Noncommutative Quantum Mechanics: The Two-Dimensional Central Field

TL;DR

This paper explores quantum mechanics on a noncommutative two-dimensional plane, providing perturbative solutions for large noncommutativity, explicit eigenstates and eigenvalues, and analyzing the behavior of Green functions and specific limits.

Contribution

It offers explicit solutions and Green functions for noncommutative quantum systems with central potentials, including the Landau problem, highlighting differences between polynomial and non-polynomial potentials.

Findings

Perturbative solutions valid for large noncommutative parameter $ heta$.

Explicit eigenstates and eigenvalues derived.

Green function expressed as an infinite series.

Abstract

Quantum mechanics in a noncommutative plane is considered. For a general two dimensional central field, we find that the theory can be perturbatively solved for large values of the noncommutative parameter ($\theta$) and explicit expressions for the eigenstates and eigenvalues are given. The Green function is explicitly obtained and we show that it can be expressed as an infinite series. For polynomial type potentials, we found a smooth limit for small values of $\theta$ and for non-polynomial ones this limit is necessarily abrupt. The Landau problem, as a limit case of a noncommutative system, is also considered.