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

TL;DR

This paper explores a deformed quantum harmonic oscillator with nonzero minimal uncertainties in position and momentum, using supersymmetric quantum mechanics and shape invariance, revealing connections to q-deformed oscillators and special functions.

Contribution

It introduces a two-parameter deformation framework for the harmonic oscillator, deriving its spectrum and eigenvectors using SUSYQM, and links to q-deformed oscillators and special functions.

Findings

Spectrum reduces to quadratic form when one parameter vanishes

Eigenvectors are expressed as linear combinations of q-boson states

Special case with equal parameters corresponds to q-deformed oscillator

Abstract

In the context of a two-parameter $(\alpha, \beta)$ 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 techniques of supersymmetric quantum mechanics 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 $\alpha$, $\beta$ vanishes, in which case shape invariance under parameter translation occurs. In the special case where $\alpha = \beta \ne 0$, 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 $0 \ne \alpha \ne \beta \ne 0$, 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.