On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ = ǫ k ij θk and a momentum noncommutativity matrix parameter βij = ǫ k ij βk, is showed to be equivalent to QuantumMechanics (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 to see a free particle on NCQPS as equivalent to a harmonic oscillator with Larmor frequency depending on β, representing the same particle in presence of a magnetic field ~ B = q~ β. 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 perceived with opposite sign. PACS : 03.65.ca and 11.10.Nx