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In this paper the effective mass approximation and k.p multi-band models,describing quantum evolution of electrons in a crystal lattice, are discussed.Electrons are assumed to move in both a periodic potential and a macroscopicone. The typical period of the periodic potential is assumed to be very small,while the macroscopic potential acts on a much bigger length scale. Suchhomogenization asymptotic is investigated by using the envelope-functiondecomposition of the electron wave function. If the external potential issmooth enough, the k.p and effective mass models, well known in solid-statephysics, are proved to be close (in strong sense) to the exact dynamics.Moreover, the position density of the electrons is proved to converge weakly toits effective mass approximation.

Quantum transport in crystals: effective-mass theorem and k.p Hamiltonians