Wigner Phase-space Distribution Function Research Articles

Five momenta

A quantum or classical wavefunction depending on position can be associated with a local momentum in at least five apparently different ways: first, as the phase gradient of the wavefunction; second, as the local expectation value of the momentum operator; third, via the local current; fourth, via the Wigner phase-space distribution function; and fifth, as the weak value of momentum with position postselected. The different formulas are all equivalent, but give different insights into the underlying physics. Momenta one through three are largely familiar, but the fourth and fifth are less so.

Wigner Function for Klein—Gordon Landau Problem

First we calculate the Wigner phase-space distribution function for the Klein—Gordan Landau problem on a commmutative space. Then we study the modifications introduced by the coordinate-coordinate noncommuting and momentum-momentum noncommuting, namely, by using a generalized Bopp's shift method we construct the Wigner function for the Klein—Gordan Landau problem both on a noncommutative space (NCS) and a noncommutative phase space (NCPS).

Closure of quantum hydrodynamic moment equations

The hydrodynamic formulation of mixed quantum states involves a hierarchy of coupled equations of motion for the momentum moments of the Wigner function. In this work a closure scheme for the hierarchy is developed. The closure scheme uses information contained in the lower known moments to expand the Wigner phase-space distribution function in a Gauss-Hermite orthonormal basis. The higher moment required to terminate the hierarchy is then easily obtained from the reconstructed approximate Wigner function by a straightforward integration over the momentum space. Application of the moment closure scheme is demonstrated for the dissipative and nondissipative dynamics of two different systems: (i) double-well potential, (ii) periodic potential.

Deformation quantization of noncommutative quantum mechanics

The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantization. In this paper, we investigate the basic aspects of deformation quantization for noncommutative quantum mechanics (NCQM). We first prove some general relations of the Weyl correspondence in non-commuting phase-space. Then we derive explicit form of the Wigner Function (WF) for NCQM starting from fundamental principle of the Weyl correspondence, and show that it satisfies a generalized -genvalue equation. We also demonstrate that the new WFs possess orthonormality and completeness, so they can be used as a basis to expand all phase-space functions. Some example is discussed to support our results.

Comment on “Why the `coarse-graining' of Wigner function is always coarse” [Opt. Commun. 178 (2000) 147

Incorrect statements and assumptions undermining the validity of the results presented in the above paper are pointed out and shortly discussed. It is stressed that a proper `coarse graining' of a Wigner phase-space distribution function cannot be in general performed over finite phase-space cells and that any `smearing' over infinite phase-space areas cannot violate the respective uncertainty relations, which furnishes an answer to the question asked in the title.

Density matrix of anharmonic oscillators by a thermodynamic variation–perturbation technique, a closed form approach

A general approximation technique for the density matrix of coupled anharmonic oscillators is developed. Starting with the Moyal–Weyl trace formalism for the Wigner phase space distribution function, a combination of the Gibbs–Bogoliubov inequality for the partition function and an operator perturbation technique is applied to the Moyal–Weyl trace. Introduction of generating functions and of operator techniques allows the closed form evaluation of the Moyal–Weyl formula for anharmonic oscillators. The Fourier transforms relating the Moyal–Weyl formula and the Wigner function as well as those relating the Wigner function and the density matrix can be given in terms of Hermite polynomials. The Moyal–Weyl formula as the crucial part of the technique is worked out in the first order thermodynamic variation perturbation approximation for the anharmonic quartic oscillator as well as for two coupled anharmonic oscillators as an example for arbitrary coupled oscillators.

On constructing the wave function of a quantum particle from its Wigner phase-space distribution function

For a Schrödinger wave function, the part of the phase that depends only on time disappears in the construction of the corresponding Wigner phase-space (quasi)distribution function. Despite this, it can be recovered from the Wigner function using the quantum Hamilton–Jacobi equation. This is demonstrated for three simple cases.

Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium.

Within the paraxial approximation, a closed-form solution for the Wigner phase-space distribution function is derived for diffuse reflection and small-angle scattering in a random medium. This solution is based on the extended Huygens-Fresnel principle for the optical field, which is widely used in studies of wave propagation through random media. The results are general in that they apply to both an arbitrary small-angle volume scattering function, and arbitrary (real) ABCD optical systems. Furthermore, they are valid in both the single- and multiple-scattering regimes. Some general features of the Wigner phase-space distribution function are discussed, and analytic results are obtained for various types of scattering functions in the asymptotic limit s >> 1, where s is the optical depth. In particular, explicit results are presented for optical coherence tomography (OCT) systems. On this basis, a novel way of creating OCT images based on measurements of the momentum width of the Wigner phase-space distribution is suggested, and the advantage over conventional OCT images is discussed. Because all previous published studies regarding the Wigner function are carried out in the transmission geometry, it is important to note that the extended Huygens-Fresnel principle and the ABCD matrix formalism may be used successfully to describe this geometry (within the paraxial approximation). Therefore for completeness we present in an appendix the general closed-form solution for the Wigner phase-space distribution function in ABCD paraxial optical systems for direct propagation through random media, and in a second appendix absorption effects are included.

