Construction of Dirac spinors for electron vortex beams in background electromagnetic fields

TL;DR

This paper introduces a novel method using geometric algebras to construct exact Dirac spinor solutions for electron vortex beams in various electromagnetic fields, addressing a gap in solutions for time-dependent quantum systems.

Contribution

It develops a new approach for deriving exact Dirac solutions with defined angular momentum, including non-stationary cases involving complex electromagnetic fields.

Findings

Derived stationary solutions for free electrons and magnetic fields

Developed new non-stationary solutions with electromagnetic waves and inhomogeneous magnetic fields

Provided insights into the dynamics and self-consistent fields in these configurations

Abstract

Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics exists. Given the growing number of applications of high energy electron beams interacting with a variety of quantum systems in laser fields, novel methods for finding exact solutions to the Dirac equation are called for. We present a method for building up solutions to the Dirac equation employing a recently introduced approach for the description of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. We illustrate the method by developing several stationary as well as non-stationary solutions of the Dirac equation with well defined orbital angular momentum along the electron's propagation direction. The first set of solutions describe free electron beams in terms of Bessel functions as well as stationary solutions for both a homogeneous and an inhomogeneous magnetic field. The second set of solutions are new and involve a plane electromagnetic wave combined with a generally inhomogeneous longitudinal magnetic field. Moreover, the developed technique allows us to derive general physical properties of the dynamics in such field configurations, as well as provides physical predictions on the self-consistent electromagnetic fields induced by the dynamics.