On Quantum Mechanics on Noncommutative Quantum Phase Space

TL;DR

This paper develops a framework linking Noncommutative Quantum Mechanics to standard Quantum Mechanics on transformed phase spaces, revealing a fundamental noncommutativity parameter and novel phenomena like free particles mimicking harmonic oscillators.

Contribution

It introduces a unified framework for NCQM on transformed phase spaces, identifying a fundamental noncommutativity parameter and analyzing physical systems within this context.

Findings

Lower bound on noncommutativity parameters related to Heisenberg uncertainty

Free particle on NCQPS behaves like a harmonic oscillator with a magnetic field

Opposite charges experience effective noncommutativity with opposite signs

Abstract

In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM) is showed to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraints on this particular transformation, we firstly find that the product of the two noncommutativity parameters possesses a lower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on the physical system under study. This means that "noncommutativity" is represented by a unique parameter which may play the role of a "fundamental constant" characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Among the obtained results, we discover a new phenomenon which consists to see a free particle on NCQPS as equivalent to a harmonic oscillator with Larmor frequency depending on one noncommutativity parameter, representing the same particle in the presence of a magnetic field. For the other examples, additional correction terms appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively perceived with opposite sign.