Abstract
We show how to relate the full quantum dynamics of a spin-½ particle on \({\mathbb{R}^d}\) to a classical Hamiltonian dynamics on the enlarged phase space \({\mathbb{R}^{2d} \times \mathbb{S}^{2}}\) up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for \({\mathbb{R}^d}\) (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.
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Communicated by Jens Marklof.
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Gat, O., Lein, M. & Teufel, S. Semiclassics for Particles with Spin via a Wigner–Weyl-Type Calculus. Ann. Henri Poincaré 15, 1967–1991 (2014). https://doi.org/10.1007/s00023-013-0294-0
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DOI: https://doi.org/10.1007/s00023-013-0294-0