Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics

TL;DR

This paper introduces two supersymmetry-based methods to analyze two-dimensional quantum models that resist standard separation of variables, enabling spectrum and wave function determination through shape invariance and partner Hamiltonian relations.

Contribution

It presents novel supersymmetry-inspired techniques for separating variables in complex 2D quantum systems, expanding analytical tools beyond traditional methods.

Findings

Successfully applied methods to a 2D Morse potential example.

Derived parts of the spectrum using supercharge separation.

Obtained full spectrum and wave functions for partner Hamiltonians.

Abstract

Two different approaches are formulated to analyze two-dimensional quantum models which are not amenable to standard separation of variables. Both methods are essentially based on supersymmetrical second order intertwining relations and shape invariance - two main ingredients of the supersymmetrical quantum mechanics. The first method explores the opportunity to separate variables in the supercharge, and it allows to find a part of spectrum of the Schr\"odinger Hamiltonian. The second method works when the standard separation of variables procedure can be applied for one of the partner Hamiltonians. Then the spectrum and wave functions of the second partner can be found. Both methods are illustrated by the example of two-dimensional generalization of Morse potential for different values of parameters.