  We study the quantum evolution in dimension three of a system composed by a
test particle interacting with an environment made of $N$ harmonic oscillators.
At time zero the test particle is described by a spherical wave, i.e. a highly
correlated continuous superposition of states with well localized position and
momentum, and the oscillators are in the ground state. Under suitable
assumptions on the physical parameters characterizing the model, we give an
asymptotic expression of the solution of the Schr\"odinger equation of the
system with an explicit control of the error. The result shows that the
approximate expression of the wave function is the sum of two terms, orthogonal
in $L^2(\erre^{3(N+1)})$ and describing rather different situations. In the
first one all the oscillators remain in their ground state and the test
particle is described by the free evolution of a slightly deformed spherical
wave. The second one consists of a sum of $N$ terms where in each term there is
only one excited oscillator and the test particle is correspondingly described
by the free evolution of a wave packet, well concentrated in position and
momentum. Moreover the wave packet emerges from the excited oscillator with an
average momentum parallel to the line joining the oscillator with the center of
the initial spherical wave. Such wave packet represents a semiclassical state
for the test particle, propagating along the corresponding classical
trajectory. The main result of our analysis is to show how such a semiclassical
state can be produced, starting from the original spherical wave, as a result
of the interaction with the environment.
