Shape invariance and the exactness of quantum Hamilton-Jacobi formalism

TL;DR

This paper demonstrates that shape invariance, a key concept in supersymmetric quantum mechanics, can also be used to determine eigenvalues within the quantum Hamilton-Jacobi formalism, providing an alternative analytical approach.

Contribution

It establishes that shape invariance is sufficient for eigenvalue determination in quantum Hamilton-Jacobi theory, extending its applicability beyond SUSYQM.

Findings

Shape invariance enables eigenvalue calculation in quantum Hamilton-Jacobi formalism.

The method provides an iterative algorithm for quantum momentum functions.

Shape invariance's role in eigenvalue determination is generalized beyond SUSYQM.

Abstract

Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schr\"odinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory.