  We show how to relate the full quantum dynamics of a spin-1/2 particle on R^d
to a classical Hamiltonian dynamics on the enlarged phase space R^d x S^2 up to
errors of second order in the semiclassical parameter. This is done via an
Egorov-type theorem for normal Wigner-Weyl calculus for R^d [Lei10,Fol89]
combined with the Stratonovich-Weyl calculus for SU(2) [VGB89]. For a specific
class of Hamiltonians, including the Rabi- and Jaynes-Cummings model, we prove
an Egorov theorem for times much longer than the semiclassical time scale. We
illustrate the approach for a simple model of the Stern-Gerlach experiment.
