Something new about radial wave functions of fermions in the repulsive Coulomb field

TL;DR

This paper introduces new solutions to the Dirac equation in a repulsive Coulomb field by applying a boundary condition that excludes the wave function inside a certain radius, showing these solutions closely resemble standard Coulomb functions.

Contribution

It proposes a novel boundary condition approach for the Dirac equation in Coulomb fields, leading to new solutions that align with standard functions at certain distances.

Findings

New solutions nearly match standard Coulomb functions at specific distances

Matrix elements with new solutions are similar to standard matrix elements

Methodological approach aids in quantum theory development

Abstract

An impermeable barrier at $r=r_{cl}$ in the effective potential of the relativistic Schr\"odinger-type equation leads to exclusion of the range $0 \leq r < r_{cl}$ from the wave function domain. Based on duality of the Schr\"odinger-type equation and the Dirac equation, a similar exclusion should be made in the wave functions domain of the Dirac equation. As a result, we obtain new solutions to the Dirac equation in the Coulomb repulsive field. Calculations show that depending on working parameters, at distances of several fractions or units of the Compton wavelength of the fermion from $r=r_{cl}$ new solutions almost coincide with the standard Coulomb functions of the continuous spectrum. Practically, matrix elements with new solutions will coincide to a good accuracy with standard matrix elements with the Coulomb functions of the continuous spectrum. Our consideration is methodological and helpful for discussing further development of quantum theory.