that gives Dirac gauginos. The configuration is related to ''deconstructed gaugino mediation''. A high energy completion is provided based on existing ISS-like models of deconstructed gaugino mediation. This provides a complete picture of Dirac gauginos that includes the necessary extra adjoint fermions (generated as magnetic quarks of the ISS theory) and supersymmetry breaking (via the ISS mechanism). Moreover the screening of the scalar masses means that they can similar to or less than the gaugino masses, even though the supersymmetry breaking is driven by F-terms. (orig.)
Modulational instability in dispersion-kicked optical fibers
Nodari, S Rota; Dujardin, G; Kudlinski, A; Mussot, A; Trillo, S; De Bièvre, S
2015-01-01
We study, both theoretically and experimentally, modulational instability in optical fibers that have a longitudinal evolution of their dispersion in the form of a Dirac delta comb. By means of Floquet theory, we obtain an exact expression for the position of the gain bands, and we provide simple analytical estimates of the gain and of the bandwidths of those sidebands. An experimental validation of those results has been realized in several microstructured fibers specifically manufactured for that purpose. The dispersion landscape of those fibers is a comb of Gaussian pulses having widths much shorter than the period, which therefore approximate the ideal Dirac comb. Experimental spontaneous MI spectra recorded under quasi continuous wave excitation are in good agreement with the theory and with numerical simulations based on the generalized nonlinear Schr\\"odinger equation.
Weyl, Majorana and Dirac fields from a unified perspective
Aste, Andreas
2016-01-01
A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory which are commonly not included in introductory courses treating fermionic fields via the Dirac equation in the first place.
Weyl, Majorana and Dirac Fields from a Unified Perspective
Andreas Aste
2016-08-01
Full Text Available A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory, which are commonly not included in introductory courses when treating fermionic fields via the Dirac equation in the first place.
Bezak, V
2002-01-01
The Waxman-Peck theory of the population genetics is discussed in regard of soil bacteria. Each bacterium is understood as a carrier of a phenotypic parameter p. The central aim is the calculation of the probability density with respect to p of the carriers living at time t>0. The theory involves two small parameters: the mutation probability $\\mu$ and a parameter $\\gamma$ involved in a function w(p) defining the fitness of the bacteria to survive the generation time $\\tau$ and give birth to offspring. The mutation from a state p to a state q is defined by a Gaussian. The author focuses attention on an equation generalizing Waxman's equation. The author solves this equation in the standard style of a perturbation theory and discusses how the solution depends on the choice of the fitness function w(p). In a sense, the function $c(p)=1-w(p)/w(0)$ is analogous to the dispersion function E(p) of fictitious quasiparticles. With a general function c(p), the distribution function ${\\mathit\\Phi}(p,t;0)$ is composed o...
Jing Yin
2015-07-01
Full Text Available A total variation diminishing-weighted average flux (TVD-WAF-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer (HLL Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth-order monotone upstream-centered scheme for conservation laws (MUSCL. The time marching scheme based on the third-order TVD Runge-Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.
Dirac Induction for loop groups
Posthuma, H.
2011-01-01
Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl-Kac character formula and show compatibility with Dirac induc
Classical electromagnetic radiation of the Dirac electron
Lanyi, G.
1973-01-01
A wave-function-dependent four-vector potential is added to the Dirac equation in order to achieve conservation of energy and momentum for a Dirac electron and its emitted electromagnetic field. The resultant equation contains solutions which describe transitions between different energy states of the electron. As a consequence it is possible to follow the space-time evolution of such a process. This evolution is shown in the case of the spontaneous emission of an electromagnetic field by an electron bound in a hydrogen-like atom. The intensity of the radiation and the spectral distribution are calculated for transitions between two eigenstates. The theory gives a self-consistent deterministic description of some simple radiation processes without using quantum electrodynamics or the correspondence principle.
