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Journal of Modern Optics

Wigner equation of motion for time-dependent potentials

Lorenzo Galleani, Leon W. Cohen

Published Mar 1, 2002• 3 citations
Publisher

Abstract

The equation of motion is obtained for the Wigner distribution for time-dependent potentials. It is shown that one cannot obtain an equation of motion for the standard Wigner distribution but one can do so for the fourdimensional distribution of the variables position, momentum, time, and frequency. Three forms are given and it is shown that for time-independent potentials the new form reduces to the equations originally obtained by Wigner and Moyal.

DOI
10.1080/09500340110088515

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The equation of motion is obtained for the Wigner distribution for time-dependent potentials. It is shown that one cannot obtain an equation of motion for the standard Wigner distribution but one can do so for the fourdimensional distribution of the variables position, momentum, time, and frequency. Three forms are given and it is shown that for time-independent potentials the new form reduces to the equations originally obtained by Wigner and Moyal.
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.
The formalism of the Wigner distribution function is reviewed. In addition to the Liouville equation, which expresses the time rate of change of this function in terms of its Moyal bracket with the Hamiltonian, and its expression as a projection operator, a third equation is proposed with the aid of an auxiliary variable s, to which a formal solution is constructed in terms of known quantum-mechanical eigenfunctions and eigenvalues. In addition, an ab initio solution to the three equations in terms of an error function is found for the free particle in one dimension. Two views are advanced: the orthodox, that this new equation is merely a consistency requirement, and the speculative, that the measurement process has something to do with the choice of s.
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.