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The Effect Of The Phase-Space Noncommutativity On The Nonrelativistic Limit Of The Dirac Equation

Ilyas Haouam

Published Aug 3, 2014• 0 citations
OpenAlex

Abstract

In this paper we are interested in the effect of the phase-space noncommutativity on the nonreltivistic limit of the Dirac equation in intereaction with an electromagnetic potential, the nonreltivistic limit is done by the large and small wavefunction components approach, knowing that the nonrelativistic limit gives the Schr\odinger-pauli equation.

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In this paper we are interested in the effect of the phase-space noncommutativity on the nonreltivistic limit of the Dirac equation in intereaction with an electromagnetic potential, the nonreltivistic limit is done by the large and small wavefunction components approach, knowing that the nonrelativistic limit gives the Schr\odinger-pauli equation.
In this article, we investigate Ehrenfest's theorem from the Dirac equation in a noncommutative phase-space where we calculate the time derivative of the position and the kinetic momentum operators for Dirac particles in interaction with electromagnetic field and within a noncommutative setting. This allows examining the effect of the phase-space noncommutativity on Ehrenfest's theorem. Knowing that with both the linear Bopp-Shift and Moyal-Weyl product, the noncommutativity is inserted.
The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrödinger-like equation obtained by Dirac equation with the nonrelativistic solvable models is the most efficient method. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped Oscillator Potentials are accessible, when the scalar potential is equal to the vector potential. Using solvable nonrelativistic quantum mechanics systems as a basic model and considering the physical conditions provide the changes in the restrictions of relativistic parameters based on the nonrelativistic definitions of parameters.
In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in a noncommutative phase-space as well as the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic fields are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.
"Symplectic unitary representations for the Poincar ́ e group are studied. The formalism is based on the noncommutative structure of the star–product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quan- tum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. In this context, we derive the Klein–Gordon and Dirac equations. As an application, we study the Dirac equation with electromagnetic interaction in phase space."