Propagator and Slater sum in one-body potential theoryPropagator and Slater sum in one-body potential theory
Faculty of Sciences. Physics

Department of Physics

article

2003Berlin, 2003

Physics

Physica status solidi: B: basic research. - Berlin

237(2003):1, p. 265-273

0370-1972

000182801800025

E

English (eng)

University of Antwerp

For a one-body potential V(r) generating eigenfunctions psi(i)(r) and corresponding eigenvalues epsilon(i), the Feynman propagator K(r, r, t) is simply related to the canonical density matrix C(r,r',beta) by beta --> it. The diagonal element S(r,r,beta) 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 sech(2) potential, and some physical properties of interest such as the local density of states are derived by way of illustration. Then, the Coulomb potential -Ze(2)/r is next considered, and it is shown that what is essentially the inverse Laplace transform of S(r,beta)/beta 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.

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