The $q$-deformed Schr\"odinger Equation of the Harmonic Oscillator on the Quantum Euclidian Space

TL;DR

This paper develops a $q$-deformed version of the Schr"odinger equation for the harmonic oscillator in quantum Euclidean space, introducing new operators and solutions using $q$-analysis techniques.

Contribution

It introduces a systematic method to derive energy levels and eigenfunctions of the $q$-deformed harmonic oscillator using $q$-difference equations and $q$-polynomials.

Findings

Derived creation and annihilation operators for the $q$-deformed system

Obtained $q$-series representations of eigenfunctions

Solved the $q$-difference Schr"odinger equation using $q$-analysis

Abstract

We consider the $q$-deformed Schr\"odinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space. The creation and annihilation operator are found, which systematically produce all energy levels and eigenfunctions of the Schr\"odinger equation. In order to get the $q$-series representation of the eigenfunction, we also give an alternative way to solve the Schr\"odinger equation which is based on the $q$-analysis. We represent the Schr\"odinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions.