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Journal of Shaanxi University of Technology

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

Yan Jiang-feng

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

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.

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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.
The quantum harmonic oscillator is a basic model for many complex ones.After integration,the Wigner function of quantum harmonic oscillator can be expressed by a simple form,which can be used to solve many related problems.This paper has placed the harmonic oscillator in non-commutative phase space,and studied the Wigner function of two-dimensional charged harmonic oscillator in electric field.The results show that when the non-commutative parameter is zero,the Wigner function of two-dimensional charged harmonic oscillator is equivalent to its form in commutative space.
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.