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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Wigner equations of motion for classical systems

Lorenzo Galleani, Leon W. Cohen

Published Nov 13, 2000• 3 citations
Publisher

Abstract

We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.

DOI
10.1117/12.406530

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We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
Abstract Some of the non-classical distribution functions defined on the phase space can be unified owing to specific parameterization. The latter allows one to think about the general form of the equation of motion (EOM) for such parameterized distribution functions. Motivated by this idea, we derive the EOM for so-called the τ-Wigner distribution function (WDF). This parameterization directly results from a modification of the linear transformation of the spatial variables which is used to derive the original WDF and its EOM. The derived equation is analytically solved for the case of linear potential. The point symmetries of this last equation are also analyzed.
We have previously presented a method to write equations for the Wigner distribution corresponding to the solution of a linear differential equation. We extend this method to arbitrary time-frequency distributions, the Short Time Fourier Transform, and the wavelet transform. Also, we apply the method to stochastic signals and address a number of applications including Brownian motion. We obtain the Wigner distribution for Brownian motion by solving the Langevin equation and also by using our method. We obtain exact solutions.
A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.