Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/9811027

q-Deformed Harmonic Oscillator in Phase Space

A. K. Aringazin, K. M. Aringazin, Sotirios Baskoutas, G. N. Brodimas, A. D. Jannussis, Vlachos, E.

Published Nov 2, 1998• 1 citation
arXivHTMLPDF

Abstract

Relation between Bopp-Kubo formulation and Weyl-Wigner-Moyal symbol calculus, and non-commutative geometry interpretation of the phase space representation of quantum mechanics are studied. Harmonic oscillator in phase space via creation and annihilation operators, both the usual and $q$-deformed, is investigated. We found that the Bopp-Kubo formulation is just non-commuting coordinates representation of the symbol calculus. The Wigner operator for the $q$-deformed harmonic oscillator is shown to be proportional to the 3-axis spherical angular momentum operator of the algebra $su_{q}(2)$. The relation of the Fock space for the harmonic oscillator and double Hilbert space of the Gelfand-Naimark-Segal construction is established. The quantum extension of the classical ergodiicity condition is proposed.

DOI
10.48550/arxiv.hep-th/9811027

Related papers

Quantum and classical mechanics of q-deformed systems

Sergei V. Shabanov

Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.

Journal of Physics A Mathematical and General•Jun 7, 1993•21 citations

Coordinate and momentum representations of theq-deformed oscillator and their interpretation

A. K. Rajagopal

A derivation of the coordinate representation of the q-deformed oscillator is given. The Hilbert space associated with it is defined on a finite coordinate space y: (-\ensuremath{\pi}/2s,\ensuremath{\pi}/2s), where s is related to the deformation q via q=exp(-${\mathit{s}}^{2}$), with the respective ``coordinate'' and ``momentum'' operators given by X${\mathrm{^}}_{\mathit{s}}$=y-${\mathit{A}}_{\mathit{s}}$(y), P${\mathrm{^}}_{\mathit{s}}$=-id/dy-${\mathit{iA}}_{\mathit{s}}$(y), where ${\mathit{A}}_{\mathit{s}}$(y)=1/2d ln${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$(y)/dy, ${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$ being a measure in this space. The commutation rule obeyed by them is X${\mathrm{^}}_{\mathit{s}}$P${\mathrm{^}}_{\mathit{s}}$-P${\mathrm{^}}_{\mathit{s}}$X${\mathrm{^}}_{\mathit{s}}$=i${\mathrm{\ensuremath{\Elzxh}}}_{\mathit{s}}$(y)=i{1-${\mathit{dA}}_{\mathit{s}}$ /dy}. After constructing a suitable representation in the momentum space, we also set up the Wigner distribution function in this phase space. A brief discussion of these results and their interpretation are given.

Physical Review A•May 1, 1993•4 citations

First, Second Quantization and Q-Deformed Harmonic Oscillator

Man Van Ngu, Ngo Gia Vinh, Nguyen Tri Lan +2 more

Relations between the first, the second quantized representations and deform algebra are investigated. In the case of harmonic oscillator, the axiom of first quantization (the commutation relation between coordinate and momentum operators) and the axiom of second quantization (the commutation relation between creation and annihilation operators) are equivalent. We shown that in the case of q -deformed harmonic oscillator, a violence of the axiom of second quantization leads to a violence of the axiom of first quantization, and inverse. Using the coordinate representation, we study fine structures of the vacuum state wave function depend in the deformation parameter q . A comparison with fine structures of Cooper pair of superconductivity in the coordinate representation is also performed.

Journal of Physics Conference Series•Jun 26, 2015•5 citations

DEFORMED HARMONIC OSCILLATOR AND NONLINEAR COHERENT STATES: NONCOMMUTATIVE QUANTUM SPACE APPROACH

M. H. Naderi, M. Soltanolkotabi, R. Roknizadeh

In this paper, by using the Wess–Zumino formalism of noncommutative differential calculus, we show that the concept of nonlinear coherent states originates from noncommutative geometry. For this purpose, we first formulate the differential calculus on a GL p, q(2) quantum plane. By using the commutation relations between coordinates and their interior derivatives, we then construct the two-parameter (p, q)-deformed quantum phase space together with the associated deformed Heisenberg commutation relations. Finally, by applying the obtained results for the quantum harmonic oscillator we construct the associated coherent states, which can be identified as nonlinear coherent states. Furthermore, we show that some of the well-known deformed (nonlinear) coherent states, such as two-parameter (p, q)-deformed coherent states, Maths-type q-deformed coherent states, Phys-type q-deformed coherent states and Quesne deformed coherent states, can be easily obtained from our treatment.

