A class of exactly solvable models to illustrate supersymmetry and test approximation schemes in quantum mechanics

TL;DR

This paper introduces a family of exactly solvable quantum potentials, explores their supersymmetric partners, and uses these solutions to evaluate the accuracy of various approximation methods in quantum mechanics.

Contribution

It derives analytical solutions for a new class of exponential potentials and demonstrates their use in testing approximation schemes like variational and semiclassical methods.

Findings

Exact eigenvalues and eigenstates for FPWEF potentials

Validation of semiclassical quantization formulas against exact solutions

Insights into the applicability of approximation schemes in quantum mechanics

Abstract

We derive the analytical eigenvalues and eigenstates of a family of potentials wells with exponential form (FPWEF). We provide a brief summary of the supersymmetry formalism applied to quantum mechanics and illustrate it by producing from the FPWEF another class of exact solutions made of their isospectral partners. Interestingly, a subset of the supersymmetric partners provides a class of exactly solvable double well potentials. We use the exact solutions of the FPWEF to test the robustness and accuracy of different approximation schemes. We determine (i) the ground state through variational method applied to an approriate set of trial functions and (ii) the whole spectrum using three semiclassical quantization formula: the WKB, JWKB and its supersymmetric extension, the SWKB quantization formula. We comment on the importance of Maslov index and on the range of validity of these different semiclassical approaches.