Long time asymptotics for quantum particles in a periodic potential

We study a quantum particle in a periodic potential and subject to slowly varying electromagnetic potentials. It is proved that, in the Heisenberg picture, the scaled position operator εx(ε^-1t) has a limit as ε→0. The limit operator is determined by the semiclassical equations of motion, which implies that for long times the wave packet is well approximated by the semiclassical evolution. From our result we infer the hydrodynamic limit, q→0, t→8, qt = const, of the structure function S(q, t) of a fluid of noninteracting fermions in a crystal potential.