Semiclassical approximation of the Wigner function for the canonical ensemble

The Weyl-Wigner representation of quantum mechanics allows one to map the density operator in a function in phase space - the Wigner function - which acts like a probability distribution. In the context of statistical mechanics, this mapping makes the transition from the classical to the quantum regimes very clear, because the thermal Wigner function tends to the Boltzmann distribution in the high temperature limit. We approximate this quantum phase space representation of the canonical density operator for general temperatures in terms of classical trajectories, which are obtained through a Wick rotation of the semiclassical approximation for the Weyl propagator. A numerical scheme which allows us to apply the approximation for a broad class of systems is also developed. The approximation is assessed by testing it against systems with one and two degrees of freedom, which shows that, for a considerable range of parameters, the thermodynamic averages are well reproduced

Metaplectic sheets and caustic traversals in the Weyl representation

The quantum Hamiltonian generates in time a family of evolution operators. Continuity of this family holds within any choice of representation and, in particular, for the Weyl propagator, even though its simplest semiclassical approximation may develop caustic singularities. The phase jumps of the Weyl propagator across caustics have not been previously determined. The semiclassical approximation relies on individual classical trajectories together with their neighbouring tangent map. Based on the latter, one defines a continuous family of unitary tangent propagators, with an exact Weyl representation that is close to the full semiclassical approximation in an appropriate neighbourhood. The phase increment of the semiclassical Weyl propagator, as a caustic is crossed, is derived from the facts that the corresponding family of tangent operators belong to the metaplectic group and that the products of the tangent propagators are obtained from Gaussian integrals. The Weyl representation of the metaplectic group is presented here, with the correct phases determined within an intrinsic ambiguity for the overall sign. The elements that fully determine the phase increment across a particular caustic are then analysed

Boletín del Servicio Meteorológico Nacional: Epoca 2ª Año XI Número 111 - 1962 Abril 21

Numeración errónea en el original, se ha corregido