Coherent States of the Deformed Heisenberg-Weyl Algebra in Noncommutative Space

TL;DR

This paper clarifies the non-equivalence of deformed and undeformed Heisenberg-Weyl algebras in noncommutative space, highlighting the challenges in constructing certain coherent states and their physical interpretations.

Contribution

It demonstrates the non-equivalence of deformed and undeformed algebras in noncommutative space and explores the physical realization of deformed coherent states.

Findings

Deformed algebra not fully equivalent to undeformed algebra in noncommutative space

No straightforward method to construct deformed coherent states from undeformed ones

Deformed states correspond to physical states of Rydberg atoms and free particles

Abstract

In two-dimensional space a subtle point that for the case of both space-space and momentum-momentum noncommuting, different from the case of only space-space noncommuting, the deformed Heisenberg-Weyl algebra in noncommutative space is not completely equivalent to the undeformed Heisenberg-Weyl algebra in commutative space is clarified. It follows that there is no well defined procedure to construct the deformed position-position coherent state or the deformed momentum-momentum coherent state from the undeformed position-momentum coherent state. Identifications of the deformed position-position and deformed momentum-momentum coherent states with the lowest energy states of a cold Rydberg atom in special conditions and a free particle, respectively, are demonstrated.