  In this note, we will consider semiclassical scattering for compactly
supported non-trapping perturbations of the Laplacian on $\mathbb{R}^d$. We
will define a family of Gaussian states on $\mathbb{S}^{d-1}$, parametrized by
points in $T^*\mathbb{S}^{d-1}$, and show that the action of the scattering
matrix on a Gaussian state of parameter $\rho\in T^*\mathbb{S}^{d-1}$ is still
a Gaussian state, with parameter $\kappa(\rho)$, where $\kappa$ is the
(classical) scattering map. This is one way of saying that \emph{the scattering
matrix quantizes the scattering map}, complementary to a previous result of
Alexandrova in terms of Fourier Integral Operators.