Arditi, Roger; Lobry, Claude; Sari, Tewfik
2015-12-01
The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations. First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been completely analysed. Here, we complete this analysis and we delineate the conditions under which fragmentation associated to dispersal is either beneficial or detrimental to total population abundance. Therefore, this is a contribution to the SLOSS question. Importantly, we also show that, depending on the underlying mechanism, there is no unique way to generalize the logistic model to a patchy situation. In many cases, the standard model is not the correct generalization. We analyse several alternative models and compare their predictions. Finally, we emphasize the shortcomings of the logistic model when written in the r-K parameterization and we explain why Verhulst's original polynomial expression is to be preferred.
Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation
Ak, Turgut; Battal Gazi Karakoc, S.; Triki, Houria
2016-10-01
In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable.
Gravity, torsion, Dirac field and computer algebra using MAPLE and REDUCE
Vulcanov, D N
2002-01-01
The article presents computer algebra procedures and routines applied to the study of the Dirac field on curved spacetimes. The main part of the procedures is devoted to the construction of Pauli and Dirac matrices algebra on an anholonomic orthonormal reference frame. Then these procedures are used to compute the Dirac equation on curved spacetimes in a sequence of special dedicated routines. A comparative review of such procedures obtained for two computer algebra platforms (REDUCE + EXCALC and MAPLE + GRTensorII) is carried out. Applications for the calculus of Dirac equation on specific examples of spacetimes with or without torsion are pointed out.
One real function instead of the Dirac spinor function
Akhmeteli, Andrey
2010-01-01
Schr\\"{o}dinger (Nature, v.169, p.538(1952)) noted that for each solution of the equations of scalar electrodynamics (the Klein-Gordon-Maxwell electrodynamics) there is a physically equivalent (i.e. coinciding with it up to a gauge transform) solution with a real matter field, despite the widespread belief about charged fields requiring complex representation. Surprisingly, the same result is true for spinor electrodynamics (the Dirac-Maxwell electrodynamics): the Dirac equation for the four complex components of the spinor function can be replaced by a fourth-order equation for one of those components, and this component can be made real by a gauge transform.
P. G. L. Leach
2014-04-01
Full Text Available Dirac devised his theory of Quantum Mechanics and recognised that his operators resembled the canonical coordinates of Hamiltonian Mechanics. This gave the latter a new lease of life. We look at what happens to Dirac’s Quantum Mechanics if one starts from Hamiltonian Mechanics.
Trzetrzelewski, Maciej
2011-01-01
In c=1 units the product (mass x radius) for the neutron and the proton is about 4.7\\hbar assuming their radii equal to 1fm. We show that the corresponding products for the Dirac neutral and charged membrane coincide and are equal 1.6\\hbar.
Guignard, G
2005-01-01
The DIRAC project aims to the design and development of one of the key aspects of the international Facility for Antiproton and Ion Research (FAIR) planned for construction at GSI in Darmstadt, Germany: the broad implementation and optimization of ion storage/cooler rings and of in-ring experimentation with internal targets and secondary beams.
'Parabolic' trapped modes and steered Dirac cones in platonic crystals.
McPhedran, R C; Movchan, A B; Movchan, N V; Brun, M; Smith, M J A
2015-05-08
This paper discusses the properties of flexural waves governed by the biharmonic operator, and propagating in a thin plate pinned at doubly periodic sets of points. The emphases are on the design of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates for a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that these Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.
SU Ting; WANG Zhi-Wei
2010-01-01
@@ By using the generalized version of the dressing method,we consider a Dirac system.The types of nonlinear evo-lution equations discussed involve the integrable variable-coefficient Dirac equation and the defocusing nonlinear Schrodinger equation.As an application,their explicit solutions and Lax pairs are given.
Trogdon, Thomas, E-mail: trogdon@cims.nyu.edu [Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 (United States); Deconinck, Bernard [Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA 98195 (United States)
2014-01-31
All solutions of the Korteweg–de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that are either spatially localized or (quasi-)periodic. In this paper, we discuss a class of solutions that is a nonlinear superposition of these two cases: their asymptotic state for large |x| is (quasi-)periodic, but they may contain solitons, with or without dispersive tails. Such scenarios might occur in the case of localized perturbations of previously present sea swell, for instance. Such solutions have been discussed from an analytical point of view only recently. We numerically demonstrate different features of these solutions.
