Noncommutative quantum mechanics

Abstract

A general noncommutative 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 $\ensuremath{\theta},$ 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 $V=V({H}_{\mathrm{HO}},{L}_{z}),$ where ${H}_{\mathrm{HO}}$ is the Hamiltonian of the two-dimensional harmonic oscillator and ${L}_{z}$ is the z component of the angular momentum. For other finite values of $\ensuremath{\theta}$ the model can be solved by using perturbation theory.