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arXiv:hep-th/9208073

Deformation of Supersymmetric and Conformal Quantum Mechanics Through Affine Transformations

Vyacheslav Pavlovich Spiridonov

Published Aug 27, 1992• 8 citations
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Abstract

Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional $N=2$ supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are $q$-isospectral, i.e. the spectrum of one can be obtained from another (with possible exception of the lowest level) by $q^2$-factor scaling. This construction allows easily to rederive a special self-similar potential found by Shabat and to show that for the latter a $q$-deformed harmonic oscillator algebra of Biedenharn and Macfarlane serves as the spectrum generating algebra. A general class of potentials related to the quantum conformal algebra $su_q(1,1)$ is described. Further possibilities for $q$-deformation of known solvable potentials are outlined. Talk presented at the workshop on Harmonic Oscillators, College Park, 25-28 March 1992.

DOI
10.48550/arxiv.hep-th/9208073

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Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional N=2N=2N=2 supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are qqq-isospectral, i.e. the spectrum of one can be obtained from another (with possible exception of the lowest level) by q2q^2q2-factor scaling. This construction allows easily to rederive a special self-similar potential found by Shabat and to show that for the latter a qqq-deformed harmonic oscillator algebra of Biedenharn and Macfarlane serves as the spectrum generating algebra. A general class of potentials related to the quantum conformal algebra suq(1,1)su_q(1,1)suq​(1,1) is described. Further possibilities for qqq-deformation of known solvable potentials are outlined. Talk presented at the workshop on Harmonic Oscillators, College Park, 25-28 March 1992.
Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e., the spectrum of one can be obtained from another (with possible exception of the lowest level) by the q2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra su q(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane.
In the context of a two-parameter (α, β) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters α, β vanishes, in which case shape invariance under parameter translation occurs. In the special case where α = β ≠ 0, the oscillator Hamiltonian is shown to coincide with that of the q -deformed oscillator with q > 1 and its eigenvectors are therefore n - q -boson states. In the general case where 0 ≠ α ≠ β ≠ 0, the eigenvectors are constructed as linear combinations of n - q -boson states by resorting to a Bargmann representation of the latter and to q -differential calculus. They are finally expressed in terms of a q -exponential and little q -Jacobi polynomials.
Generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show that this generalized formalism is able (a) to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system, (b) to reproduce, as particular cases, results already known for shape-invariant systems (such as standard harmonic oscillator and Pöschl–Teller potentials as well as quantum deformed harmonic oscillator models) and (c) point to a formalism that provides a unified description of the different kind of coherent states for quantum systems, deformed or not deformed.
In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.