Discrete quantum model of the harmonic oscillator

TL;DR

This paper introduces a novel discrete quantum harmonic oscillator model with a deformed spectrum, explicit wavefunctions, and a fractional Fourier transform, which converges to the classical oscillator as the deformation parameter approaches 1.

Contribution

The paper presents a new discrete quantum oscillator model with explicit wavefunctions and a fractional Fourier transform framework, extending the traditional harmonic oscillator.

Findings

Energy spectrum is equally spaced and bounded from below.

Position and momentum spectra are discrete and depend on the parameter q.

Model reduces to the standard harmonic oscillator as q approaches 1.

Abstract

We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bounded, whereas the spectra of position and momentum are a denumerable non-degenerate set of points in [-1,1] that depends on the deformation parameter q from (0,1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q-Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit q to 1, one recovers the ordinary quantum harmonic oscillator.