Two-Dimensional Pauli Equation in Noncommutative Phase-Space

We study the Pauli equation in a two-dimensional noncommutative phase-space by considering a constant magnetic field perpendicular to the plane. The noncommutative problem is related to the equivalent commutative one through a set of two-dimensional Bopp-shift transformations. The energy spectrum and the wave function of the two-dimensional noncommutative Pauli equation are found, where the problem in question has been mapped to the Landau problem. In the classical limit, we have derived the noncommutative semiclassical partition function for one- and N- particle systems. The thermodynamic properties such as the Helmholtz free energy, mean energy, specific heat and entropy in noncommutative and commutative phasespaces are determined. The impact of the phase-space noncommutativity on the Pauli system is successfully examined.

ON THE THREE-DIMENSIONAL PAULI EQUATION IN NONCOMMUTATIVE PHASE-SPACE

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.

Zitterbewegung in noncommutative geometry

We have considered the effects of space and momentum noncommutativity separately on the zitterbewegung (ZBW) phenomenon. In the space noncommutativity scenario, it has been expressed that, due to the conservation of momentum, the Fourier decomposition of the expectation value of position does not change. However, the noncommutative (NC) space corrections to the magnetic dipole moment of electron, that was traditionally perceived to come into play only in the first-order of perturbation theory, appear in the leading-order calculations with the similar structure and numerically the same order, but with an opposite sign. This result may explain why for large lumps of masses, the Zeeman effect due to the noncommutativity remains undetectable. Moreover, we have shown that the x- and y-components of the electron magnetic dipole moment, contrary to the commutative (usual) version, are nonzero and with the same structure as the z-component. In the momentum noncommutativity case, we have indicated that, due to the relevant external uniform magnetic field, the energy spectrum and also the solutions of the Dirac equation are changed in 3[Formula: see text]+[Formula: see text]1 dimensions. In addition, our analysis shows that in 2[Formula: see text]+[Formula: see text]1 dimensions, the resulted NC field makes electrons in the zero Landau level rotate not only via a cyclotron motion, but also through the ZBW motion with a frequency proportional to the field which doubles the amplitude of the rotation. In fact, this is a hallmark of the ZBW in graphene that provides a promising way to be tested experimentally.

Phase–space noncommutativity and the thermodynamics of the Landau system

Thermodynamics of the Landau system in noncommutative phase–space (NCPS) has been studied in this paper. The analysis involves the use of generalized Bopp-shift transformations to map the noncommutative (NC) system to its commutative equivalent system. The partition function of the system is computed and from this, the magnetization and the susceptibility of the Landau system are obtained. The results reveal that the magnetization and the susceptibility get modified by both the spatial and momentum NC parameters [Formula: see text] and [Formula: see text]. We then investigate the de Hass–van Alphen effect in NCPS. Here, the oscillation of the magnetization and the susceptibility get corrected by both the spatial and momentum NC parameters [Formula: see text] and [Formula: see text].

Two-dimensional noncommutative quantum mechanics with the central potential

Quantum mechanics in a noncommutative plane with both space noncommutativity and momentum noncommutativity is considered. For a general two-dimensional central field, we show that the theory can be perturbatively solved for large values of the space noncommutative parameter [Formula: see text] when the momentum noncommutative parameter [Formula: see text] is proportional to [Formula: see text]. We obtain the expressions for the eigenstates and eigenvalues. We also discuss the more general noncommutative algebra which have the nonvanishing commutator for [Formula: see text] for different [Formula: see text], [Formula: see text].

Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

A phase-space noncommutative picture of nuclear matter

Noncommutative features are introduced into a relativistic quantum field theory model of nuclear matter, the quantum hadrodynamics-I nuclear model (QHD-I). It is shown that the nuclear matter equation of state (NMEoS) depends on the fundamental momentum scale, [Formula: see text], introduced by the phase-space noncommutativity (NC). Although it is found that NC geometry does not affect the nucleon fields up to [Formula: see text], it affects the energy density, the pressure and other derivable quantities of the NMEoS, such as the nucleon effective mass. Under the conditions of saturation of the symmetric NM under consideration, the estimated value for the noncommutative parameter is [Formula: see text].