Mapping Wigner distribution functions into semiclassical distribution functions

A mapping that relates the Wigner phase-space distribution function of a given stationary quantum mechanical wave function, a solution of the Schr\"odinger equation, to a specific solution of the Liouville equation, both subject to the same potential, is studied. By making this mapping, bound states are described by semiclassical distribution functions still depending on Planck's constant, whereas for elastic scattering of a particle by a potential they do not depend on it, the classical limit being reached in this case. Following this method, the mapped distributions of a particle bound in the P\"oschl-Teller potential and also in a modified oscillator potential are obtained.

A simple semiclassical approach to the Kramers’ problem

We show that the Wigner–Leggett–Caldeira equation for Wigner phase space distribution function which describes the quantum Brownian motion of a particle in a force field in a high temperature, ohmic environment can be identified as a semiclassical version of Kramers’ equation. Based on an expansion in powers of ℏ, we solve this equation to calculate the semiclassical correction to Kramers’ rate.

Comment on “Wigner phase-space distribution function for the hydrogen atom”

We object to the proposal that the mapping of the three-dimensional hydrogen atom into a four-dimensional harmonic oscillator can be readily used to determine the Wigner phase-space distribution function for the hydrogen atom.

Wigner trajectory characteristics in phase space and field theory

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple - indeed, classical - for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.

Phase-Space Quantization of Field Theory

In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple--indeed, classical--for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published and reported at the Yukawa Institute Workshop ''Gauge Theory and Integrable Models'', 26-29 January, 1999.

Features of time-independent Wigner functions

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.

Classical distribution functions derived from Wigner distribution functions

A mapping which relates the Wigner phase-space distribution function associated with a given stationary quantum-mechanical wavefunction to a specific solution of the time-independent Liouville transport equation is obtained. Two examples are studied.

Inelastic neutron scattering from classical systems

The scattering law for inelastic neutron scattering from classical systems is derived by treating the wavefunctions of the scattering system in the stationary phase approximation and taking the classical limit of the Wigner phase space distribution function. In this way recoil effects are properly accounted for and the scattering law fulfils the relation of detailed balance. It is shown why classical van Hove correlation functions do not properly describe neutron scattering from classical systems.

Operator equation of motion in phase space: Application to time-dependent systems possessing invariants.

We have derived an equation of motion for a Wigner operator in phase space, which is the phase-space analog of the Heisenberg equation of motion for a quantum-mechanical operator. An application of this operator equation to time-dependent systems possessing invariants is considered, and the solution for the corresponding Wigner phase-space distribution function is obtained.

Thermal quasiparticles with phase-space distribution.

A new Wigner phase-space distribution function is introduced for thermal quasiparticles currently in use in thermo field dynamics. The properties of this phase-space distribution are studied and compared with those of the corresponding zero-temperature distribution. Generalized uncertainty relations are implemented in order to study the explicit spreading in phase space as a function of temperature and quantum number of the thermal excitation, thereby permitting the investigation of quantum and thermal fluctuations.

Squeezed states and thermally excited states in the wigner phase-space picture of quantum mechanics

The phase-space picture of thermal excitations is presented starting from the formalism of two-mode coherent and squeezed states. It is shown that the thermal degree of freedom can be augmented into the existing formalism of coherent and squeezed states. It is shown further that the thermal excitation leads to a diametric expansion of the Wigner phase-space distribution function.

Linear canonical transformations of coherent and squeezed states in the Wigner phase space. II. Quantitative analysis.

It is possible to calculate expectation values and transition probabilities from the Wigner phase-space distribution function. Based on the canonical transformation properties of the Wigner function, an algorithm is developed for calculating these quantities in quantum optics for coherent and squeezed states. It is shown that the expectation value of a dynamical variable can be written in terms of its vacuum expectation value of the canonically transformed variable. Parallel-axis theorems are established for the photon number and its variant. It is also shown that the transition probability between two squeezed states can be reduced to that of the transition from one squeezed state to vacuum.