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Physical Review C

Semiclassical quantization of the extended time-dependent Hartree-Fock equation: Schematic two-level model

Kazunari Kaneko

Published Mar 1, 1985• 4 citations
Publisher

Abstract

A semiclassical method of evaluating the path integral over collective (Bose) and independent-particle (Fermi) fields is represented. As an illustration, the schematic two-level model is adopted. The semiclassical calculations of the energy spectra is performed and the result obtained is compared with the exact one.
DOI
10.1103/physrevc.31.1039

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Working within a time-dependent Hartree-Fock framework, one develops a semiclassical approximation appropriate for large systems. It is demonstrated that the standard semiclassical approach, the Thomas-Fermi approximation, is inconsistent with Hartree-Fock theory when the basic two-body interaction is short-ranged (as in nuclear systems, for example). However, by introducing a simple extension of the Thomas-Fermi approximation, one overcomes this problem. One also discusses the infinite nuclear matter problem and point out that time-dependent Hartree-Fock theory yields collective modes of the zero sound variety instead of ordinary hydrodynamic (first) sound. One thus emphasizes that one should be extremely circumspect when attempting to cast the equations of motion of time-dependent Hartree-Fock theory into a hydrodynamic-like form.
In a previous paper, a quantum-mechanical formulation involving both mean fields and independent-particle fields in many-fermion systems was proposed using the path-integral technique. Then the semiclassical calculation of the energy spectra was performed, and the quantization rule was derived by applying a stationary phase approximation on the path integral. In this paper, a boson-fermion expansion is derived from our formulation using Dirac quantization. As an illustration, the Lipkin model is utilized.