Propagator and Slater sum in one-body potential theory
For a one-body potential V(r) generating eigenfunctions ψi(r) and corresponding eigenvalues εi, the Feynman propagator K(r, r′, t) is simply related to the canonical density matrix C(r, r′, β) by β → it. The diagonal element S(r, r, β) of C is the so-called Slater sum of statistical mechanics. Differential equations for the Slater sum are first briefly reviewed, a quite general equation being available for a one-dimensional potential V(x). This equation can be solved for a sech2 potential, and some physical properties of interest such as the local density of states are derived by way of illustration. Then, the Coulomb potential −Ze2/r is next considered, and it is shown that what is essentially the inverse La-place transform of S(r, β) / β can be calculated for an arbitrary number of closed shells. Blinder has earlier determined the Feynman propagator in terms of Whittaker functions and contact is here established with his work. The currently topical case of Fermion vapours which are harmonically confined is then treated, for both two and three dimensions. Finally, in an Appendix, a perturbation series for the Slater sum is briefly summarized, to all orders in the one-body potential V(r). The corresponding kinetic energy is thereby accessible.
