Non-Central Potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance

TL;DR

This paper demonstrates how supersymmetric quantum mechanics and shape invariance simplify the derivation of spherical harmonics and analyze bound states in non-central potentials like Coulomb, Aharonov-Bohm, and monopole fields.

Contribution

It introduces operator methods from supersymmetry to derive properties of spherical harmonics and solve non-central potential problems more straightforwardly.

Findings

Simplified derivation of spherical harmonics properties.

Analysis of bound states in Coulomb plus Aharonov-Bohm and monopole fields.

Application of supersymmetric techniques to non-central potentials.

Abstract

It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with non-central vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.