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Propagator and Slater sum in one‐body potential theory

N. H. March, I. A. Howard

Published Apr 24, 2003• 8 citations

Abstract

Abstract 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 Laplace 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.

DOI: 10.1002/pssb.200301779

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Abstract 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 Laplace 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.

Hilton, March, and Curtis have earlier focused on the utility of the effective potential U(r,β) in determining the Slater sum Z(r,β)=Z0(β)exp(−βU(r,β)). Here an explicit, though highly nonlinear, partial differential equation is derived for determining the effective potential U(x,β) in one-dimensional problems. Direct solution of this equation by power series expansion in β leads readily to Husimi’s results obtained from off-diagonal density matrix calculations. Perturbation theory in the potential is also developed, and thereby it is shown that an infinite subseries of the Husimi expansion is readily summed, and that a scaling property is exhibited.

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.

The one-dimensional equation of motion for the canonical or Bloch density matrix is solved by the method of separation of variables for independent particles moving in a potential V(x). The solution is exact when third and higher derivatives of V(x) are zero, embracing thereby the examples of (a) harmonic potentials and (b) a constant electric field of arbitrary strength. The form of the Slater sum obtained involves then two functions of temperature only, to be determined. One obvious relation between these two functions is via the partition function. A further relation comes from the orthonormality of the wave functions appearing in the Slater sum: the analog of idempotency for the ground-state Dirac density matrix. These two conditions again are shown to lead back to the exact result for harmonic potentials. The results presented generalize the one-dimensional Thomas-Fermi theory to apply to a temperature range that is significantly wider than for the original method.