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arXiv:math-ph/0306047 · Journal of Physics A Mathematical and General

Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework

C. Quesne, V. M. Tkachuk

Published Oct 1, 2003• 69 citations

Abstract

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.

DOI: 10.1088/0305-4470/36/41/009

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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.

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator associated with a position-dependent effective mass by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique, we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass and uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution for the corrections of the GUP.

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.

Abstract Liénard-type nonlinear one dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schrödinger equation is exactly solved in momentum space via the approach of supersymmetric quantum mechanics (SUSYQM). The bound-states energy spectra and corresponding wave functions are given explicitly in terms of the ambiguity parameters. The limiting case of no deformation agrees exactly with the eigenenergies and eigenfunctions of the ordinary quantum harmonic oscillator.

We consider the minimal bosonization realization of supersymmetric shape-invariant systems where generalized supercharge operators are constructed using the partner supersymmetric operators, the parameter potential translation formalism and the reflection operator. We obtain the solution of the eigenvalue equation and study the quantum dynamics of the supersymmetric system including terms in the Hamiltonian which are constructed using the combination of the bosonized supercharge operators. The connections between the bosonized supersymmetric formalism, the Bose–Fermi transformation and the generalization of the -deformed Heisenberg algebra are discussed. As an illustration, we apply the generalized formalism for the case of the trigonometric Rosen–Morse potential.

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.

Abstract We discussed transformations of supersymmetric quantum mechanics by considering a general six-parameter function, from which the superpotential and the supersymmetric partner potentials V − ( r ) and V + ( r ) were constructed. By specific choice of the parameters, V − ( r ) reproduced the radial harmonic oscillator, or its rationally extended version, with solutions containing X 1 -type Laguerre polynomials. The choice of parameters also determined which of the four possible SUSY transformations T i , i = 1, ...4 were employed. Within this formalism, the relation between the radial harmonic oscillator and its rational extension can be discussed in a straightforward way, furthermore, several aspects of SUSYQM and exactly solvable potentials can be demonstrated in a unified framework on a level suitable for basic quantum mechanics courses.