The $q$-deformed Schrödinger Equation of the Harmonic Oscillator on the Quantum Euclidian Space

Abstract

We consider the $q$-deformed Schrödinger 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 Schrödinger equation. In order to get the $q$-series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the $q$-analysis. We represent the Schrödinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions.