Quasi-classical path integral approach to supersymmetric quantum mechanics

TL;DR

This paper develops a quasi-classical path integral method for supersymmetric quantum mechanics, deriving new quantization rules that improve energy spectrum estimates, especially in broken SUSY cases, compared to traditional WKB methods.

Contribution

It introduces a novel quasi-classical quantization approach for SUSY-QM, including a new formula for broken SUSY scenarios, enhancing spectral calculation accuracy.

Findings

New SUSY path integral quantization rules derived.

The broken SUSY formula often overestimates energy levels.

The method can outperform WKB in certain bound state problems.

Abstract

{}From Feynman's path integral, we derive quasi-classical quantization rules in supersymmetric quantum mechanics (SUSY-QM). First, we derive a SUSY counterpart of Gutzwiller's formula, from which we obtain the quantization rule of Comtet, Bandrauk and Campbell when SUSY is good. When SUSY is broken, we arrive at a new quantization formula, which is found as good as and even sometime better than the WKB formula in evaluating energy spectra for certain one-dimensional bound state problems. The wave functions in the stationary phase approximation are also derived for SUSY and broken SUSY cases. Insofar as a broken SUSY case is concerned, there are strong indications that the new quasi-classical approximation formula always overestimates the energy eigenvalues while WKB always underestimates.