4 research outputs found
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
BoletĂn del Servicio MeteorolĂłgico Nacional: Epoca 2ÂȘ Año XI NĂșmero 111 - 1962 Abril 21
NumeraciĂłn errĂłnea en el original, se ha corregido