  We analyse the dynamics of expectation values of quantum observables for the
time-dependent semiclassical Schr\"odinger equation. To benefit from the
positivity of Husimi functions, we switch between observables obtained from
Weyl and Anti-Wick quantization. We develop and prove a second order Egorov
type propagation theorem with Husimi functions by establishing transition and
commutator rules for Weyl and Anti-Wick operators. We provide a discretized
version of our theorem and present numerical experiments for Schr\"odinger
equations in dimensions two and six that validate our results.
