Quantum Theory of Kinetic Equations for Electrons in Phonon Fields

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Abstract
A method is presented which yields a kinetic equation for the time development of the one-electron density matrix for dynamically independent electrons in phonon and other arbitrary static fields of force under the influence of a weak but otherwise general external dynamical disturbance. The exclusion principle, the finiteness of the phonon energies, and the quantum-mechanical nature of the processes involved are taken into account rigorously. The effects of the electron-phonon interaction are described only in the Born approximation. A kinetic equation for the steady state is also derived in an arbitrary one-electron representation. An iterative solution of the kinetic equation is discussed and the power absorption due to direct and indirect transitions is derived from the general kinetic equation. The case of a uniform but oscillating electric field is considered in detail for free, Landau, and Bloch electrons, for which special kinetic equations are derived. A "local density matrix"—useful in the calculation of space-varying densities of various physical observables—is introduced and a kinetic equation for it is obtained from the general theory. Applications of it are made to the case of free and Landau electrons for disturbances of arbitrary space-time variation. The applicability of the method to more general situations is indicated.