Paraxial Dirac equation

TL;DR

This paper explores the paraxial approximation of the free Dirac equation, deriving solutions for various beam types using superpositions and direct methods, under standard paraxial assumptions.

Contribution

It introduces a systematic derivation of the paraxial Dirac equation and solutions for Gaussian, Bessel-Gaussian, modified Bessel-Gaussian, and Laguerre-Gaussian beams.

Findings

Wave functions match previous superposition results

Solutions are valid under standard paraxial approximation

Provides explicit forms for different beam profiles

Abstract

In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms, that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.