On the Electrical Conductivity of Metals

The exact formula for the electrical conductivity of a metal, derived by Kubo, Greenwood, and others, is evaluated using Van Hove's methods for quantum mechanical transport problems. The evaluation is restricted to the case of elastic scattering by impurities or lattice vibrations, but completely avoids the use of the customary random phase assumption. In the limit of weak coupling our method yields formulae for the conductivity that are identical with those derived from the Boltzmann equation. The accuracy of the weak coupling approximation is investigated and it is found to be accurate as long as h/τ<<η, where τ is the collision time and η is the Fermi energy. A general treatment of the conductivity in an anisotropic metal is given and an explicit expression for the conductivity is written down in terms of the eigenvalues and eigenvectors of an integral equation. A general and rigorous derivation of the conventional Boltzmann equation is given using Van Hove's results on the quantum mechanical master equation.