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.

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

Acta Polytechnica ◽

10.14311/ap.2021.61.0230 ◽

2021 ◽

Vol 61 (1) ◽

pp. 230-241

Author(s):

Ilyas Haouam

Keyword(s):

Phase Space ◽

Continuity Equation ◽

Helmholtz Free Energy ◽

Three Dimensional ◽

Noncommutative Phase Space ◽

Pauli Equation ◽

Knowing That ◽

The One ◽

Magnetization Current ◽

Space Noncommutativity

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.

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Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space

Journal of Applied Mathematics and Physics ◽

10.4236/jamp.2020.812207 ◽

2020 ◽

Vol 08 (12) ◽

pp. 2801-2823

Author(s):

Martin Tcoffo ◽

Germain Yinde Deuto ◽

Issofa Nsangou ◽

Armel Azangue Koumetio ◽

Lylyane S. Yonya Tchapda ◽

...

Keyword(s):

Magnetic Field ◽

Phase Space ◽

Harmonic Oscillator ◽

Two Dimensional ◽

Magnetic Field Effects ◽

Noncommutative Phase Space ◽

Field Effects

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Noncommutative Phase Space Schrödinger Equation with Minimal Length

Advances in High Energy Physics ◽

10.1155/2014/459345 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 6

Author(s):

H. Hassanabadi ◽

Z. Molaee ◽

S. Zarrinkamar

Keyword(s):

Magnetic Field ◽

Phase Space ◽

External Magnetic Field ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Jacobi Polynomials ◽

Minimal Length ◽

Two Dimensional ◽

Noncommutative Phase Space ◽

Energy Eigenvalues

We consider the Schrödinger equation under an external magnetic field in two-dimensional noncommutative phase space with an explicit minimal length relation. The eigenfunctions are reported in terms of the Jacobi polynomials, and the explicit form of energy eigenvalues is reported.

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

Open Systems & Information Dynamics ◽

10.1142/s1230161215500213 ◽

2015 ◽

Vol 22 (04) ◽

pp. 1550021 ◽

Cited By ~ 1

Author(s):

Fabio Benatti ◽

Laure Gouba

Keyword(s):

Phase Space ◽

Configuration Space ◽

Classical Limit ◽

Coherent States ◽

Point Of View ◽

Quantum Mechanical ◽

Wigner Functions ◽

Quantum Mechanical Oscillators ◽

Mechanical Oscillators ◽

Space Noncommutativity

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.

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On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/49/5/055202 ◽

2016 ◽

Vol 49 (5) ◽

pp. 055202 ◽

Cited By ~ 3

Author(s):

H Falomir ◽

P A G Pisani ◽

F Vega ◽

D Cárcamo ◽

F Méndez ◽

...

Keyword(s):

Phase Space ◽

Algebraic Structure ◽

Two Dimensional ◽

Noncommutative Phase Space ◽

Rotationally Invariant

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Phenomenology of noncommutative phase space via the anomalous Zeeman effect in hydrogen atom

International Journal of Modern Physics A ◽

10.1142/s0217751x14501772 ◽

2014 ◽

Vol 29 (31) ◽

pp. 1450177 ◽

Cited By ~ 2

Author(s):

Willien O. Santos ◽

Andre M. C. Souza

Keyword(s):

Phase Space ◽

Hydrogen Atom ◽

Zeeman Effect ◽

Nonrelativistic Limit ◽

Noncommutative Phase Space ◽

G Factor ◽

First Order ◽

Anomalous Zeeman Effect ◽

Space Noncommutativity ◽

First Order Perturbation

The Hamiltonian describing the anomalous Zeeman effect for the hydrogen atom on noncommutative (NC) phase space is studied using the nonrelativistic limit of the Dirac equation. To preserve gauge invariance, space noncommutativity must be dropped. By using first-order perturbation theory, the correction to the energy is calculated for the case of a weak external magnetic field. We also obtained the orbital and spin g-factors on the NC phase space. We show that the experimental value for the spin g-factor puts an upper bound on the magnitude of the momentum NC parameter of the order of [Formula: see text], 34 μ eV /c. On the other hand, the experimental value for the spin g-factor was used to establish a correction introduced by NC phase space to the presently accepted value of Planck's constant with an uncertainty of 2 part in 1035.

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The Transformation of Commutative Phase Space to Noncommutative One, and its Lorentz Transformation-Like Forms

Geometry Integrability and Quantization ◽

10.7546/giq-22-2021-188-198 ◽

2021 ◽

pp. 188-198

Author(s):

Makoto Nakamura ◽

Hiroshi Kakuhata ◽

Kouichi Toda

Keyword(s):

Phase Space ◽

Lorentz Transformation ◽

Linear Transformation ◽

Arbitrary Dimension ◽

Two Dimensional ◽

Exact Form ◽

Noncommutative Phase Space ◽

Two Dimension ◽

Direct Sum

Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to co-ordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp's shift.

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