Established pseudo solution of second-order Dirac-Coulomb equation with position-dependent mass

TL;DR

This paper clarifies the correct solutions to the second-order Dirac-Coulomb equation with position-dependent mass, highlighting previous inaccuracies and emphasizing the lack of experimental validation for the concept.

Contribution

It provides the correct solutions to the Dirac-Coulomb equation with position-dependent mass, correcting prior misconceptions and clarifying the mathematical derivation.

Findings

Previous solutions were pseudo and not mathematically rigorous

The correct solutions are derived from the original first-order Dirac equations

No experimental data currently supports the position-dependent mass hypothesis

Abstract

We show that one of exact solutions of the second-order Dirac-Coulomb equation were pseudo. In the corresponding original literature, it was considered that the mass of the electron with a Coulomb potential was position-dependent, but the obtained eigenvalues set was not the inevitable mathematical deduction of the given second-order Dirac equation, and the second-order Dirac equations were not the inevitable mathematical deduction of the given couplet first-order Dirac equation with the position-dependent mass of the electron. In the present paper, we obtain the correct solution of the introduced first-order differential equations. This new solution would be tenable only when the wave equation is correct, but there is not any experiment date to validate the so-called position-dependent of the electron in the Coulomb field.