skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Wigner functions of s waves

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.

Authors:
 [1];  [2];  [3];  [2]; ;  [4]
  1. Chemical Physics, Department of Chemistry, Technical University of Denmark, DTU 207, DK-2800 Lyngby (Denmark)
  2. (Germany)
  3. Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, (Hungary)
  4. Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm (Germany)
Publication Date:
OSTI Identifier:
20982459
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052107; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENFUNCTIONS; EIGENVALUES; PARTICLES; QUANTUM ENTANGLEMENT; S WAVES; WAVE FUNCTIONS; WIGNER DISTRIBUTION

Citation Formats

Dahl, J. P., Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm, Varro, S., Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm, Wolf, A., and Schleich, W. P. Wigner functions of s waves. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.75.052107.
Dahl, J. P., Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm, Varro, S., Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm, Wolf, A., & Schleich, W. P. Wigner functions of s waves. United States. doi:10.1103/PHYSREVA.75.052107.
Dahl, J. P., Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm, Varro, S., Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm, Wolf, A., and Schleich, W. P. Tue . "Wigner functions of s waves". United States. doi:10.1103/PHYSREVA.75.052107.
@article{osti_20982459,
title = {Wigner functions of s waves},
author = {Dahl, J. P. and Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm and Varro, S. and Institut fuer Quantenphysik, Universitaet Ulm, D-89069 Ulm and Wolf, A. and Schleich, W. P.},
abstractNote = {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},
journal = {Physical Review. A},
number = 5,
volume = 75,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
  • In the functional-integral technique an auxiliary field, coupled to appropriate operators such as spins, linearizes the interaction term in a quantum many-body system. The partition function is then averaged over this time-dependent stochastic field. Quantum Monte Carlo methods evaluate this integral numerically, but suffer from the [ital sign] (or [ital phase]) [ital problem]: the integrand may not be positive definite (or not real). It is shown that, in certain cases that include the many-band Hubbard model and the Heisenberg model, the sign problem is inevitable on fundamental grounds. Here, Monte Carlo simulations generate a distribution of incompatible operators---a [ital Wigner]more » [ital function]---from which expectation values and correlation functions are to be calculated; in general no positive-definite distribution of this form exists. The distribution of time-averaged auxiliary fields is the convolution of this operator distribution with a Gaussian of variance proportional to temperature, and is interpreted as a Boltzmann distribution exp([minus][beta][ital V][sub eff]) in classical configuration space. At high temperatures and large degeneracies this [ital classical] [ital effective] [ital Hamiltonian] [ital V][sub eff] tends to the static approximation as a classical limit. In the low-temperature limit the field distribution becomes a Wigner function, the sign problem occurs, and [ital V][sub eff] is complex. Interpretations of the distributions, and a criterion for their positivity, are discussed. The theory is illustrated by an exact evaluation of the Wigner function for spin [ital s] and the effective classical Hamiltonian for the spin-1/2 van der Waals model. The field distribution can be negative here, more noticeably if the number of spins is odd.« less
  • The Wigner phase-space distribution function provides the basis for Moyal{close_quote}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 ({open_quotes}star{close_quotes}) eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ({open_quotes}supersymmetric{close_quotes}) isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P{umlt o}schl-Teller potential, and the Liouville potential. {copyright} {ital 1998} {ital The American Physical Society}
  • In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasiprobability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuouslymore » indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.« less