On Quantum Mechanics on Noncommutative Quantum Phase Space

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...

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Bibliographic Details
Published in:arXiv.org 2003-10
Main Authors: Djemai, A E F, Smail, H
Format: Article
Language:English
Subjects:
Charged particles
Energy spectra
Equivalence
Harmonic oscillators
Hydrogen storage
Lower bounds
Parameters
Phase transitions
Quantum mechanics
Quantum physics
Quantum theory
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.0309006