Unheeded pseudo solution of Dirac-Coulomb equations with an indirect transformation of functions

TL;DR

This paper critiques a published incorrect solution to the Dirac-Coulomb equations, demonstrating that the transformation used leads to pseudo solutions and invalid energy eigenvalues due to mathematical inconsistencies.

Contribution

It exposes the mathematical errors in the previous solution and clarifies that the transformation results in non-physical pseudo solutions and eigenvalues.

Findings

The transformation yields inconsistent energy eigenvalues.

The second-order equations have no valid solutions.

The original solutions are pseudo solutions.

Abstract

We open out one of incorrect solutions of the Driac equation in the Coulomb field given in a published paper. By introducing a transformation of function, the paper transformed the original radial first-order Dirac-Coulomb equation into two second-order Dirac-Coulomb equation. However, each of the second-order differential equations has differential energy eigenvalues set. The original paper wrote the two differential equations into one of forms, and then gave the distinguished energy eigenvalues. The mathematical procedure is not correct. For the same quantum system, introducing a transformation of function yields two different energy eigenvaluse, the result violates the uniqueness of solution. It actually shows that the given second-order differential equations have no solution. On the other hand, the given formal solutions of the second-order Dirac-Coulomb equations violate the conditions for determining solution. Consequently, the solutions given by the author are pseudo solution, and the corresponding energy eigenvalues set is also a pseudo eigenvalues set.