We extend the region of applicability of phase-space techniques, for the study of quantum systems, by developing an expansion for the Wigner distribution function and correlation functions, in powers of an interaction potential. These results are not restricted to the near-classical regime (expansion in powers of \ensuremath{\Elzxh}), in contrast to the Wigner-Kirkwood expansion. The relation to other perturbation methods (Lindstedt-Poincar'e, Green's function) is explored.
A third order expansion for Wigner-Kirkwood commutation function, a complex function in classical phase space that accounts for the Heisenberg uncertainty relation, is approximated and integrated over momentum to give a real function in position configuration space. Metropolis Monte Carlo computer simulation results are given for liquid Lennard-Jones
4He below 10\,K.
The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner phase space formalism. For this purpose the Wigner series expansion for the partition function is generalized to include relativistic corrections. The new series for partition functions and all thermodynamic potentials yield quantum corrections in terms of powers of h(sup 2) and relativistic corrections given by Kelvin functions (modified Hankel functions) K(sub nu)(mc(sup 2)/kT). As applications, the symmetric Toda oscillator, isotonic and singular anharmonic oscillators, and hindered rotators, i.e. oscillators with cosine potential, are addressed.
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.
A numerical method, suitable for the simulation of the time evolution of quantum spin models of arbitrary lattice dimension, is presented. The method combines sampling of the Wigner function with evolution equations obtained from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Quantum correlations are treated through both the Wigner function sampling and the BBGKY evolution, bringing about highly accurate estimates of correlation functions. The method is particularly suitable for long-range interacting systems, and we demonstrate its power by comparing with exact results as well as other numerical methods. As an application we compute spin squeezing in a frustrated two-dimensional lattice with antiferromagnetic power-law interactions and a transverse field, making predictions pertinent to future ion trap experiments.
For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength
λ. This result permits one to generate an analogous
λ-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type
exp(−2x2/λ2) (
x denotes the distance from the wall boundary) which are non-analytic in
λ. Under some condition, the analyticity in
λ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of
λ, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the
ν-dimensional (integer
ν≥2) space.