Dirac and Klein-Gordon equations with equal scalar and vector potentials

TL;DR

This paper analyzes the Dirac and Klein-Gordon equations with equal scalar and vector potentials, revealing that their relativistic spectra differ from nonrelativistic counterparts and emphasizing the importance of correct relativistic extensions.

Contribution

It provides a detailed study of relativistic equations with noncentral potentials, showing the non-uniqueness of relativistic extensions despite identical nonrelativistic limits.

Findings

Relativistic spectra differ from nonrelativistic counterparts for equal scalar and vector potentials.

Nonrelativistic limit is well-defined, but relativistic extension is not unique.

Correct relativistic extensions are identified for spherically symmetric exponential potentials.

Abstract

We study the three-dimensional Dirac and Klein-Gordon equations with scalar and vector potentials of equal magnitudes as an attempt to give a proper physical interpretation of this class of problems which has recently been accumulating interest. We consider a large class of these problems in which the potentials are noncentral (angular-dependent) such that the equations separate completely in spherical coordinates. The relativistic energy spectra are obtained and shown to differ from those of well-known problems that have the same nonrelativistic limit. Consequently, such problems should not be misinterpreted as the relativistic extension of the given potentials despite the fact that the nonrelativistic limit is the same. The Coulomb, Oscillator and Hartmann potentials are considered. This shows that although the nonrelativistic limit is well-defined and unique, the relativistic extension is not. Additionally, we investigate the Klein-Gordon equation with uneven mix of potentials leading to the correct relativistic extension. We consider the case of spherically symmetric exponential-type potentials resulting in the s-wave Klein-Gordon-Morse problem.