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Physical Review Letters

Semiclassical Calculation of One-Particle, One-Hole and Two-Particle, Two-Hole Level Densities

Gautam Kumar Ghosh, Rainer W. Hasse, Peter Schuck, J. Winter

Published Apr 25, 1983• 20 citations
Publisher

Abstract

One-particle, one-hole and two-particle, two-hole level densities are calculated in the local-density, i.e., Thomas-Fermi, approach. For the harmonic oscillator this leads to analytical expressions which are in excellent agreement with the exact quantum results.
DOI
10.1103/physrevlett.50.1250

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Using an analytic approximation to the quantum momentum previously derived to calculate single particle reduced density matrices in the semiclassical approximation, we have been able to construct equations of motion with which to perform trajectory calculations. Information calculated along these trajectories is used to calculate single particle reduced density matrices for one-dimensional systems. Charge densities for N fermions in a harmonic oscillator potential calculated using this quantum momentum approximation are in excellent agreement with the exact quantum mechanical calculations.
The semiclassical $\ensuremath{\hbar}$ expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of B. Grammaticos and A. Voros [Ann. Phys. (N.Y.) 123, 359 (1979)], originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virtue of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\ensuremath{\hbar}$ expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a finite, second-order gradient correction to the Hartree-Fock energy, which takes the form $\ensuremath{\varepsilon}{(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}/\sqrt{\ensuremath{\rho}}$, with $\ensuremath{\varepsilon}$ being small ($|\ensuremath{\varepsilon}|\ensuremath{\ll}1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.
We consider the density functional theory from the point of view of one-dimensional two-particle systems, where the particles are either spin-1/2 fermions or spin-0 bosons. The basic equations are derived and applied to the system of two coupled harmonic oscillators. An exact density functional solution is constructed from the exactly known single-particle density for this system. We demonstrate how this solution may be applied to other two-particle systems to calculate the exact single-particle density and energy.