Europhysics Letters (EPL)

Density for Fermions in a Harmonic-Oscillator Potential

J. Cugnon, O. Harouna

Published Jun 1, 1986• 2 citations

Abstract

We show explicitly that the ground-state density of a system of noninteracting fermions in a harmonic-oscillator potential well can be given in terms of the wave functions of the single-particle states around the Fermi level.

DOI: 10.1209/0295-5075/1/11/001

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Reply to ‘‘Comment on ‘Kinetic-energy density functional for a special shape-invariant potential of a one-dimensional two-level system’ ’’

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For a harmonic potential, which is the special shape-invariant potential discussed in our paper [1], the relation between the ground state and the first-excited-state wave functions [Eq. (2)] should be . . . Therefore, we have shown that one cannot find a closed form of the kinetic-energy density for the harmonic oscillator.

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Europhysics Letters (EPL)

Density for Fermions in a Harmonic-Oscillator Potential

J. Cugnon, O. Harouna

Published Jun 1, 1986• 2 citations

Publisher

Abstract

DOI: 10.1209/0295-5075/1/11/001

Related papers

Exact Level Densities for the Harmonic Oscillator

P. Van Isacker

Physical Review LettersDec 12, 20027 citations

One-Dimensional Fermion Density Formula

Herbert Payne

The Journal of Chemical PhysicsApr 15, 19633 citations

Reply to ‘‘Comment on ‘Kinetic-energy density functional for a special shape-invariant potential of a one-dimensional two-level system’ ’’

Jiushu Shao, John J. Stezowski

Physical Review ASep 1, 19911 citation

Unitary Fermi gas in a harmonic trap

S. Y. Chang, G. F. Bertsch

We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers $N=2--22$ using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ${E}_{\mathit{TF}}$ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers $N=2--22$ using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ${E}_{\mathit{TF}}$ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.

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Interacting fermions in a two-dimensional trap and fractional exclusion statistics

M. K. Srivastava, R. K. Bhaduri, J. Law +1 more

We consider N fermions in a two-dimensional harmonic oscillator potential interacting with a very short-range repulsive pair-wise potential. The ground-state energy of this system is obtained by performing a Thomas-Fermi as well as a self-consistent Hartree-Fock calculation. The two results are shown to agree even for a small number of particles. We next use the finite-temperature Thomas-Fermi method to demonstrate that in the local density approximation, these interacting fermions are equivalent to a system of noninteracting particles obeying the Haldane-Wu fractional exclusion statistics. It is also shown that mapping onto a system of N noninteracting quasiparticles enables us to predict the energies of the ground and excited states of the N-body system. PACS Nos.: 05.30-d, 73.20DxWe consider N fermions in a two-dimensional harmonic oscillator potential interacting with a very short-range repulsive pair-wise potential. The ground-state energy of this system is obtained by performing a Thomas-Fermi as well as a self-consistent Hartree-Fock calculation. The two results are shown to agree even for a small number of particles. We next use the finite-temperature Thomas-Fermi method to demonstrate that in the local density approximation, these interacting fermions are equivalent to a system of noninteracting particles obeying the Haldane-Wu fractional exclusion statistics. It is also shown that mapping onto a system of N noninteracting quasiparticles enables us to predict the energies of the ground and excited states of the N-body system. PACS Nos.: 05.30-d, 73.20Dx

Canadian Journal of PhysicsJan 1, 20007 citations

Non-local kinetic energy functional for an arbitrary number of Fermions moving independently in one-dimensional harmonic oscillator potential

N. H. March, Patrick Senet, V. E. Van Doren

Physics Letters AMay 1, 200021 citations

N=2--22

using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where

{E}_{\mathit{TF}}

is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.