Ma-Xu quantization rule and exact WKB condition for translationally shape invariant potentials

TL;DR

This paper demonstrates that the Ma-Xu quantization rule for translationally shape invariant potentials can be derived directly from the exact WKB condition, providing a simplified calculation method for these quantum systems.

Contribution

It introduces a straightforward alternative approach to compute corrections for all translationally shape invariant potentials, linking Ma-Xu rule with the exact WKB condition.

Findings

Ma-Xu rule is a consequence of the exact WKB condition.

A direct method to calculate corrections for shape invariant potentials.

Simplifies the analysis of quantum systems with shape invariance.

Abstract

For translationally shape invariant potentials, the exact quantization rule proposed by Ma and Xu is a direct consequence of exactness of the modified WKB quantization condition proved by Barclay. We propose here a very direct alternative way to calculate the appropriate correction for the whole class of translationally shape invariant potentials.