Approximate Solutions of Dirac Equation with Hyperbolic-type Potential

TL;DR

This paper presents an approximate method to compute energy eigenvalues of a Dirac particle in a hyperbolic-type potential using confluent Heun functions and numerical zero-finding techniques.

Contribution

It introduces a novel approach to solving the Dirac equation with hyperbolic potentials by deriving a transcendental energy function involving confluent Heun functions.

Findings

Derived a transcendental energy function $\\mathcal{F}(E)$ for the Dirac equation

Numerically obtained energy eigenvalues by zero-finding on complex functions

Provided approximate solutions for specific hyperbolic-type potentials

Abstract

The energy eigenvalues of a Dirac particle for the hyperbolic-type potential field have been computed approximately. It is obtained a transcendental function of energy, $\mathcal{F}(E)$, by writing in terms of confluent Heun functions. The numerical values of energy are then obtained by fixing the zeros on "$E$-axis" for both complex functions $Re[\mathcal{F}(E)]$ and $Im[\mathcal{F}(E)]$.