Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions

Abstract

We construct the relationship between the radial equations of the $d$-dimensional hydrogen atom and the $D$-dimensional simple harmonic oscillator. The supersymmetric partners of each of these systems are obtained, and a series of maps between the various systems is delineated. We present an explicit physical example of our maps. Finally, we generalize to D dimensions the recent work of Balantekin on the supersymmetric oscillator with spin-orbit coupling, and we demonstrate that this supersymmetry is different from the one we study.