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.

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We investigate equilibration processes shortly after sudden perturbations are applied to ultracold trapped superfluids. We show the similarity of phase imprinting and localized density depletion perturbations, both of which initially are found to produce "phase walls". These planar defects are associated with a sharp gradient in the phase. Importantly they relax following a quite general sequence. Our studies, based on simulations of the complex time-dependent Ginzburg-Landau equation, address the challenge posed by these experiments: how a superfluid eventually eliminates a spatially extended planar defect. The processes involved are necessarily more complex than equilibration involving simpler line vortices. An essential mechanism for relaxation involves repeated formation and loss of vortex rings near the trap edge.

The optical absorption or emission by an electron localized at a point imperfection in a crystal is broadened by lattice vibrations. In a fundamental discussion the normal modes of vibration of the whole crystal must be considered. The effect of a mode depends on the displacements it produces in ions near the imperfection. The presence of the imperfection will modify the modes, and, in particular, change the displacements of ions near the imperfection even when the modes do not become localized at the imperfection. In the case of the F center the temperature dependence of the breadth of the F band suggests that this effect must enhance the broadening produced by low-frequency modes (which will certainly not be localized). That this is to be expected from the dynamical theory of normal modes will be shown by discussing: (1) the vibrations of a one-dimensional chain of particles with two adjacent weakened links, (2) the spherical vibrations of an elastic continuum with a spherical cavity, (3) the vibrations with cubic symmetry round the F center in NaCl if only the first eleven shells of ions round the center are allowed to vibrate. Results obtained with this last model will be compared with experimental results for the breadth of the F band in NaCl as a function of temperature.

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