Semiclassical evaluation of expectation values

Kush Mohan Mittal, Olivier Giraud, Denis Ullmo

Published 2019-11-22Version 1

Semiclassical Mechanics comprises of a description of quantum systems which preserves their phase information while using only the system's classical dynamics as an input. Over time an identification has been developed between stationary phase approximation and semiclassical mechanics. Although it is true that in most of the cases in semiclassical mechanics the significant contributions come from the neighborhood of the stationary points, there are some important exceptions to it. In this paper, we address one of these exceptions, namely, the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions that are not in the neighborhood of stationary point and provide a new semiclassical expression for the evolution of the expectation values. For our analysis, we employ and discuss two major semiclassical tools. The first being the association of quantum evolution of a wavefunction to the classical evolution of a Lagrangian manifold as done by Maslov and the second being the derivation for semiclassical Wigner function, here the derivation follows the footprints of Berry's original work but our final expression will be bit different and explicitly canonically invariant. Using the canonical invariance of the formalism, we derive an expression for the expectation value of the observable for a one-dimensional case and then further generalize it to higher dimensions. We find that the expression can be written as a sum of classical contribution which corresponds to Truncated Wigner Approximation (TWA) and a term corresponding to oscillatory contributions. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.