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

Abstract

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(x), 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 wellknown 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 selfconsistent framework is pointed out.

Temperature dependence of persistent spin currents in a spin-orbit-coupled electron gas: A density-matrix approach

“…The Bloch density matrix is of particular interest since its knowledge enables the Dirac density matrix ͑r ជ , ; r ជ Ј, Ј͒ to be found, through the inverse Laplace transform. 20,21 In fact, at T = 0, the single particle density operator, E F = ͚ all i ͉⌿ i ͗͘⌿ i ͉͑E F − ⑀ i ͒ can be rewritten, for a given fermi energy E F , as…”

Propagator and Slater sum in one‐body potential theory

“…It is relevant here to add that, using the early work of March and Murray [4], analytical progress can be made in deriving the local density of states Nðr; EÞ in the continuum for the bare Coulomb potential. For its s-wave component N s ðr; EÞ, the result is simple in terms of the Whittaker function MðÀiZ=k; 1=2; 2ikrÞ with energy E ¼ k 2 =2, as shown by Howard et al [13], the r-dependence of N s having the form N s ðr; EÞ / M 2 ðÀiZ=k; 1=2; 2ikrÞ…”

Nonlinear theory of scattering by localized potentials in metals

“…March and Murray [1] gave a partial differential equation for the lth partial wave component, ρ l (r, E), of the total density of electrons at r lying below energy E, say ρ(r, E). Putting l = 0 in their equation (4.12), one can remove their energy integration by considering the local density of s-states:…”

“…Thus, in section 2 immediately below, we study analytically a central repulsive exponential potential. Reverting to the early investigation of March and Murray [1], we utilize their treatment of central potentials to study analytically the s-wave component ρ s (r, E) of the density generated by such an exponential potential. Some contact can then be established with the ZNE result.…”