Klein-Gordon equation for a charged particle in space varying electromagnetic fields-A systematic study via Laplace transform

TL;DR

This paper systematically derives exact solutions to the Klein-Gordon equation for a charged particle in three types of spatially varying electromagnetic fields using Laplace transforms, expressing solutions with hypergeometric and Bessel functions.

Contribution

It introduces a systematic Laplace transform method to solve the Klein-Gordon equation in complex, spatially varying electromagnetic fields, providing explicit solutions in terms of special functions.

Findings

Exact solutions for three electromagnetic field configurations

Solutions expressed via hypergeometric and Bessel functions

Method applicable to differential equations with linear coefficients

Abstract

Exact solutions of the Klein-Gordon equation for a charged particle in the presence of three spatially varying electromagnetic fields, namely, (i) $\vec{E}=\alpha\beta_0e^{-\alpha x_2}\hat{x}_2$, $\vec{B}=\alpha\beta_1e^{-\alpha x_2}\hat{x}_3$ (ii) $\vec{E}=\frac{\beta_0^{'}}{x_2^2}\hat{x}_2$, $\vec{B}=\frac{\beta_1^{'}}{x_2^2}\hat{x}_3$, and (iii) $\vec{E}=\frac{2\beta_0^{'}}{x_2^3}\hat{x}_2$, $\vec{B}=\frac{2\beta_1^{'}}{x_2^3}\hat{x}_3$, are studied. All these fields are generated from a systematic study of a particular type of differential equation whose coefficients are linear in independent variable. The Laplace transform approach is used to find the solutions and the corresponding eigenfunctions are expressed in terms of the hypergeometric functions $\,_{1}F_{1}(a', b'; x)$ for first two cases of the above configurations while the same are expressed in terms of the Bessel functions of first kind, $J_{n}(x)$, for the last case