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| 2004 41 (06): 837-844 ISSN: 0253-6102 CN: 11-2592/O3 |
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On Quantum Mechanics on Noncommutative Quantum Phase Space
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| A.E.F. Djemaï1 and H. Smail2 |
1 Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy
2 Département de Physique, Institut d'Hydraulique,
Centre Universitaire Mustapha Stambouli, Mascara 29000, Algeria
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Received
2003-8-27
Revised
Online
Accepted
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Abstract In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=εijk
θk and a momentum noncommutativity matrix parameter
β=εijk
βk, is shown 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 parameters θ and β 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 of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,
representing the same particle in presence of a magnetic field
$\vec{B}=q^{-1}\vec{\beta}$. For the other examples, additional
correction terms depending on β 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 feeled with opposite sign.
Key words
noncommutative space
quantum mechanics
Moyal product
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