Gauge invariant energy spectra in 2-dimensional noncommutative quantum mechanics

Abstract

In this paper, we consider an electron moving on a 2D noncommutative plane immersed in a constant magnetic field under the influence of a polynomial potential. We obtain gauge invariant energy spectra of this system using 2-parameter family of unitarily equivalent irreducible representations of the nilpotent Lie group GNC that were worked out in detail in Chowdhury (2017). We work out the cases of anisotropic harmonic potential and the Hall potential as physical applications of the proposed method. We also show subsequently that straightforward generalization of the Landau problem and the quantum Hall effect in the noncommutative setting using minimal coupling prescription as is done naively on many occasions in the literature violate gauge invariance of energy spectra.