Relation Between Dirac and Canonical Density Matrices, with Applications to Imperfections in Metals
Abstract
It is shown that the canonical density matrix in a singleparticle framework may be related directly to the generalized canonical density matrix, containing the FermiDirac 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. Freeelectron density matrices are fully worked out and a perturbation theory based on these freeelectron forms is presented. It is further shown that for a nonspherical potential energy V(r), the work of Green on the quantummechanical partition function may be utilized to yield a perturbation theory for the Dirac matrix. In this way, the correct formulation to replace Mott's wellknown firstorder 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 selfconsistent framework is pointed out.
 Publication:

Physical Review
 Pub Date:
 November 1960
 DOI:
 10.1103/PhysRev.120.830
 Bibcode:
 1960PhRv..120..830M