Motion of an Electron in a Perturbed Periodic Potential

Wannier's theorem concerning the motion of an electron in a perturbed periodic potential is generalized so as to take into account transitions between energy bands. The generalized form of the theory is placed on a rigorous quantum-mechanical foundation and the basic approximation of Wannier's form of the theorem elucidated. A compact matrix formulation is developed which permits the multiband character of the formalism to be handled expeditiously. It is shown that corrections to Wannier's approximate equation of motion arising from variations of the potential within a unit cell require a multiband formalism, since these corrections affect the motion primarily by causing transitions between bands. The correction terms are expressed in such a form that they may be conveniently treated as perturbations. A generalized form of Wannier's theorem is presented in which the perturbing potential may depend upon both the coordinate and momentum of the electron. The special case that the perturbative potential is due to slowly varying combined electric and magnetic fields is discussed in detail. Applicability to the problem of the electron-phonon interaction is discussed briefly.