Annales Henri Poincaré

, Volume 15, Issue 10, pp 1967–1991 | Cite as

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 \({\mathbb{R}^d}\) 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.


Principal Symbol Semiclassical Limit Semiclassical Approximation Linear Polynomial Extend Phase Space 
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© Springer Basel 2014

Authors and Affiliations

  1. 1.Racah Institute of PhysicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Department of Mathematics and Fields InstituteUniversity of TorontoTorontoCanada
  3. 3.Mathematisches InstitutEberhard Karls Universität TübingenTübingenGermany

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