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.

scholarly journals Harmonic Oscillator Chain in Noncommutative Phase Space with Rotational Symmetry

2019 ◽  
Vol 64 (2) ◽  
pp. 131
Author(s):  
Kh. P. Gnatenko
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Rotational Symmetry ◽  
Center Of Mass ◽  
Quantum Space ◽  
Noncommutative Phase Space ◽  
Noncommutative Algebra ◽  
A Chain ◽  
Rotationally Invariant

We consider a quantum space with a rotationally invariant noncommutative algebra of coordinates and momenta. The algebra contains the constructed tensors of noncommutativity involving additional coordinates and momenta. In the rotationally invariant noncommutative phase space, the harmonic oscillator chain is studied. We obtain that the noncommutativity affects the frequencies of the system. In the case of a chain of particles with harmonic oscillator interaction, we conclude that, due to the noncommutativity of momenta, the spectrum of the center-of-mass of the system is discrete and corresponds to the spectrum of a harmonic oscillator.

scholarly journals Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space

2018 ◽  
Vol 33 (16) ◽  
pp. 1850091 ◽  
Author(s):  
Kh. P. Gnatenko ◽  
O. V. Shyiko
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Rotational Symmetry ◽  
Free Particle ◽  
Second Order ◽  
Noncommutative Phase Space ◽  
Noncommutative Algebra ◽  
Ordinary Space ◽  
Canonical Type ◽  
Rotationally Invariant

We consider rotationally invariant noncommutative algebra with tensors of noncommutativity constructed with the help of additional coordinates and momenta. The algebra is equivalent to the well-known noncommutative algebra of canonical type. In the noncommutative phase space, rotational symmetry influence of noncommutativity on the spectrum of free particle and the spectrum of harmonic oscillator is studied up to the second-order in the parameters of noncommutativity. We find that because of momentum noncommutativity, the spectrum of free particle is discrete and corresponds to the spectrum of harmonic oscillator in the ordinary space (space with commutative coordinates and commutative momenta). We obtain the spectrum of the harmonic oscillator in the rotationally invariant noncommutative phase space and conclude that noncommutativity of coordinates affects its mass. The frequency of the oscillator is affected by the coordinate noncommutativity and the momentum noncommutativity. On the basis of the results, the eigenvalues of squared length operator are found and restrictions on the value of length in noncommutative phase space with rotational symmetry are analyzed.

scholarly journals Noncommutative phase space with rotational symmetry and hydrogen atom

2017 ◽  
Vol 32 (26) ◽  
pp. 1750161 ◽  
Author(s):  
Kh. P. Gnatenko ◽  
V. M. Tkachuk
Keyword(s):  
Phase Space ◽  
Hydrogen Atom ◽  
Rotational Symmetry ◽  
Energy Levels ◽  
Upper Bounds ◽  
Second Order ◽  
Noncommutative Phase Space ◽  
Noncommutative Algebra ◽  
Canonical Type ◽  
Rotationally Invariant

We construct algebra with noncommutativity of coordinates and noncommutativity of momenta which is rotationally invariant and equivalent to noncommutative algebra of canonical type. Influence of noncommutativity on the energy levels of hydrogen atom is studied in rotationally invariant noncommutative phase space. We find corrections to the levels up to the second order in the parameters of noncommutativity and estimate the upper bounds of these parameters.

scholarly journals Kinematic variables in noncommutative phase space and parameters of noncommutativity

2017 ◽  
Vol 32 (31) ◽  
pp. 1750166 ◽  
Author(s):  
Kh. P. Gnatenko
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Relative Motion ◽  
Equivalence Principle ◽  
Composite System ◽  
Center Of Mass ◽  
Weak Equivalence ◽  
Noncommutative Phase Space ◽  
Weak Equivalence Principle ◽  
Particle Composite

We consider a space with noncommutativity of coordinates and noncommutativity of momenta. It is shown that coordinates in noncommutative phase space depend on mass therefore they cannot be considered as kinematic variables. Also, noncommutative momenta are not proportional to a mass as it has to be. We find conditions on the parameters of noncommutativity on which these problems are solved. It is important that on the same conditions the properties of kinetic energy are preserved, the motion of the center-of-mass of composite system and relative motion are independent, the trajectory of a particle (composite system) is independent of its mass and composition therefore the weak equivalence principle is recovered in four-dimensional (2D configurational space and 2D momentum space) noncommutative phase space.

scholarly journals Two-Particle System in Noncommutative Space with Preserved Rotational Symmetry

2016 ◽  
Vol 61 (5) ◽  
pp. 432-439 ◽  
Author(s):  
Kh.P. Gnatenko ◽  
◽  
V.M. Tkachuk ◽  
Keyword(s):  
Particle System ◽  
Rotational Symmetry ◽  
Noncommutative Space

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 Features of free particles system motion in noncommutative phase space and conservation of the total momentum

2018 ◽  
Vol 33 (23) ◽  
pp. 1850131 ◽  
Author(s):  
Kh. P. Gnatenko ◽  
H. P. Laba ◽  
V. M. Tkachuk
Keyword(s):  
Phase Space ◽  
Composite System ◽  
Free Particle ◽  
Total Momentum ◽  
Initial Moment ◽  
Integrals Of Motion ◽  
Integral Of Motion ◽  
Noncommutative Phase Space ◽  
Free Particles ◽  
Canonical Type

Influence of noncommutativity on the motion of composite system is studied in noncommutative phase space of canonical type. A system composed by N free particles is examined. We show that because of the momentum noncommutativity free particles of different masses with the same velocities at the initial moment of time do not move together. The trajectory and the velocity of a free particle in noncommutative phase space depend on its mass. So, a system of the free particles flies away. Also, it is shown that the total momentum defined in the traditional way is not integral of motion in a space with noncommutativity of coordinates and noncommutativity of momenta. We find that in the case when parameters of noncommutativity corresponding to a particle are determined by its mass, the trajectory and the velocity of the free particles are independent of the mass. Also, the total momenta as integrals of motion can be introduced in noncommutative phase space.

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