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Journal of Physics A Mathematical and General

Coherent states of a relativistic particle in an external electromagnetic field

Vladislav G. Bagrov, I. L. Buchbinder, Dmitri M Gitman

Published Nov 1, 1976• 21 citations
Publisher

Abstract

The problem of the construction of coherent states of relativistic particles (bosons and fermions) in electromagnetic fields has been solved. The states discovered are exact solutions to the Klein-Gordon and Dirac equations; the wavefunctions of the particles are found in the homogeneous magnetic and electric fields and the Redmond configuration field, as well as in the plane-wave field. The formulation of relativistic quantum mechanics is used on the 'null' plane.

DOI
10.1088/0305-4470/9/11/021

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The problem of the construction of coherent states of relativistic particles (bosons and fermions) in electromagnetic fields has been solved. The states discovered are exact solutions to the Klein-Gordon and Dirac equations; the wavefunctions of the particles are found in the homogeneous magnetic and electric fields and the Redmond configuration field, as well as in the plane-wave field. The formulation of relativistic quantum mechanics is used on the 'null' plane.
Abstract Some general properties of coherent states of a particle in quantum mechanics are discussed and examples of relativistics coherent states of an electron moving in a steady and homogeneous field and in a non‐stationary external field of special kind are presented. In particular cases this field may be a plane wave, a Redmond field (a superposition of plane wave and homogenous magnetic field), a superposition of a Redmond field and a longitudinal electric field or other physically interesting fields.
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.
We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein–Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov–Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special nonuniform fields. The solutions obtained can be useful to study the Aharonov–Bohm effect in the corresponding electromagnetic fields.
We use the formulation of the quantum mechanics of first quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits. 1
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.