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arXiv:2010.09844 · Physica Scripta

Degenerate solutions to the Dirac equation for massive particles and their applications in quantum tunneling

G. Tsigaridas, Aristides I. Kechriniotis, Christos Tsonos, Konstantinos K. Delibasis

Published Mar 29, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract In a recent work we have proven the existence of degenerate solutions to the Dirac equation, corresponding to an infinite number of different electromagnetic fields, providing also some examples regarding massless particles. In the present article our results are extended significantly, providing degenerate solutions to the Dirac equation for particles with arbitrary mass, which, under certain conditions, could be interpreted as pairs of particles (or antiparticles) moving in a potential barrier with energy equal to the height of the barrier and spin opposite to each other. We calculate the electromagnetic fields corresponding to these solutions, providing also some examples regarding both spatially constant electromagnetic fields and electromagnetic waves. Further, we discuss some potential applications of our work, mainly regarding the control of the particles outside the potential barrier, without affecting their state inside the barrier. Finally, we study the effect of small perturbations to the degenerate solutions, showing that our results are still valid, in an approximate sense, provided that the amplitude of the electromagnetic fields corresponding to the exact degenerate solutions is sufficiently small.

DOI
10.1088/1402-4896/abf304

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Open Physics•Jan 1, 2017•1 citation
Abstract In a recent work we have proven the existence of degenerate solutions to the Dirac equation, corresponding to an infinite number of different electromagnetic fields, providing also some examples regarding massless particles. In the present article our results are extended significantly, providing degenerate solutions to the Dirac equation for particles with arbitrary mass, which, under certain conditions, could be interpreted as pairs of particles (or antiparticles) moving in a potential barrier with energy equal to the height of the barrier and spin opposite to each other. We calculate the electromagnetic fields corresponding to these solutions, providing also some examples regarding both spatially constant electromagnetic fields and electromagnetic waves. Further, we discuss some potential applications of our work, mainly regarding the control of the particles outside the potential barrier, without affecting their state inside the barrier. Finally, we study the effect of small perturbations to the degenerate solutions, showing that our results are still valid, in an approximate sense, provided that the amplitude of the electromagnetic fields corresponding to the exact degenerate solutions is sufficiently small.
In this article, we present a novel class of degenerate solutions to the massless Dirac equation, corresponding to a wide variety of electromagnetic 4-potentials and fields, including both zero field and circularly polarized electromagnetic waves. An interesting property of these solutions is that the spin of the particles rotates in synchronization with the electric and magnetic fields of the electromagnetic waves. These results could be utilized for the development of optical memories based on materials supporting massless Dirac fermions, such as graphene.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.
This paper investigates the problem of a relativistic Dirac half-integer spin free particle tunneling through a rectangular quantum-mechanical barrier. If the energy difference between the barrier and the particle is positive, and the barrier width is large enough, there is proof that the tunneling is always superluminal. For antiparticle states, the tunneling may be either subluminal or superluminal instead, depending on the barrier width. These results derive from studying the tunneling time in terms of phase time. For particle states these are always negatives while for antiparticle states they are always positives, whatever the height and width of the barrier. The scattering also leads to an anomalous distortion of the Dirac spinor that tends to disappear as the particle velocity approaches the speed of light. Moreover, the phase time tends to zero, increasing the potential barrier both for particle and antiparticle states. This agrees with the interpretation of quantum tunneling that the Heisenberg uncertainty principle provides. This study’s results are innovative with respect to those available in the literature. Moreover, they show that the superluminal behaviour of particles occurs in those processes with high-energy confinement.
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.
Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the interaction constant of a particle with a field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion and the expression for the force acting on the charge during motion in an external electromagnetic field are derived. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, problems for a particle in a potential well, and penetration of a particle through a potential barrier, as well as problem of a hydrogen atom are presented.
Abstract This paper investigates the problem of a relativistic Dirac half-integer spin free particle tunneling through a rectangular quantum-mechanical barrier. If the energy difference between the barrier and the particle is positive, and the barrier width is large enough, there is proof that the tunneling may be superluminal. For first spinor components of particle and antiparticle states, the tunneling is always superluminal regardless the barrier width. Conversely, the second spinor components of particle and antiparticle states may be either subluminal or superluminal depending on the barrier width. These results derive from studying the tunneling time in terms of phase time. For the first spinor components of particle and antiparticle states, it is always negative while for the second spinor components of particle and antiparticle states, it is always positive, whatever the height and width of the barrier. In total, the tunneling time always remains positive for particle states while it becomes negative for antiparticle ones. Furthermore, the phase time tends to zero, increasing the potential barrier both for particle and antiparticle states. This agrees with the interpretation of quantum tunneling that the Heisenberg uncertainty principle provides. This study’s results are innovative with respect to those available in the literature. Moreover, they show that the superluminal behaviour of particles occurs in those processes with high-energy confinement.