A q-Deformed Schr\"odinger Equation

TL;DR

This paper introduces a q-deformed Schrödinger equation using hermitian realizations of fundamental operators within the $su_q(2)$ algebra, providing solutions for Coulomb and harmonic oscillator potentials.

Contribution

It presents a novel formulation of the Schrödinger equation based on q-deformation and explores its solutions for specific quantum potentials.

Findings

Solutions for Coulomb potential are derived.

Solutions for harmonic oscillator potential are derived.

The q-deformed framework maintains vector behavior under $su_q(2)$.

Abstract

We found hermitian realizations of the position vector $\vec{r}$, the angular momentum $\vec{\Lambda}$ and the linear momentum $\vec{p}$, all behaving like vectors under the $su_q(2)$ algebra, generated by $L_0$ and $L_\pm$. They are used to introduce a $q$-deformed Schr\" odinger equation. Its solutions for the particular cases of the Coulomb and the harmonic oscillator potentials are given and briefly discussed.