Relation Between Dirac and Canonical Density Matrices, with Applications to Imperfections in Metals

It is shown that the canonical density matrix in a single-particle framework may be related directly to the generalized canonical density matrix, containing the Fermi-Dirac function, and to the Dirac density matrix.

A study is then made of density matrices in central field problems. A new differential equation is derived, from the Bloch equation, for the diagonal element of the canonical density matrix. In the case of a continuum of energy levels, this is shown to lead directly to a differential equation for the diagonal element of the Dirac matrix, that is, the particle density. Free-electron density matrices are fully worked out and a perturbation theory based on these free-electron forms is presented.

It is further shown that for a nonspherical potential energy $V\left(\mathrm{r}\right)$, the work of Green on the quantum-mechanical partition function may be utilized to yield a perturbation theory for the Dirac matrix. In this way, the correct formulation to replace Mott's well-known first-order approximation for dealing with imperfections in metals is obtained. A brief discussion of the way in which this removes qualitatively the difficulties of the Mott treatment is given and the possibility of direct numerical application in a self-consistent framework is pointed out.