DEFORMATION QUANTIZATION FOR COUPLED HARMONIC OSCILLATORS ON A GENERAL NONCOMMUTATIVE SPACE

2008 ◽  
Vol 23 (06) ◽  
pp. 445-456 ◽  
Author(s):  
BINGSHENG LIN ◽  
SICONG JING ◽  
TAIHUA HENG
Keyword(s):  
Phase Space ◽  
Deformation Quantization ◽  
Harmonic Oscillators ◽  
Energy Spectra ◽  
Noncommutative Space ◽  
Wigner Functions ◽  
Noncommutative Phase Space ◽  
Classical Systems ◽  
Coupled Harmonic Oscillators ◽  
Special Cases

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two-coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.

THE EIGENVALUES OF TWO MODES COUPLED HARMONIC OSCILLATORS IN NONCOMMUTATIVE PHASE-SPACE

2011 ◽  
Vol 26 (09) ◽  
pp. 1561-1567 ◽  
Author(s):  
XIN-FENG DIAO ◽  
CHAO-YUN LONG ◽  
GUANG-JIE GUO ◽  
ZHENG-WEN LONG
Keyword(s):  
Phase Space ◽  
Coordinate Transformation ◽  
Algebraic Method ◽  
Harmonic Oscillators ◽  
Noncommutative Phase Space ◽  
Coupled Harmonic Oscillators

The Hamiltonian of two modes coupled harmonic oscillators in noncommutative phase-space can be expressed in the standard quadric form by introducing a coordinate transformation. As a result, the coupling items are eliminated and the Hamiltonian can be simplified to the two modes independent harmonic oscillators. Then, the corresponding energy spectrums are obtained with an algebraic method.

scholarly journals Berry Phase for Time-Dependent Coupled Harmonic Oscillators in the Noncommutative Phase Space via Path Integral Techniques

2020 ◽  
pp. 58-70
Author(s):  
Leila Khiari ◽  
Tahar Boudjedaa ◽  
Abdenacer Makhlouf ◽  
Mohammed Tayeb Meftah
Keyword(s):  
Phase Space ◽  
Path Integral ◽  
Adiabatic Approximation ◽  
Berry Phase ◽  
Quadratic System ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Noncommutative Phase Space ◽  
Integral Formalism ◽  
Coupled Harmonic Oscillators

The purpose of this paper is the description of Berry’s phase, in the Euclidean Path Integral formalism, for 2D quadratic system: two time dependent coupled harmonic oscillators. This treatment is achieved by using the adiabatic approximation in the commutative and noncommutative phase space

scholarly journals Composite system in rotationally invariant noncommutative phase space

2018 ◽  
Vol 33 (07) ◽  
pp. 1850037 ◽  
Author(s):  
Kh. P. Gnatenko ◽  
V. M. Tkachuk
Keyword(s):  
Phase Space ◽  
Composite System ◽  
Particle System ◽  
Rotational Symmetry ◽  
Center Of Mass ◽  
Noncommutative Space ◽  
Effective Parameters ◽  
Noncommutative Phase Space ◽  
Noncommutative Algebra ◽  
Rotationally Invariant

Composite system is studied in noncommutative phase space with preserved rotational symmetry. We find conditions on the parameters of noncommutativity on which commutation relations for coordinates and momenta of the center-of-mass of composite system reproduce noncommutative algebra for coordinates and momenta of individual particles. Also, on these conditions, the coordinates and the momenta of the center-of-mass satisfy noncommutative algebra with effective parameters of noncommutativity which depend on the total mass of the system and do not depend on its composition. Besides, it is shown that on these conditions the coordinates in noncommutative space do not depend on mass and can be considered as kinematic variables, the momenta are proportional to mass as it has to be. A two-particle system with Coulomb interaction is studied and the corrections to the energy levels of the system are found in rotationally invariant noncommutative phase space. On the basis of this result the effect of noncommutativity on the spectrum of exotic atoms is analyzed.

