Finite q-oscillator

TL;DR

The paper introduces a finite q-oscillator model based on su_q(2) algebra, with discrete spectra, coherent states, and wave functions involving dual q-Kravchuk polynomials, unifying classical and quantum oscillator limits.

Contribution

It presents a novel finite q-oscillator framework with explicit spectra, wave functions, and a fractional Fourier transform, connecting classical, finite, and standard quantum oscillators.

Findings

Discrete position and momentum spectra with 2j+1 points

Wave functions involve dual q-Kravchuk polynomials

System supports coherent states and a fractional Fourier-q-Kravchuk transform

Abstract

The finite q-oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q-algebra su_q(2). The spectrum of position in this discrete system, in a fixed representation j, consists of 2j+1 "sensor"-points x_s=(1/2)[2s]_q, s=-j, -j+1,..., j, and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wave functions involve dual q-Kravchuk polynomials, which are solutions to a finite-difference Schrodinger equation. Time evolution (times a phase) defines the fractional Fourier-q-Kravchuk transform. In the classical limit q -> 1 we recover the finite oscillator Lie algebra, the N=2j -> infinity limit returns the Macfarlane-Biedenharn q-oscillator and both limits contract the generators to the standard quantum-mechanical oscillator.