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 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.

Invariance Groups in Classical and Quantum Mechanics

1973 ◽  
Vol 28 (3-4) ◽  
pp. 538-540 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Classical Mechanics ◽  
Symmetry Groups ◽  
Casimir Invariants ◽  
Classical Phase Space ◽  
Invariance Groups ◽  
Classical And Quantum Mechanics

AbstractThis is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

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 Towards an Observable Test of Noncommutative Quantum Mechanics

2019 ◽  
Vol 64 (11) ◽  
pp. 983 ◽  
Author(s):  
Liang Shi-Dong ◽  
T. Harko
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Quantum Physics ◽  
Planck Scale ◽  
Noncommutative Phase Space ◽  
A Value ◽  
Noncommutative Quantum Mechanics ◽  
Noncommutative Spacetime ◽  
Aharonov Bohm ◽  
Noncommutative Quantum

The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.

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.

BOUND STATES FOR SCHRÖDINGER HAMILTONIANS: PHASE SPACE METHODS AND APPLICATIONS

1996 ◽  
Vol 08 (04) ◽  
pp. 503-547 ◽  
Author(s):  
PH. BLANCHARD ◽  
J. STUBBE
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Dynamical Systems ◽  
Bound States ◽  
Energy Levels ◽  
Schrödinger Operators ◽  
Particle Systems ◽  
Existence And Nonexistence ◽  
Phase Space Methods ◽  
Special Case

Properties of bound states for Schrödinger operators are reviewed. These include: bounds on the number of bound states and on the moments of the energy levels, existence and nonexistence of bound states, phase space bounds and semi-classical results, the special case of central potentials, and applications of these bounds in quantum mechanics of many particle systems and dynamical systems. For the phase space bounds relevant to these applications we improve the explicit constants.

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 Elliptical orbits in the phase-space quantization

2016 ◽  
Vol 38 (3) ◽  
Author(s):  
Leonardo Andreta de Castro ◽  
Carlos Alexandre Brasil ◽  
Reginaldo de Jesus Napolitano
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Phase Space ◽  
Energy Levels ◽  
Radial Component ◽  
Original Method ◽  
Direct Integration ◽  
Old Quantum Theory ◽  
Complex Integration ◽  
Elliptical Orbits

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.

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.

scholarly journals COMMUTATOR ANOMALY IN NONCOMMUTATIVE QUANTUM MECHANICS

2006 ◽  
Vol 21 (39) ◽  
pp. 2971-2976 ◽  
Author(s):  
SAYIPJAMAL DULAT ◽  
KANG LI
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
Noncommutative Phase Space ◽  
Noncommutative Quantum Mechanics ◽  
Physical Observable ◽  
Noncommutative Quantum

In this paper, the Schrödinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space–space and space–momentum as well as momentum–momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.

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