Eur. Phys. J. B 58, 137-148 (2007)
https://doi.org/10.1140/epjb/e2007-00212-6

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

¹ Institut Fourier, UMR 5582 CNRS-UJF, UFR de Mathématiques, Université Grenoble I, BP 74, 38402 Saint Martin d'Hères Cedex, France
² Laboratoire de Physique Moléculaire et des Collisions, ICPMB-FR CNRS 2843, Université Paul Verlaine-Metz, 57078 Metz Cedex 3, France

Corresponding author: ^a mohrbach@univ-metz.fr

Received: 16 August 2006
Revised: 26 June 2007
Published online: 10 August 2007

Abstract

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.