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Physical Review A

Wigner functions of s waves

Joachim Dahl, Sándor Varró, A. Wolf, Wolfgang P. Schleich

Published May 11, 2007• 2 citations
Publisher

Abstract

We derive explicit expressions for the Wigner function of wave functions in $D$ dimensions which depend on the hyperradius---that is, of $s$ waves. They are based either on the position or the momentum representation of the $s$ wave. The corresponding Wigner function depends on three variables: the absolute value of the $D$-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementary $s$ wave and the energy eigenfunction of a free particle.

DOI
10.1103/physreva.75.052107

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We derive explicit expressions for the Wigner function of wave functions in DDD dimensions which depend on the hyperradius---that is, of sss waves. They are based either on the position or the momentum representation of the sss wave. The corresponding Wigner function depends on three variables: the absolute value of the DDD-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementary sss wave and the energy eigenfunction of a free particle.
We show that sss waves, that is wave functions that only depend on a hyperradius, are entangled if and only if the corresponding Wigner functions exhibit negative domains. We illustrate this feature using a special class of sss waves which allows us to perform the calculations analytically. This class includes a Gaussian, a maximally entangled as well as a ``shell'' state.
Using the Wigner-Vlasov formalism, an exact three-dimensional solution of the Schr\"odinger equation for a scalar particle in an electromagnetic field is constructed. Electric and magnetic fields are nonuniform. According to the exact expression for the wave function, the search for two types of Wigner function is conducted. The first function is the usual Wigner function with a kinetic momentum. The second Wigner function is constructed on the basis of the Weyl-Stratonovich transform [Javanainen et al., Phys. Rev. A 35, 2791 (1987); Nedjalkov et al., Phys. Rev. B 99, 014423 (2019)]. It turns out that the second function, unlike the first one, has areas of negative values for wave functions with the Gaussian distribution (Hudson's theorem). An example of an electromagnetic quantum system described by a non-Gaussian wave function has successfully been found. The second Wigner function is positive over the whole phase space for the non-Gaussian wave function. This result is analogous to the Hudson theorem for the gauge invariant Wigner function. On the one hand, knowing the Wigner functions allows one to find the distribution of the mean momentum vector field and the energy spectrum of the quantum system. On the other hand, within the framework of the Wigner-Vlasov formalism, the mean momentum distribution and the magnitude of the energy are initially known. Consequently, the mean momentum distributions and energy values obtained according to the Wigner functions can be compared with the exact momentum distribution and energy values. This paper presents this comparison and describes the differences. The Vlasov-Moyal approximation of average acceleration flow has been built in phase space for a quantum system with an electromagnetic field. The obtained approximation makes it possible to cut the Vlasov chain off at the second equation and also to analyze the Boltzmann H-function evolution. By averaging the Vlasov-Moyal approximation over momentum space we can derive an expression for the electromagnetic force of the classical system. This averaging makes the high-order quantum terms disappear from the ``motion equation.''
The hydrogen atom is a fundamental exactly soluble system for which the Wigner function, being a quantum analogue of the joint probability distribution of position and momentum, is unknown. In this paper, we present an effective method of calculating the Wigner function, for all bound states of the nonrelativistic hydrogen atom. The formal similarity between the eigenfunctions of the nonrelativistic hydrogen atom in the momentum representation and the Klein–Gordon propagator has allowed the calculation of the Wigner function for an arbitrary bound state of the hydrogen atom, using a simple atomic integral as a generator. These Wigner functions for some low-lying states are depicted and discussed.