Exactness of Bohr-Sommerfeld quantisation for two non-central potentials

TL;DR

This paper proves that the Bohr-Sommerfeld quantisation method precisely matches the quantum bound state spectra for two specific non-central potentials, with one potential's analysis being novel.

Contribution

It demonstrates the exactness of Bohr-Sommerfeld quantisation for two non-central potentials, including a new analysis for one potential in spherical coordinates.

Findings

Bohr-Sommerfeld quantisation reproduces quantum spectra exactly.

Complete solutions for classically bound orbits in both potentials.

One potential's analysis is original and previously unexamined.

Abstract

In this paper we demonstrate the integrability of the Hamilton-Jacobi equation for two non-central potentials in spherical polar coordinates, and present complete solutions for the classically bound orbits. We then show that the semiclassical method of Bohr-Sommerfeld quantisation exactly reproduces the bound state spectra of the corresponding quantum mechanical Schr\"odinger equations. One of these potentials has previously been analysed in parabolic coordinates; the results for the other are, to the authors' best knowledge, original.