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Yuanzihe wuli pinglun

Solution of Wigner Function’s Star-product for 3-D Harmonic Oscillator

Desheng Chen, Yahui Wang, Yao Wang, Kai Ma

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

As a quasi-probability distribution function in phase-space and a special representation of the density matrix, Wigner function has great significance in physics. In this paper, first, Wigner function’s characteristic and calculation approach are introduced. Then, with Star-eigen equation we obtain the Wigner function for three-dimensional harmonic oscillator. In the end, we discuss the method describing the interaction between phonons and electrons (or photons) and obtain the equation that transition probability satisfies in phase space.

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

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Extended Wigner function for the harmonic oscillator in the phase space

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.

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Analytical solution for quantum quartic oscillator in phase space

A. X. Martins, T. M. Rocha Filho, R. G. G. Amorim +3 more

In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.

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Relationship between the Wigner function and the probability density function in quantum phase space representation

Qian Shu Li, Gong Min Wei, Li Qiang Lü

This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.

Physical Review A•Aug 20, 2004•26 citations

Universal density matrix for the phase space

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form $W=\operatorname{Sp}\left[ ρ\mathcal{W} \right]$, where $ρ$ is the density matrix, and $\mathcal{W}$ is the universal density matrix. The density matrix $ρ$ for each quantum system is different, and the universal matrix $\mathcal{W}$ is the same for any quantum system. Thus, the matrix $\mathcal{W}$ has a fundamental character. In the work, the elements of the universal matrix $\mathcal{W}$ were found explicitly and their properties were investigated. The diagonal elements of the matrix $\mathcal{W}$ are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix $\mathcal{W}$ contain frequency oscillations responsible for dissipations in the quantum systems.

arXiv:1904.04950•Apr 10, 2019•2 citations
As a quasi-probability distribution function in phase-space and a special representation of the density matrix, Wigner function has great significance in physics. In this paper, first, Wigner function’s characteristic and calculation approach are introduced. Then, with Star-eigen equation we obtain the Wigner function for three-dimensional harmonic oscillator. In the end, we discuss the method describing the interaction between phonons and electrons (or photons) and obtain the equation that transition probability satisfies in phase space.
As Wigner quasi-probability distribution function,the Wigner function contains the entire information in the evolution of the quantum states in the phase space.The function also has great significance in physics.This paper introduces the definition and nature of the Wigner function,as well as the general expression.Then,it used the integral method given Wigner function of a one-dimensional infinitely deep square potential well.Finally,we discuss the characteristics of the distribution of the function and point out that it is a real function of the phase space.
New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.
In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.
This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.
In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form W=Sp⁡[ρW]W=\operatorname{Sp}\left[ ρ\mathcal{W} \right]W=Sp[ρW], where ρρρ is the density matrix, and W\mathcal{W}W is the universal density matrix. The density matrix ρρρ for each quantum system is different, and the universal matrix W\mathcal{W}W is the same for any quantum system. Thus, the matrix W\mathcal{W}W has a fundamental character. In the work, the elements of the universal matrix W\mathcal{W}W were found explicitly and their properties were investigated. The diagonal elements of the matrix W\mathcal{W}W are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix W\mathcal{W}W contain frequency oscillations responsible for dissipations in the quantum systems.