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Physical Review A

Analytical relations between kinetic-energy and particle densities for one-dimensional harmonically confined Fermi vapors

Norman Henry March, Luis Miguel Nieto

Published Mar 16, 2001• 17 citations
Publisher

Abstract

Analytical relations between kinetic-energy and particle densities are set out here, and used for numbers of harmonically confined independent fermions up to 100 particles. The use of positive definite kinetic-energy density is advocated in the present context, and this is then shown to tend to the von Weizs\"acker inhomogeneity kinetic-energy density near the tunneling regime for the harmonic potential barrier, the example of 50 fermions being studied specifically.

DOI
10.1103/physreva.63.044502

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Analytical relations between kinetic-energy and particle densities are set out here, and used for numbers of harmonically confined independent fermions up to 100 particles. The use of positive definite kinetic-energy density is advocated in the present context, and this is then shown to tend to the von Weizs"acker inhomogeneity kinetic-energy density near the tunneling regime for the harmonic potential barrier, the example of 50 fermions being studied specifically.
The search for the single-particle kinetic energy functional TS[n] continues to be of major interest for density functional theory. Since it is expected to be generally applicable, exactly solvable models are of obvious interest. Here we focus on one, which is also of interest experimentally in magnetic trapping of ultracold fermion vapours. This is the model of independent harmonically trapped fermions in two dimensions. Here, the role of the von Weizsacker inhomogeneity kinetic energy is a focal point, prompted also by the work of Delle Site (2005 J. Phys. A: Math. Gen. 38 7893).
In earlier work, expressions have been constructed for the single-particle kinetic-energy functional ${T}_{s}[\ensuremath{\rho}]$ for independent fermions subject to harmonic confinement in low dimensions, with $\ensuremath{\rho}$ the particle density. Here, the differential equation for $\ensuremath{\rho}$ is first obtained in d dimensions for an arbitrary number of closed shells. Then, by using the known Euler-Lagrange equation, the functional derivative $\ensuremath{\delta}{T}_{s}/\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r})$ is constructed. ${T}_{s}[\ensuremath{\rho}]$ itself is proved to take the form of a linear combination of three pieces: (i) a von Weizs\"acker inhomogeneity kinetic energy, but with the original coefficient reduced by a dimensionality factor 1/d,1/d,1/d, (ii) a Thomas-Fermi kinetic energy, and (iii) a truly nonlocal contribution that, however, is shown to involve only the density $\ensuremath{\rho}$ itself and its first derivative. Thus, for this model, which is currently highly relevant to the interpretation of experiments on the evaporative cooling of dilute, and hence almost noninteracting, fermions, a complete density-functional theory now exists.
We present a Green's-function method for the evaluation of the particle density profile and of the higher moments of the one-body density matrix in mesoscopic systems containing a large number N of Fermi particles moving independently on a line under an external potential. The usefulness of the method is illustrated by applications to a Fermi gas confined in a harmonic potential well, for which we evaluate the momentum flux density, entering the equations of generalized hydrodynamics, and the kinetic energy density, which is relevant to the density-functional theory. As a further application of the method we explicitly display the quantum mean square fluctuations of these quantities. We also study some properties of the kinetic energy functional Ekin[n(x)]{E}_{\mathrm{kin}}[n(x)]Ekin​[n(x)] in the same system. Whereas a local approximation to the kinetic energy density is demonstrably wrong, an exact single-valued relationship between the density derivative of Ekin[n(x)]{E}_{\mathrm{kin}}[n(x)]Ekin​[n(x)] and the particle density n(x)n(x)n(x) is demonstrated and evaluated for various values of the number of particles in the system.