Emergence of Geometric phase shift in Planar Non-commutative Quantum Mechanics

TL;DR

This paper investigates how geometric phase shifts emerge in noncommutative quantum mechanics using an exactly solvable 2D harmonic oscillator model, revealing a direct relation to classical Hannay angles.

Contribution

It provides an explicit, non-perturbative expression for the geometric phase in noncommutative quantum systems and links it to classical analogs.

Findings

Explicit geometric phase expression derived without perturbation

Phase shift related to classical Hannay angle

Model demonstrates effects of spatial and momentum non-commutativity

Abstract

Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of 2D simple harmonic oscilator in Moyal plane, where momentum non-commutativity are also considered along with spatial noncommutativity. After finding a suitable Bopp shift, that bridges the noncommutative phase space operators with their effective commutative counterparts, having their dependence on the non-commutative parameters, we study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to the Hannay angle of a classically analogous system, by studying the evolution of the coherent state of this system.