Trogdon, Thomas; Deconinck, Bernard
2014-01-01
All solutions of the Korteweg-de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that are either spatially localized or (quasi-)periodic. In this paper, we discuss a class of solutions that is a nonlinear superposition of these two cases: their asymptotic state for large |x| is (quasi-)periodic, but they may contain solitons, with or without dispersive tails. Such scenarios might occur in the case of localized perturbations of previously present sea swell, for instance. Such solutions have been discussed from an analytical point of view only recently. We numerically demonstrate different features of these solutions.
Geometrization of the Dirac theory of the electron
Fock, V.
1977-01-01
Using the concept of parallel displacement of a half vector, the Dirac equations are generally written in invariant form. The energy tensor is formed and both the macroscopic and quantum mechanic equations of motion are set up. The former have the usual form: divergence of the energy tensor equals the Lorentz force and the latter are essentially identical with those of the geodesic line.
A non-uniqueness problem of the Dirac theory in a curved spacetime
Arminjon, Mayeul
2009-01-01
The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. We find that, for each among the three versions of the equation, the changes that lead to an equivalent operator H are determined by initial data. Thus, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, whence a lack of uniqueness. In particular, the Dirac energy spectrum is not unique.
Transport Phenomena in Multilayered Massless Dirac Fermion System α-(BEDT-TTF2I3
Naoya Tajima
2012-06-01
Full Text Available A zero-gap state with a Dirac cone type energy dispersion was discovered in an organic conductor α-(BEDT-TTF2I3 under high hydrostatic pressures. This is the first two-dimensional (2D zero-gap state discovered in bulk crystals with a layered structure. In contrast to the case of graphene, the Dirac cone in this system is highly anisotropic. The present system, therefore, provides a new type of massless Dirac fermion system with anisotropic Fermi velocity. This system exhibits remarkable transport phenomena characteristic to electrons on the Dirac cone type energy structure.
Dirac-Point Solitons in Nonlinear Optical Lattices
Xie, Kang; Boardman, Allan D; Guo, Qi; Shi, Zhiwei; Jiang, Haiming; Hu, Zhijia; Zhang, Wei; Mao, Qiuping; Hu, Lei; Yang, Tianyu; Wen, Fei; Wang, Erlei
2015-01-01
The discovery of a new type of solitons occuring in periodic systems without photonic bandgaps is reported. Solitons are nonlinear self-trapped wave packets. They have been extensively studied in many branches of physics. Solitons in periodic systems, which have become the mainstream of soliton research in the past decade, are localized states supported by photonic bandgaps. In this Letter, we report the discovery of a new type of solitons located at the Dirac point beyond photonic bandgaps. The Dirac point is a conical singularity of a photonic band structure where wave motion obeys the famous Dirac equation. These new solitons are sustained by the Dirac point rather than photonic bandgaps, thus provides a sort of advance in conceptual understanding over the traditional gap solitons. Apart from their theoretical impact within soliton theory, they have many potential uses because such solitons have dramatic stability characteristics and are possible in both Kerr material and photorefractive crystals that poss...
Time-dependent massless Dirac fermions in graphene
Khantoul, Boubakeur, E-mail: bobphys@gmail.com [Department of Mathematics, City University London, Northampton Square, London EC1V 0HB (United Kingdom); Department of Physics, University of Jijel, BP 98, Ouled Aissa, 18000 Jijel (Algeria); Fring, Andreas, E-mail: a.fring@city.ac.uk [Department of Mathematics, City University London, Northampton Square, London EC1V 0HB (United Kingdom)
2015-10-30
Using the Lewis–Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasi-particles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a time-dependent vector potential. The eigenvalue equations for the two spinor components of the Lewis–Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians. - Highlights: • An explicit analytical solution for a massless 2+1 dimensional time-dependent Dirac equation is found. • All steps of the Lewis–Riesenfeld method have been carried out.
