  We analyze the Stern-Gerlach experiment in phase space with the help of the
matrix Wigner function, which includes the spin degree of freedom. Such
analysis allows for an intuitive visualization of the quantum dynamics of the
apparatus. We include the interaction with the environment, as described by the
Caldeira-Leggett model. The diagonal terms of the matrix provide us with
information about the two components of the state, that arise from interaction
with the magnetic field gradient. In particular, from the marginals of these
components, we obtain an analytical formula for the position and momentum
probability distributions in presence of decoherence, that show a diffusive
behavior for large values of the decoherence parameter. These features limit
the dynamics of the present model. We also observe the decay of the
non-diagonal terms with time, and use this fact to quantify the amount of
decoherence, from the norm of those terms in phase space. From here, we can
define a decoherence time scale, which differs from previous results that make
use of the same model.
