Non-Commutative Quantum Mechanics

TL;DR

This paper explores non-commutative quantum mechanics in two dimensions with central potentials, providing explicit eigenvalues for large non-commutative parameters and showing equivalence to commutative systems through a modified potential.

Contribution

It introduces a method to solve non-commutative quantum systems by relating them to commutative ones with modified potentials, especially for large non-commutative parameters.

Findings

Eigenvalues explicitly derived for large non-commutative parameter 

Equivalence established between non-commutative and commutative systems

Perturbation theory applicable for finite  values

Abstract

A general non-commutative quantum mechanical system in a central potential $V=V(r)$ in two dimensions is considered. The spectrum is bounded from below and for large values of the anticommutative parameter $\theta $, we find an explicit expression for the eigenvalues. In fact, any quantum mechanical system with these characteristics is equivalent to a commutative one in such a way that the interaction $V(r)$ is replaced by $V = V ({\hat H}_{HO}, {\hat L}_z)$, where ${\hat H}_{HO}$ is the hamiltonian of the two-dimensional harmonic oscillator and ${\hat L}_z$ is z- component of the angular momentum. For other finite values of $\theta$ the model can be solved by using perturbation theory.