Ab Initio Method for Obtaining Exactly Solvable Quantum Mechanical Potentials

TL;DR

This paper introduces a novel ab initio method for deriving exactly solvable quantum potentials by transforming the shape invariance condition into a nonlinear PDE, enabling systematic generation of superpotentials without trial and error.

Contribution

It presents the first ab initio approach to generate shape invariant superpotentials by converting the shape invariance condition into a nonlinear PDE.

Findings

The difference-differential equation is equivalent to a nonlinear PDE.

The method allows systematic generation of shape invariant superpotentials.

It eliminates the need for trial and error in finding solvable potentials.

Abstract

The shape invariance condition is the integrability condition in supersymmetric quantum mechanics (SUSYQM). It is a difference-differential equation connecting the superpotential W and its derivative at two different values of parameters. We show that this difference equation is equivalent to a non-linear partial differential equation whose solutions are translational shape invariant superpotentials. In lieu of trial and error, this method provides the first ab initio technique for generating shape invariant superpotentials.