ON THE QUANTUM MECHANICAL GENERALIZATION OF BLOCH'S TRANSPORT EQUATION

A simple derivation is given of a kinetic equation for a system of free electrons moving in uniform electric and magnetic fields, and interacting with fixed scattered. The kinetic equation describes the asymptotic behavior of the single-electron density operator if it approaches a steady value, or the asymptotic behavior of its average over oscillations if the density operator oscillates in time. This equation, which is effectively the quantum mechanical generalization of Bloch's transport equation, is identical with the one recently derived by Kosevich and Andreev using Bogolyubov's method. More general considerations show that this asymptotic equation is valid, and describes the approach to the steady state for weak magnetic fields when the relaxation time is much longer than the atomic time. For strong magnetic fields, the same statement holds if the density operator is averaged over its oscillations, whereas the unaveraged approach towards the steady state is governed by a somewhat different equation. The solutions of these two equations become identical in the most important limiting cases. The results obtained previously by different authors follow from the kinetic equation when further assumptions are introduced.