International Journal of Modern Physics A•Apr 15, 2009•4 citations

Wigner function of charged harmonic oscillator in non-commutative phase spaces

Yan Jiang-feng

In this paper,Moyal-Weyl multiplcation sign and Bopp shift are introduced.By considering the relation of non-commutative in coordinate-coordinate and momentum-momentum,the Wigner function of charged harmonic oscillator in non-commutative phase spaces are discussed.By using algebraic relation of quantum mechanics in non-commutative phase space,it has been found that the Wigner function depends on the addition item of non-commutative parameters,compared with the commutative space.

Journal of Shaanxi University of Technology•Jan 1, 2010•0 citations

New Creation and Annihilation Operators in Phase Space

A. D. Jannussis, N. Patargias, G. N. Brodimas

New creation and annihilation operators of the harmonic oscillator in the phase space are considered. The common base of these operators is a generalized Wigner distribution and their coherent states are determined. Using these operators Weyl's displacement operators and Bopp's translation operators are also determined.

Journal of the Physical Society of Japan•Jul 1, 1978•9 citations
Relation between Bopp-Kubo formulation and Weyl-Wigner-Moyal symbol calculus, and non-commutative geometry interpretation of the phase space representation of quantum mechanics are studied. Harmonic oscillator in phase space via creation and annihilation operators, both the usual and qqq-deformed, is investigated. We found that the Bopp-Kubo formulation is just non-commuting coordinates representation of the symbol calculus. The Wigner operator for the qqq-deformed harmonic oscillator is shown to be proportional to the 3-axis spherical angular momentum operator of the algebra suq(2)su_{q}(2)suq​(2). The relation of the Fock space for the harmonic oscillator and double Hilbert space of the Gelfand-Naimark-Segal construction is established. The quantum extension of the classical ergodiicity condition is proposed.
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
A derivation of the coordinate representation of the q-deformed oscillator is given. The Hilbert space associated with it is defined on a finite coordinate space y: (-\ensuremath{\pi}/2s,\ensuremath{\pi}/2s), where s is related to the deformation q via q=exp(-s2{\mathit{s}}^{2}s2), with the respective ``coordinate'' and ``momentum'' operators given by X${\mathrm{^}}_{\mathit{s}}$=y-As{\mathit{A}}_{\mathit{s}}As​(y), P${\mathrm{^}}_{\mathit{s}}$=-id/dy-iAs{\mathit{iA}}_{\mathit{s}}iAs​(y), where As{\mathit{A}}_{\mathit{s}}As​(y)=1/2d ln${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$(y)/dy, ${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$ being a measure in this space. The commutation rule obeyed by them is X${\mathrm{^}}_{\mathit{s}}$P${\mathrm{^}}_{\mathit{s}}$-P${\mathrm{^}}_{\mathit{s}}$X${\mathrm{^}}_{\mathit{s}}$=i${\mathrm{\ensuremath{\Elzxh}}}_{\mathit{s}}$(y)=i{1-dAs{\mathit{dA}}_{\mathit{s}}dAs​ /dy}. After constructing a suitable representation in the momentum space, we also set up the Wigner distribution function in this phase space. A brief discussion of these results and their interpretation are given.
Relations between the first, the second quantized representations and deform algebra are investigated. In the case of harmonic oscillator, the axiom of first quantization (the commutation relation between coordinate and momentum operators) and the axiom of second quantization (the commutation relation between creation and annihilation operators) are equivalent. We shown that in the case of q -deformed harmonic oscillator, a violence of the axiom of second quantization leads to a violence of the axiom of first quantization, and inverse. Using the coordinate representation, we study fine structures of the vacuum state wave function depend in the deformation parameter q . A comparison with fine structures of Cooper pair of superconductivity in the coordinate representation is also performed.
In this paper, by using the Wess–Zumino formalism of noncommutative differential calculus, we show that the concept of nonlinear coherent states originates from noncommutative geometry. For this purpose, we first formulate the differential calculus on a GL p, q(2) quantum plane. By using the commutation relations between coordinates and their interior derivatives, we then construct the two-parameter (p, q)-deformed quantum phase space together with the associated deformed Heisenberg commutation relations. Finally, by applying the obtained results for the quantum harmonic oscillator we construct the associated coherent states, which can be identified as nonlinear coherent states. Furthermore, we show that some of the well-known deformed (nonlinear) coherent states, such as two-parameter (p, q)-deformed coherent states, Maths-type q-deformed coherent states, Phys-type q-deformed coherent states and Quesne deformed coherent states, can be easily obtained from our treatment.