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arXiv:1103.1451 · Symmetry Integrability and Geometry Methods and Applications

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

Véronique Hussin

Published Mar 8, 2011• 12 citations
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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.

DOI
10.3842/sigma.2011.024

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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.
The authors elucidate the utility of a super-realised Wigner-Heisenberg oscillator algebra with its inherent built-in direct generalisation of the usual oscillator ladder operators, but satisfying a generalised quantum commutation rule, as an effective algebraic tool for the easier spectral resolution of general oscillator-related potentials. To illustrate the procedure they consider here the full 3D isotropic harmonic oscillator problem and also the problems of the 1D-isotonic and d-dimensional radial oscillator systems. They also point out the intimate connection between the Wigner-Heisenberg algebra and the quantum mechanical SUSY algebra associated with these systems.
The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator (QMHO) carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general – except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects.We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators.In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation.A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest.As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators.We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.
Based on an equations-of-motion method in the framework of Heisenberg's matrix mechanics, we investigate the conditions under which a one-dimensional quantum mechanical system becomes exactly solvable. By linearizing the equation of motion which is a double-commutation relation of some appropriately chosen function of the position operator with the Hamiltonian, we obtained a set of nontrivial exactly solvable potentials in one dimension. These potentials not only can be solved analytically in closed forms but also contain both the Morse potential and the Poschl-Teller potential as their limiting cases. They may thus be valuable for some potential model calculations as well as for testing various approximation schemes. We also examine these potentials in the framework of supersymmetric quantum mechanics, which is particularly useful for studying exactly solvable potentials.
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.
Morse oscillator is one of the known solvable potentials which attracts many applications in quantum mechanics especially in quantum chemistry. One of the interesting results of this study is the generation of ladder operators for Morse potential. The operators are a representation of the shifting energy levels of the states exhibited by the wavefunction. From this result, we manipulate and deform the operators in such a way that it gives a noncommutative property to promote noncommutative quantum mechanics (NCQM). The resultant NC feature can be shown in the spatial coordinates and finally the Hamiltonian. In this study, we consider two-dimensional Morse potential where the ladder operators are in the form of the corresponding 2D Morse.