Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field

Following up an earlier communication, wave functions are constructed which satisfy the Schr\"odinger equation for a potential which is a sum of a periodic and a uniform field term. The wave functions are Houston modifications of Bloch type functions; the Bloch functions form an orthogonal set whose members are fully determined except for phase. The theory exhibits them in the form of power series in the field strength; the unmodified Bloch band functions form the zero order term of that series. The solutions themselves do not allow for a Zener effect, but the fact that they are only given as power series in E may imply that there is a remainder term causing interband transitions; it would have to be asymptotically smaller than any power of E. Instead of constructing time dependent solutions of the Schr\"odinger equation one can take the time independent functions to construct an effective Hamiltonian for electrons in one band; it has the form (16). Certain indeterminacies are attached to this form of representation; it is shown, however, that final physical answers are unique. The study furnishes an incidental proof that k-space is a finite space consisting in its entirety of what is customarily called the first Brillouin zone. An appendix treats the case of degenerate bands; such bands have singularities in k-space even in the absence of a field. The difficulty is circumvented by working with a set which is not yet diagonalized but free of singularities; these intermediate functions can be continued as power series in E in the same way as nondegenerate band functions.