Annales Henri Poincaré

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

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

Article

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.

Keywords

Principal Symbol Semiclassical Limit Semiclassical Approximation Linear Polynomial Extend Phase Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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