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On Quantum Mechanics on Noncommutative Quantum Phase Space

Communications in Theoretical Physics 2004年第6期41卷 837-844页

作者： A.E.F.DjemaI H.Smail AbdusSalamInternationalCentreforTheoreticalPhysics 34100TriesteItaly DepartementdePhysique Institutd＇HydrauliqueCentreUniversitaireMustaphaStambouliMascara29000Algeria

In this work,we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=∈_(ji)~kθ_k and a momentum noncommutativity matrix parameter β_(ij)=∈_(ij)~kβ_k,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 *** means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole *** our framework,we treat some physical systems on NCQPS:free particle,harmonic oscillator,system of two-charged particles,Hydrogen *** 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=q~(-1).For the other examples,additional correction terms depending on β appear in the expression of the energy ***,in the two-particle system case,we emphasize the fact that for two opposite charges noncornmutativity is effectively feeled with opposite sign.

关键词： noncommutative space quantum mechanics moyal product

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The Foldy-Wouthuysen Transformation of the Dirac Equation in Noncommutative Phase-Space

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Journal of Modern Physics 2018年第11期9卷 2021-2034页

作者： Ilyas Haouam Lyazid Chetouani Département de Physique Facultédes sciences exactesUniversitédes Frères MentouriConstantineAlgeria

A method of Foldy-Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed;in the present work the basic properties of the Dirac hamiltonian in the FW representation in the noncommutative phase-space are investigated and the Schrödinger-Pauli equation is found, knowing that the used methods for extracting the full phase-space noncommutative Dirac equation are, the Bopp-shift linear translation method, and the moyal-Weyl product (*-product).

关键词： Foldy-Wouthuysen Transformation Nonrelativistic Limit Noncommutative Schrodinger-Pauli Equation Phase-Space Noncommutativity Noncommutative Dirac Equation moyal product Bopp-Shift Translation

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Continuity Equation in Presence of a Non-Local Potential in Non-Commutative Phase-Space

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Open Journal of Microphysics 2019年第3期9卷 15-28页

作者： Ilyas Haouam Laboratoire de Physique Mathematique et de Physique Subatomique(LPMPS) Universite Freres MentouriConstantineAlgeria

We studied the continuity equation in presence of a local potential, and a non-local potential arising from electron-electron interaction in both commutative and non-commutative phase-space. Furthermore, we examined the influence of the phase-space non-commutativity on both the locality and the non-locality, where the definition of current density in commutative phase-space cannot satisfy the condition of current conservation, but with the steady state, in order to solve this problem, we give a new definition of the current density including the contribution due to the non-local potential. We showed that the calculated current based on the new definition of current density maintains the current. As well for the case when the non- commutativity in phase-space considered, we found that the conservation of the current density completely violated;and the non-commutativity is not suitable for describing the current density in presence of non-local and local potentials. Nevertheless, under some conditions, we modified the current density to solve this problem. Subsequently, as an application we studied the Frahn-Lemmer non-local potential, taking into account that the employed methods concerning the phase-space non-commutativity are both of Bopp-shift linear transformation through the Heisenberg-like commutation relations, and the moyal-Weyl product.

关键词： Continuity Equation Non-Local Potential Non-Commutative Schrodinger Equation Phase-Space Non-Commutativity Frahn-Lemmer Potential moyal product Bopp-Shift Linear Transformation

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