The Klein–Gordon equation with the Kratzer potential in the noncommutative space

2018 ◽  
Vol 33 (35) ◽  
pp. 1850203 ◽  
Author(s):  
M. Darroodi ◽  
H. Mehraban ◽  
S. Hassanabadi
Keyword(s):  
Perturbation Theory ◽  
Phase Space ◽  
Gordon Equation ◽  
Energy Shift ◽  
Laboratory Frame ◽  
Noncommutative Space ◽  
Noncommutative Phase Space ◽  
Polar Coordinate ◽  
Klein Gordon ◽  
Klein Gordon Equation

The Klein–Gordon equation is considered for the Kratzer potential in the spherical polar coordinate in laboratory frame in noncommutative space. The energy shift due to noncommutativity is obtained via the perturbation theory. After rather cumbersome algebra, we found the eigenfunctions and eigenvalues of the system for a noncommutative phase space.

scholarly journals INVERSE SQUARE PROBLEM AND so(2, 1) SYMMETRY IN NONCOMMUTATIVE SPACE

2009 ◽  
Vol 24 (14) ◽  
pp. 2655-2663 ◽  
Author(s):  
PULAK RANJAN GIRI
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Noncommutative Space ◽  
Noncommutative Phase Space

We study the quantum mechanics of a system with inverse square potential in noncommutative space. Both the coordinates and momenta are considered to be noncommutative, which breaks the original so(2, 1) symmetry. The energy levels and eigenfunctions are obtained. The generators of the so(2, 1) algebra are also studied in noncommutative phase space and the commutators are calculated, which shows that the commutators obtained in noncommutative space is not closed. However the commutative limit Θ, [Formula: see text] for the commutators smoothly go to the standard so(2, 1) algebra.

scholarly journals DEFORMATION QUANTIZATION AND WIGNER FUNCTIONS

2005 ◽  
Vol 20 (17n18) ◽  
pp. 1371-1385 ◽  
Author(s):  
N. COSTA DIAS ◽  
J. N. PRATA
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Deformation Quantization ◽  
Wigner Distribution ◽  
Sufficient Conditions ◽  
Wigner Functions ◽  
Space Function ◽  
Wigner Spectrum ◽  
Necessary And Sufficient ◽  
Phase Space Function

We review the Weyl-Wigner formulation of quantum mechanics in phase space. We discuss the concept of Narcowich-Wigner spectrum and use it to state necessary and sufficient conditions for a phase space function to be a Wigner distribution. Based on this formalism we analize the modifications introduced by the presence of boundaries. Finally, we discuss the concept of environment-induced decoherence in the context of the Weyl-Wigner approach.

scholarly journals REPRESENTATION OF NONCOMMUTATIVE PHASE SPACE

2005 ◽  
Vol 20 (28) ◽  
pp. 2165-2174 ◽  
Author(s):  
KANG LI ◽  
JIANHUA WANG ◽  
CHIYI CHEN
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Noncommutative Space ◽  
Noncommutative Phase Space ◽  
Commutative Space

The representations of the algebra of coordinates and momenta of noncommutative phase space are given. We study, as an example, the harmonic oscillator in noncommutative space of any dimension. Finally the map of Schrödinger equation from noncommutative space to commutative space is obtained.

scholarly journals Induced entanglement entropy of harmonic oscillators in non-commutative phase space

2019 ◽  
Vol 34 (33) ◽  
pp. 1950269
Author(s):  
Bingsheng Lin ◽  
Jian Xu ◽  
Taihua Heng
Keyword(s):  
Phase Space ◽  
Ground State ◽  
Entanglement Entropy ◽  
Harmonic Oscillators ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Wigner Functions ◽  
Definition Of

We study the entanglement entropy of harmonic oscillators in non-commutative phase space (NCPS). We propose a new definition of quantum Rényi entropy based on Wigner functions in NCPS. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the 2D isotropic harmonic oscillators. We find that for some values of the non-commutative parameters, the harmonic oscillators can be entangled in NCPS. This is a new entanglement-like effect caused by the non-commutativity of the phase space.

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