WKB calculation of quantum adiabatic phases and nonadiabatic corrections

2014 Berry discovered that wave functions may acquire a geometrical phase factor, in addition to the usual dynamical phase, during an adiabatic cycle with period T. The dynamical and Berry phases are here recognized as the first two terms in a systematic WKB expansion in powers of 03B5 = 1/ T. We thus provide a simple method for the calculation of adiabatic phases and for a consistent inclusion of nonadiabatic corrections. J. Phys. France 49 (1988) 561-566 AVRIL 1988, Classification Physics Abstracts 03.65 42.50 Following the original discovery of Berry [1, 2] quantum ’adiabatic phases have become the subject of intense theoretical as well as experimental investigations. In a recent paper [3] Bouchiat provided a group-theoretical derivation of the Berry phase and suggested that the same method could be used for the computation of nonadiabatic corrections. An alternative approach is discussed here based on the realization that the usual dynamical and Berry phases are the first two terms in a suitable WKB expansion. Nonadiabatic corrections are then computed systematically by including higher-order terms in the WKB series. For definiteness we consider the precession of a spin S in a time-dependent magnetic field; i. e. we assume a Hamiltonian of the form The field is conveniently parametrized by spherical coordinates with respect to a fixed reference frame XYZ, as is indicated in figure 1 which follows the conventions of reference [3]. Both the magnitude B and the direction of the field (angles 9 and Ø) are arbitrary periodic functions of time with period T. As long as the Hamiltonian is linear in the spin operators, a solution for arbitrary spin s may be obtained from the spin) solution by elementary quadratures [4-7]. We (*) Also at the Department of Physics, University of Crete, Iraklion, Greece. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904056100