Wigner Phase-space Distribution Function Research Articles

Matter, momentum and energy flow in heavy-ion collisions

We study the flow of matter, momentum and energy in low-energy heavy-ion collisions. This is done by using the Wigner phase space distribution function to calculate quantum mechanical analogs of the classical distributions of the observables. We apply the Wigner transformation to time-dependent Hartree-Fock calculations of 16O+ 16O and 40Ca+ 40Ca reactions. Both static and dynamic features of the distribution functions are demonstrated. We find a significant energy flow perpendicular to the beam direction. The one-way matter current obtained from time-dependent Hartree-Fock calculations is reproduced well by a combination of two simplified models for the relative motion and tunneling.

Shell and surface effects in the static Wigner phase-space distribution function of nuclei

Abstract Using smoothed single-particle occupation numbers, the contribution of shell and surface effects to the static Wigner phase-space distribution function for non-interacting nucleons in an external, spherical, harmonic oscillator potential well is determined. The smoothing procedure illustrates clearly the strong quantum oscillations of the distributions function around the constant value for an infinite Fermi gas, and allows the separation of the contributions arising from shell effects connected with the structure of the single-particle level spectrum near the Fermi surface, and surface effects associated with single reflections of nucleons from the surface of the potential well. The temperature dependence of these effects in the distribution function is also investigated and a thermodynamical definition of the shell contribution to the kinetic energy density is suggested.

Diffractive ray tracing of laser beams

A method is developed for extending conventional ray-tracing techniques to include diffractive effects associated with focused laser beams. The main idea is to provide rays with a distribution of directions at each point in space, as in radiative transfer calculations. The Wigner phase-space distribution function is used to obtain formulas for the spread of ray angles that yield the correct diffraction patterns for coherent and partially coherent beams. This method is readily implemented numerically by means of computer programs that are used in Monte Carlo radiative transfer calculations.

Relativistic quantum transport theory approach to multiparticle production

The field-theoretic description of multiparticle production processes is cast in a form analogous to ordinary transport theory. Inclusive differential cross sections are shown to be given by integrals of covariant phasespace distributions. The single-particle distribution function F(p, R) is defined as the Fourier transform of a suitable correlation function in analogy with the nonrelativistic (Wigner) phase-space distribution function. Its transform F(p, q) is observed to be essentially the discontinuity of a multiparticle scattering amplitude. External-field problems are studied to exhibit the physical content of the formalism. When q=0 one recovers the single-particle distribution exactly. The equation of motion for F(p, R) generates an infinite hierarchy of coupled equations for various distribution functions. In the Hartree approximation one obtains nonlinear integral equations analogous to the Vlasov equation in plasma physics. Such equations are convenient for exhibiting collective motions; in particular it appears that a collective mode exists in a φ^4 theory for a uniform infinite medium. It is speculated that such collective modes could provide a theoretical basis for clustering effects in multiparticle production.

The rotational wigner function

The quantum mechanics of angular momentum is studied using the Wigner phase-space distribution function. To accomplish this in a way which allows one to retain the bilinear form of the distribution function originally conceived by Wigner, a new set of generalized variables is introduced, viz. the Cayley-Klein (C.K.) coordinates and momenta. The equations relating the four C.K. coordinates to the three Euler angles give rise to a constraint which is relaxed in a natural way by introducing a fourth “radial” variable ϱ whose conjugate momentum is p ϱ . Physical quantities in the classical mechanics of rotational systems are re-expressed in the C.K. representation; and the canonical transformation between the C.K. and Eulerian variables is established. In a similar manner, after introducing the basic quantum mechanical operators for the C.K. coordinates and momenta, the wavefunctions and operators for rotational systems are re-expressed in these variables. With the aid of the Weyl correspondence to link the operators of quantum mechanics with the dynamical functions of classical mechanics, the phase-space formulation is established. Rotational Wigner distribution functions are constructed, their properties investigated, and their form is explicitly given for several quantum states. Expectation values are then calculated according to the procedure identical to the one in classical statistical mechanics for calculating phase space averages. The equation of motion for the distribution function is derived. It turns out to be the sine expansion of the Poisson-bracket operator expressed in the C.K. variables and operating on the Hamiltonian and distribution function. This equation reduces to Liouville's equation in the classical limit. The Heisenberg picture of the phase-space formulation is introduced, and the equations of motion for the body-fixed components of the angular momentum are studied. In the classical limit Euler's equations of motion are recovered. Finally, the thermodynamic properties of a system in equilibrium are investigated. The temperature dependence of the ensemble Wigner function is shown to be given by a cosine expansion of the Poisson bracket operator expressed in the C.K. variables, and operating on the Hamiltonian and distribution function. An ansatz is then made on the form of the distribution function, specifically, a power series in Planck's constant with coefficients which are functions of the coordinates and momenta. Substitution of this postulated form into the phase-space Bloch equation allows an explicit determination of these functions. Integration of the Wigner function over all momenta gives the configurational distribution function. A further integration over the coordinates gives the partition function. This procedure is carried out in detail for the asymmetric top, and the first quantum corrections to both the configurational distribution function and partition function are obtained. Upon setting appropriate moments of inertia equal to each other, these quantum corrections reduce to those of the symmetrical and spherical tops.