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arXiv:2003.11737 · Results in Physics

Extended Wigner function for the harmonic oscillator in the phase space

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, Evgenii V. Burlakov

Published Nov 5, 2020• 6 citations
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Abstract

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.

DOI
10.1016/j.rinp.2020.103546

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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.
Abstract This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space x , p , p ̇ , p ̈ .
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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.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type ψ(q)=e−aq2ϕ(q)ψ(q)=e^{-aq^2}ϕ(q)ψ(q)=e−aq2ϕ(q), the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples.