The Coulomb potential in quantum mechanics revisited

TL;DR

This paper revisits the quantum Coulomb problem, highlighting the importance of boundary conditions and non-standard solutions, especially when considering a finite nuclear size, to ensure a more consistent approach.

Contribution

It demonstrates that standard methods overlook boundary condition symmetry and emphasizes the significance of non-standard solutions in Coulomb quantum mechanics.

Findings

Standard solutions are not symmetric in boundary conditions.

Finite nuclear size affects Coulomb eigenstates.

Non-standard solutions are essential for a consistent approach.

Abstract

The procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at $r=0$ and $r \rightarrow \infty$ are considered. We highlight this fact by solving a model for the Coulomb potential with a cutoff (representing the finite extent of the nucleus); in the limit that the cutoff is reduced to zero we recover the standard result, albeit in a non-standard way. This example is used to emphasize that a more consistent approach to solving the Coulomb problem in quantum mechanics requires an examination of the non-standard solution. The end result is, of course, the same.