The Coulomb potential in quantum mechanics revisited

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 → ∞ 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.