Semiclassical approximations in phase space with coherent states

Abstract

We present a complete derivation of the semiclassical limit of the coherent\nstate propagator in one dimension, starting from path integrals in phase space.\nWe show that the arbitrariness in the path integral representation, which\nfollows from the overcompleteness of the coherent states, results in many\ndifferent semiclassical limits. We explicitly derive two possible semiclassical\nformulae for the propagator, we suggest a third one, and we discuss their\nrelationships. We also derive an initial value representation for the\nsemiclassical propagator, based on an initial gaussian wavepacket. It turns out\nto be related to, but different from, Heller's thawed gaussian approximation.\nIt is very different from the Herman--Kluk formula, which is not a correct\nsemiclassical limit. We point out errors in two derivations of the latter.\nFinally we show how the semiclassical coherent state propagators lead to\nWKB-type quantization rules and to approximations for the Husimi distributions\nof stationary states.\n