Semiclassics for Particles with Spin via a Wigner–Weyl-Type Calculus

We show how to relate the full quantum dynamics of a spin-½ particle on (Formula presented.)(Formula presented.) to a classical Hamiltonian dynamics on the enlarged phase space (Formula presented.)(Formula presented.)up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for (Formula presented.)(Formula presented.)(Folland, Harmonic Analysis on Phase Space, 1989; Lein, Weyl Quantization and Semiclassics, 2010) combined with the Stratonovich–Weyl calculus for SU(2) (Varilly and Gracia-Bondia, Ann Phys 190:107–148, 1989). 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.