Second-order Supersymmetric Operators and Excited States

TL;DR

This paper extends supersymmetric quantum mechanics to construct raising and lowering operators for harmonic oscillators and their partners, providing explicit formulas and avoiding complex general methods.

Contribution

It introduces a method using double supersymmetry to generate non-singular isospectral potentials and explicit ladder operators, simplifying previous approaches.

Findings

Explicit ladder operators for supersymmetric partner potentials

Construction of non-singular isospectral potentials

Simplified method avoiding complex general approaches

Abstract

Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well-developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The use of double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wave functions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.