Supersymmetry in Quantum Mechanics

TL;DR

This paper introduces supersymmetry in quantum mechanics, demonstrating how it generates new solvable potentials, explores their spectral properties, and extends methods for solving quantum systems, including shape invariant and periodic potentials.

Contribution

It provides explicit constructions of new exactly solvable potentials using supersymmetry, extending the operator method and introducing reflectionless and periodic potentials.

Findings

Constructed new exactly solvable potentials with fewer bound states.

Derived relationships between eigenvalues, eigenfunctions, and scattering matrices.

Showed the SWKB approximation is exact for shape invariant potentials.

Abstract

An elementary introduction is given to the subject of Supersymmetry in Quantum Mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct new exactly solvable n Hamiltonians having n-1,n-2,...,0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentials called shape invariant potentials. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s-matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape invariant potentials. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.