Bloch electrons in an electric field

It is used the Wigner distribution function and the Von Neumann equation to treat the problem of the evolution of Bloch electrons moving under the combined influence of a periodic potential and of an applied continuous electric field. It is shown that the analysis provides an efficient tool for the solution of the problem and incorporates, in a fairly natural way, the features of previous treatments. It is discussed analogies with other dynamical systems, ruled by a periodic potential, as those relevant to Free Electron Laser quantum interactions