Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential

TL;DR

This paper explores classical and quantum systems with generalized Heisenberg algebras, applying the framework to infinite well and Morse potentials, and explains degeneracies via ladder operators and supersymmetry.

Contribution

It introduces a generalized algebraic approach to analyze degeneracies and symmetries in 2D quantum systems, linking ladder operators to supersymmetric models.

Findings

Degeneracies explained by permutation symmetry and ladder operators.

Products of ladder operators satisfy unique commutation relations.

Two-dimensional Morse system linked to a generalized supersymmetric model.

Abstract

We consider classical and quantum one and two-dimensional systems with ladder operators that satisfy generalized Heisenberg algebras. In the classical case, this construction is related to the existence of closed trajectories. In particular, we apply these results to the infinite well and Morse potentials. We discuss how the degeneracies of the permutation symmetry of quantum two-dimensional systems can be explained using products of ladder operators. These products satisfy interesting commutation relations. The two-dimensional Morse quantum system is also related to a generalized two-dimensional Morse supersymmetric model. Arithmetical or accidental degeneracies of such system are shown to be associated to additional supersymmetry.