Dirac reduced radial equations and the problem of additional solutions

TL;DR

This paper investigates the Dirac radial equations, demonstrating that additional solutions are unphysical and should be discarded, and highlights the restrictions imposed by distribution theory on the use of reduced equations in quantum mechanics.

Contribution

It clarifies the conditions under which reduced radial equations in the Dirac framework are valid, emphasizing the removal of unphysical solutions and the role of boundary conditions.

Findings

Additional solutions in Dirac radial equations are unphysical.

Distribution theory restricts the use of reduced equations for all potentials.

Additional solutions do not survive in the two-dimensional Dirac equation under physical constraints.

Abstract

We show that additional solutions must be ignored (in differences of the Schrodinger and Klein-Gordon equations) in the Dirac equation, where usually passed the second order radial equation, called the reduced equation, instead of a system. Analogously to the Schrodinger equation, in this process the Dirac delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survives as a result of usual physical requirements.