q-oscillator from the q-Hermite Polynomial

TL;DR

This paper derives the creation and annihilation operators for a q-oscillator from the q-Hermite polynomial, showing it satisfies a q-oscillator algebra and establishing its equivalence to the harmonic oscillator.

Contribution

It introduces a new derivation of the q-oscillator operators from the q-Hermite polynomial using Hamiltonian factorization and shape-invariance, aligning it with the classical harmonic oscillator.

Findings

q-oscillator operators derived from q-Hermite polynomial

q-oscillator satisfies a q-oscillator algebra

Establishes equivalence with the ordinary harmonic oscillator

Abstract

By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of the shape-invariance of the Hamiltonian. A second set of q-oscillator is derived from the exact Heisenberg operator solution. Now the q-oscillator stands on the equal footing to the ordinary harmonic oscillator.