Koch, Herbert; Vişan, Monica
2014-01-01
The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ide...
陈少伟; 肖利琴
2016-01-01
研究了一类带大参数的周期Thomas-Fermi-Dirac-von Weizs(a)cker方程非零解的存在性问题,运用一个新的无穷维环绕定理,证明了当参数充分大时,该方程存在非零解.
DIRAC Workload Management System
Garonne, V; Stokes-Rees, I
2005-01-01
The Workload Management System is the core component of the DIRAC distributed MC production and analysis grid environment of the CERN LHCb experiment. This paper discusses the architecture, implementation and performance of this system. The WMS is a community scheduler, realizing a pull paradigm, particulary for the high troughput computing context. It has recently been used for an intensive physics simulation production involving more than 60 sites, 65 TB of data, and over 1000-GHz processor-years.
Quantum transport through 3D Dirac materials
Salehi, M. [Department of Physics, Sharif University of Technology, Tehran 11155-9161 (Iran, Islamic Republic of); Jafari, S.A., E-mail: jafari@physics.sharif.edu [Department of Physics, Sharif University of Technology, Tehran 11155-9161 (Iran, Islamic Republic of); Center of Excellence for Complex Systems and Condensed Matter (CSCM), Sharif University of Technology, Tehran 1458889694 (Iran, Islamic Republic of)
2015-08-15
Bismuth and its alloys provide a paradigm to realize three dimensional materials whose low-energy effective theory is given by Dirac equation in 3+1 dimensions. We study the quantum transport properties of three dimensional Dirac materials within the framework of Landauer–Büttiker formalism. Charge carriers in normal metal satisfying the Schrödinger equation, can be split into four-component with appropriate matching conditions at the boundary with the three dimensional Dirac material (3DDM). We calculate the conductance and the Fano factor of an interface separating 3DDM from a normal metal, as well as the conductance through a slab of 3DDM. Under certain circumstances the 3DDM appears transparent to electrons hitting the 3DDM. We find that electrons hitting the metal-3DDM interface from metallic side can enter 3DDM in a reversed spin state as soon as their angle of incidence deviates from the direction perpendicular to interface. However the presence of a second interface completely cancels this effect.
Zero Point Energy and the Dirac Equation
Forouzbakhsh, Farshid
2007-01-01
Zero Point Energy (ZPE) describes the random electromagnetic oscillations that are left in the vacuum after all other energy has been removed. One way to explain this is by means of the uncertainty principle of quantum physics, which implies that it is impossible to have a zero energy condition. I...... this article, the ZPE is explained by using a novel description of the graviton. This is based on the behavior of photons in gravitational field, leading to a new definition of the graviton. In effect, gravitons behave as if they have charge and magnetic effects. These are referred to as negative color charge...
图解法求解Lamb波频散方程%A graphical method to solve a dispersion equation of Lamb wave
许西宁; 余祖俊; 朱力强
2012-01-01
为了简化Lamb波频率方程的求解过程,采用了图解法进行求解.首先划分了Lamb波频散方程的取值区间,并对标准频散方程做了公式修正,运用图解法将抽象公式可视化,结合图形分析,确定了频散方程的求解方式,最终通过二分法求解得到Lamb波频散方程的数值解,绘制了相速度频散曲线,并运用模态跟踪算法,绘制出了群速度频散曲线.图解法可以将复杂抽象的公式用图形表示,简化了求解过程,提高了求解效率.%A graphical method is used to simplify the process of solving a dispersion equation of Lamb wave. First, solution intervals of the Lamb wave dispersion equation are divided and the standard dispersion equation is modified. Next, the graphical method is used to visualize abstract formulas, combined with graphical analysis to determine how to solve dispersion equations. Finally, the dichotomy method is used to acquire the numerical solution of dispersion equation of Lamb wave, and dispersive curves are obtained in terms of phase velocity and frequency. By modes tracking and calculation, the group velocity is acquired. Graphic method can represent complex and abstract formulas graphically, simplifies the solution process and improves the solution efficiency.
