On the Existence of Additional (Hydrino) states in the Dirac equation

TL;DR

This paper investigates additional solutions in the Dirac equation for electrons under Coulomb potential, concluding that such solutions are physically inadmissible due to issues with probability conservation and orthogonality.

Contribution

It extends the analysis of singular solutions from the Klein-Gordon equation to the Dirac equation, showing these solutions are non-physical in the relativistic case.

Findings

Additional solutions exist but are non-physical

Singularity rate is higher for spin-1/2 particles

Standard solutions are the physically valid ones

Abstract

In case of spinless particles there appear additional (singular) solutions in the framework of relativistic Klein-Gordon equation for Coulomb potential. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states (hydrino, small hydrogen) should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle (electron) in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions.