An Alternate Approach to Transition Potentials

TL;DR

This paper uses supersymmetric quantum mechanics to analyze transition potentials with singular behavior, providing a natural method to determine the spectrum and justify conventional choices in quantum systems with such potentials.

Contribution

It introduces a SUSYQM-based approach to uniquely determine spectra for transition potentials, clarifying the selection of physically relevant wave functions.

Findings

SUSYQM offers a natural prescription for spectrum determination.

The approach reproduces the conventional 'less singular' wave functions.

Illustrative examples include P"oschl-Teller II and a two-anyon system.

Abstract

We analyze transition potentials $(V(r) \stackrel{r\sim 0}{\rightarrow} {\alpha r^{-2}})$ in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range $-1/4 < \alpha < 3/4$, the eigenvalue problem becomes ill-defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM based approach picks out the same "less singular" wave functions as the conventional approach, and thus provides a simple justification for the usual practice in the literature. Two examples (the P\"oschl-Teller II potential and a two anyon system on the plane) have been worked out for illustrative purposes.