The motion of an electron in a periodic potential field, and accelerated by a uniform field, can be obtained by treating the time-dependent Schroedinger equation. The result shows that the wave vector increases linearly with the time within the bounds of a single Brillouin zone. At the boundaries of the zones transitions to other zones may take place if the accelerating field is large enough.
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W. V. Houston (1940) studied this question.
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