Classical behaviour of the Dirac bispinor
Bell, S B M; Díaz, B M; Bell, Sarah B. M.; Cullerne, John P.; Diaz, Bernard M.
2000-01-01
It is usually supposed that the Dirac and radiation equations predict that the phase of a fermion will rotate through half the angle through which the fermion is rotated, which means, via the measured dynamical and geometrical phase factors, that the fermion must have a half-integral spin. We demonstrate that this is not the case and that the identical relativistic quantum mechanics can also be derived with the phase of the fermion rotating through the same angle as does the fermion itself. Under spatial rotation and Lorentz transformation the bispinor transforms as a four-vector like the potential and Dirac current. Previous attempts to provide this form of transformational behaviour have foundered because a satisfactory current could not be derived.(14)
Octonion generalization of Pauli and Dirac matrices
Chanyal, B. C.
2015-10-01
Starting with octonion algebra and its 4 × 4 matrix representation, we have made an attempt to write the extension of Pauli's matrices in terms of division algebra (octonion). The octonion generalization of Pauli's matrices shows the counterpart of Pauli's spin and isospin matrices. In this paper, we also have obtained the relationship between Clifford algebras and the division algebras, i.e. a relation between octonion basis elements with Dirac (gamma), Weyl and Majorana representations. The division algebra structure leads to nice representations of the corresponding Clifford algebras. We have made an attempt to investigate the octonion formulation of Dirac wave equations, conserved current and weak isospin in simple, compact, consistent and manifestly covariant manner.
Photonic crystal surface-emitting lasers enabled by an accidental Dirac point
Chua, Song Liang; Lu, Ling; Soljacic, Marin
2014-12-02
A photonic-crystal surface-emitting laser (PCSEL) includes a gain medium electromagnetically coupled to a photonic crystal whose energy band structure exhibits a Dirac cone of linear dispersion at the center of the photonic crystal's Brillouin zone. This Dirac cone's vertex is called a Dirac point; because it is at the Brillouin zone center, it is called an accidental Dirac point. Tuning the photonic crystal's band structure (e.g., by changing the photonic crystal's dimensions or refractive index) to exhibit an accidental Dirac point increases the photonic crystal's mode spacing by orders of magnitudes and reduces or eliminates the photonic crystal's distributed in-plane feedback. Thus, the photonic crystal can act as a resonator that supports single-mode output from the PCSEL over a larger area than is possible with conventional PCSELs, which have quadratic band edge dispersion. Because output power generally scales with output area, this increase in output area results in higher possible output powers.
Heavy Dirac fermions in a graphene/topological insulator hetero-junction
Cao, Wendong; Zhang, Rui-Xing; Tang, Peizhe; Yang, Gang; Sofo, Jorge; Duan, Wenhui; Liu, Chao-Xing
2016-09-01
The low energy physics of both graphene and surface states of three-dimensional topological insulators (TIs) is described by gapless Dirac fermions with linear dispersion. In this work, we predict the emergence of a ‘heavy’ Dirac fermion in a graphene/TI hetero-junction, where the linear term almost vanishes and the corresponding energy dispersion becomes highly nonlinear. By combining ab initio calculations and an effective low-energy model, we show explicitly how strong hybridization between Dirac fermions in graphene and the surface states of TIs can reduce the Fermi velocity of Dirac fermions. Due to the negligible linear term, interaction effects will be greatly enhanced and can drive ‘heavy’ Dirac fermion states into the half quantum Hall state with non-zero Hall conductance.
Aloisi, A.M.; Nali, P. F.
2016-01-01
In 1931, Dirac advanced a startling prediction about the existence of a new elementary particle, characterized by a magnetic charge of a single polarity: the magnetic monopole. This prediction, that was not based on experimental reasons but on mathematical consistency considerations and the generalization of the formalism of quantum mechanics, illustrates emblematically the Dirac conception of the relationship between physics and mathematics. ----- Nel 1931 Dirac avanz\\`o una sorprendente pre...
Exact analytic solutions for a Dirac electron moving in graphene under magnetic fields.
