Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

Abstract

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations . We show that the model has two essentially different phases which is determined by . We construct a operator commuting with and discuss the harmonic oscillator model in two dimensional non-commutative space for three case kappa > 0, kappa = 0, kappa < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.