  Semiclassical Mechanics allows for a description of quantum systems which
preserves their phase information, while using only the system's classical
dynamics as an input. Over the 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, occurring in the evaluation of the time evolution of the
expectation value of an operator. We explain why it is necessary to include
contributions which are not in the neighborhood of stationary points and
provide new semiclassical expressions for the evolution of the expectation
values. For our analysis we employ and discuss two major semiclassical tools.
The first one is the association of the quantum evolution of a wavefunction to
the classical evolution of a Lagrangian manifold, as done by Maslov. The second
one is the derivation of an expression for the semiclassical Wigner function
whose properties under canonical transformation are made explicit. Using the
canonical invariance of the formalism, we derive an expression for the
expectation value of observables for the one-dimensional case and then
generalize it to higher dimensions. We find that the expression can be written
as the sum of a classical contribution which corresponds to what is referred to
as the Truncated Wigner Approximation (TWA) in the cold-atoms physics context,
or the Linearized Semiclassical Initial Value Representation(LSC-IVR) in
chemical or molecular physics, and additional terms associated with
interferences. Along the way, we get a deeper understanding of the origin of
these interference effects and an intuitive geometric picture associated with
them.
