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Proceedings of the YSU A Physical and Mathematical Sciences

INVARIANT SOLUTION OF THE DIRAC EQUATION IN THE CROSSED ELECTRIC AND MAGNETIC FIELDS $ (H \mathclose{>} E) $

R.G. Petrosyan, M.A. Davtyan

Published Aug 15, 2019• 0 citations
Publisher

Abstract

In the article exact analytical and invariant solutions for both spinless and half-spin relativistic charged particles in crossed constant electric and magnetic fields, when $ H \mathclose{>} E $ have been found. It is shown that in both cases the problem reduces to that of quantum harmonic oscillator.

DOI
10.46991/pysu:a/2019.53.2.138

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In the article exact analytical and invariant solutions for both spinless and half-spin relativistic charged particles in crossed constant electric and magnetic fields, when H>EH \mathclose{>} EH>E have been found. It is shown that in both cases the problem reduces to that of quantum harmonic oscillator.
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.