Quantum-Mechanical Approach to Thermal Transport Phenomena in Metals

A quantum-mechanical transport equation corresponding to the classical Boltzmann equation is developed. Classical transport theory in the presence of a temperature gradient involves the phase space density, which has no meaning in quantum mechanics due to the complementarity of position and momentum. The superlattice representation allows the development of a quantum transport equation from the density matrix Schr\"odinger equation, through the introduction of irreversibility by standard methods. The quantum transport equation governing the electron occupation probability in the superlattice representation is derived for impurity scattering and for phonon scattering. This equation has the same field terms as the classical Boltzmann equation, but involves the discrete coarse-grained wave number and position of the superlattice representation. The scattering terms involve transitions between different superlattice states, and thus include scattering processes which move electrons from one superlattice cell to another. The solution of the classical equation is affected very little by these quantum effects, for all temperature gradients which might reasonably be encountered.