Bound states of a uniform spherical charge distribution—revisited!

Abstract

In this note, we revisit modeling the nucleus as a uniformly charged sphere in order to examine finite nuclear size effects on atomic binding energies. We use nonrelativistic quantum mechanics to extract binding energies of a negatively charged lepton bound by a uniform-sphere potential. Energies are determined by using MATHEMATICA to match interior and exterior solutions at the nuclear radius. Muonic lead and tin binding energies are found, as are their fine-structure transition energies, and compared with experimental data. In the course of our reinvestigation of this problem, we came upon several subtle features of using infinite (especially asymptotic) series for numerical evaluation. Effective handling of these features is made practical both by using certain analytic transformations and the powerful computer programs currently available. We believe that these technical observations are of value to a wider class of problems beyond the ones considered in detail in this paper.