Equivalent power law potentials

TL;DR

This paper demonstrates a mathematical mapping between different power law potentials in quantum mechanics, revealing a general correspondence between their spectra and classical orbits, extending the known Coulomb-oscillator relationship.

Contribution

It introduces a general transformation linking power law potentials and their spectra, broadening the understanding of spectral correspondences beyond Coulomb and oscillator cases.

Findings

Spectral mapping between related power law potentials.

Classical orbit correspondence in related potentials.

Extension of Coulomb-oscillator spectral relationship.

Abstract

It is shown that the radial Schroedinger equation for a power law potential and a particular angular momentum may be transformed using a change of variable into another Schroedinger equation for a different power law potential and a different angular momentum. It is shown that this leads to a mapping of the spectra of the two related power law potentials. It is shown that a similar correspondence between the classical orbits in the two related power law potentials exists. The well known correspondence of the Coulomb and oscillator spectra is a special case of a more general correspondence between power law potentials.