Relativistic Approximate Solutions for a Two-Term Potential: Riemann-Type Equation

TL;DR

This paper derives approximate analytical solutions for relativistic wave equations with a two-term potential using a Riemann-type equation, providing energy eigenvalues and wave functions applicable to various special cases.

Contribution

It introduces a novel method to solve relativistic wave equations with a two-term potential via a Riemann-type equation, expanding analytical solutions to include special cases like Coulomb and Manning-Rosen potentials.

Findings

Derived energy eigenvalue equations for Klein-Gordon and Dirac equations.

Obtained normalized wave functions in terms of hypergeometric functions.

Unified approach applicable to multiple potential types.

Abstract

Approximate analytical solutions of a two-term potential are studied for the relativistic wave equations, namely, for the Klein-Gordon and Dirac equations. The results are obtained by solving of a Riemann-type equation whose solution can be written in terms of hypergeometric function $\,_{2}F_{1}(a,b;c;z)$. The energy eigenvalue equations and the corresponding normalized wave functions are given both for two wave equations. The results for some special cases including the Manning-Rosen potential, the Hulth\'{e}n potential and the Coulomb potential are also discussed by setting the parameters as required.