Exact solutions of the 2D Schr\"odinger equation with central potentials induced by the non-commutativity of space

TL;DR

This paper derives exact solutions for the 2D Schrödinger equation with specific central potentials in non-commutative space, revealing energy level shifts analogous to the Zeeman effect, up to first-order non-commutativity.

Contribution

It provides the first exact solutions for these potentials in non-commutative space, demonstrating energy level shifts due to non-commutativity.

Findings

Energy levels are shifted to m levels, similar to Zeeman splitting.

Exact solutions are obtained using power-series expansion method.

Results extend understanding of quantum systems in non-commutative geometry.

Abstract

We obtain exact solutions of the 2D Schr\"odinger equation with the central potentials $V(r)=ar^2+br^{-2}+cr^{-4}$ and $V(r)=ar^{-1}+br^{-2}$ in a non-commutative space up to the first order of non-commutativity parameter using the power-series expansion method similar to the 2D Schr\"odinger equation with the singular even-power and inverse-power potentials respectively in commutative space. We derive the exact non-commutative energy levels and show that the energy is shifted to $m$ levels, as in the Zeeman effect.