Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 024, 16 pages      arXiv:1103.1451      https://doi.org/10.3842/SIGMA.2011.024
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

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

Véronique Hussin a and Ian Marquette b
a) Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada
b) Department of Mathematics, University of York, Heslington, York YO10 5DD, UK

Received December 23, 2010, in final form March 01, 2011; Published online March 08, 2011

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.

Key words: generalized Heisenberg algebras; degeneracies; Morse potential; infinite well potential; supersymmetric quantum mechanics.

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