Kuru, S; Negro, J; Nieto, L M
2009-11-11
Exact analytical solutions for the bound states of a graphene Dirac electron in various magnetic fields with translational symmetry are obtained. In order to solve the time-independent Dirac-Weyl equation the factorization method used in supersymmetric quantum mechanics is adapted to this problem. The behavior of the discrete spectrum, probability and current densities are discussed.
Unified Description of Dirac Electrons on a Curved Surface of Topological Insulators
Takane, Yositake; Imura, Ken-Ichiro
2013-07-01
Existence of a protected surface state described by a massless Dirac equation is a defining property of the topological insulator. Though this statement can be explicitly verified on an idealized flat surface, it remains to be addressed to what extent it could be general. On a curved surface, the surface Dirac equation is modified by the spin connection terms. Here, in the light of the differential geometry, we give a general framework for constructing the surface Dirac equation starting from the Hamiltonian for bulk topological insulators. The obtained unified description clarifies the physical meaning of the spin connection.
Malkov, M A
1996-01-01
The asymptotic travelling wave solution of the KdV-Burgers equation driven by the long scale periodic driver is constructed. The solution represents a shock-train in which the quasi-periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions. It is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes. Two types of chaotic behavior are considered. The first type is a weak chaos, where only a small chaotic deviation from the periodic solution occurs. The second type corresponds to the developed chaos where the solution ``ignores'' the driver period and represents a random sequence of uncorrelated shocks. In the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor, while in the case of developed chaos it moves on a repellor. Both solutions depend on a parameter indicating the reference shock position in the shock-train. The structure...
J.-S. Chen
2011-04-01
Full Text Available This study presents a generalized analytical solution for one-dimensional solute transport in finite spatial domain subject to arbitrary time-dependent inlet boundary condition. The governing equation includes terms accounting for advection, hydrodynamic dispersion, linear equilibrium sorption and first order decay processes. The generalized analytical solution is derived by using the Laplace transform with respect to time and the generalized integral transform technique with respect to the spatial coordinate. Several special cases are presented and compared to illustrate the robustness of the derived generalized analytical solution. Result shows an excellent agreement. The analytical solutions of the special cases derived in this study have practical applications. Moreover, the derived generalized solution which consists an integral representation is evaluated by the numerical integration to extend its usage. The developed generalized solution offers a convenient tool for further development of analytical solution of specified time-dependent inlet boundary conditions or numerical evaluation of the concentration field for arbitrary time-dependent inlet boundary problem.
Deng, Guo; Biondini, Gino; Trillo, Stefano
2016-10-01
We study the small dispersion limit of the Korteweg-de Vries (KdV) equation with periodic boundary conditions and we apply the results to the Zabusky-Kruskal experiment. In particular, we employ a WKB approximation for the solution of the scattering problem for the KdV equation [i.e., the time-independent Schrödinger equation] to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem. We then perform a detailed analysis of the structure of said spectrum (i.e., band widths, gap widths and relative band widths) as a function of the dispersion smallness parameter ɛ. We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of ɛ. Finally, by performing an appropriate rescaling, we compare our results to those in the famous Zabusky and Kruskal's paper, showing very good agreement with the numerical results.
Production of Dirac particle in twisted Minkowsky space-time
Samary, Dine Ousmane; Kanfon, Antonin
2015-01-01
In this paper we study the Dirac equation interacting with external gravitation field. This curve background, which correspond to the deformation of Minkowsky space-time is described with the tetrad of the form $e_b^\\mu(x)=\\varepsilon(\\delta_b^\\mu+\\omega_{ba}^\\mu x^a)$, where $\\varepsilon=1$ for $\\mu=0$ and $\\varepsilon=i$ for $\\mu=1,2,3.$ Using separation of variables the corresponding Dirac equation is solved. The probability density of the vacuum-vacuum pair creation is given. In particular case of vanishing electromagnetic fields, we point out that, this external gravitation field modify weakly the well know probability of pair production of the Dirac particle given in ordinary space-time.