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

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Document pages: 14 pages

Abstract: We studied thecontinuity equation in presence of a local potential, and a non-local potential arisingfrom electron-electron interaction in both commutative and non-commutativephase-space. Furthermore, we examined the influenceof the phase-space non-commutativity on both the locality and the non-locality,where the definition of current density in commutative phase-space cannotsatisfy 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 dueto the non-local potential. We showed that the calculated current based on thenew definition of current densitymaintains the current. As well for the case when the non- commutativity in phase-space considered, we found that the conservationof the current density completely violated; and the non-commutativity is notsuitable for describing the current density in presence of non-local and localpotentials. Nevertheless, under some conditions, we modified the currentdensity to solve this problem. Subsequently, as an application we studied theFrahn-Lemmer non-local potential, takinginto account that the employed methodsconcerning the phase-space non-commutativity are both of Bopp-shift linear transformation through the Heisenberg-like commutation relations, andthe Moyal-Weyl product.

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