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

OJM > Vol.9 No.3, August 2019

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

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DOI: 10.4236/ojm.2019.93003 1,387 Downloads 3,250 Views Citations

Author(s)

Ilyas Haouam

Affiliation(s)

Laboratoire de Physique Mathématique et de Physique Subatomique (LPMPS), Université Frères Mentouri, Constantine, Algeria.

ABSTRACT

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.

KEYWORDS

Continuity Equation, Non-Local Potential, Non-Commutative Schrödinger Equation, Phase-Space Non-Commutativity, Frahn-Lemmer Potential, Moyal Product, Bopp-Shift Linear Transformation

Share and Cite:

Haouam, I. (2019) Continuity Equation in Presence of a Non-Local Potential in Non-Commutative Phase-Space. Open Journal of Microphysics, 9, 15-28. doi: 10.4236/ojm.2019.93003.

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