Journal of Physics A Mathematical and General

Exact particle and energy densities for a noninteracting Fermi gas in a harmonic trap

Xian Zhi Wang

Published Oct 28, 2002• 5 citations

Abstract

The exact particle and energy densities for a noninteracting Fermi gas at any temperature in a d -dimensional harmonic trap are obtained, and they are expressed by the gamma function and the Laguerre polynomials. The obtained analytical expressions are rapidly converging series and may be conveniently used for the numerical calculations. These expressions can be used for the comparison with the experimental results and for the test of the local density approximation.

DOI: 10.1088/0305-4470/35/45/308

Related papers

Simple Analytical Particle and Kinetic Energy Densities for a Dilute Fermionic Gas in a d -Dimensional Harmonic Trap

M. Brack, Brandon P. van Zyl

arXiv:cond-mat/0010201Physical Review LettersFeb 19, 200186 citations

Exact finite-temperature particle and energy densities for noninteracting Bose and Fermi gases in d -dimensional anisotropic harmonic traps

Xian Zhi Wang

A much simpler method is proposed for the evaluation of the exact finite-temperature particle and energy densities for noninteracting Bose and Fermi gases in d-dimensional anisotropic harmonic traps. For the Bose gas in the whole range of temperature, the exact results are obtained. For the Fermi gas with chemical potential less than the single-particle ground-state energy, the exact results are obtained.

Physical Review AApr 4, 200212 citations

Noninteracting fermions in a one-dimensional harmonic atom trap: Exact one-particle properties at zero temperature

F. Gleisberg, W. Wonneberger, U. Schlöder +1 more

arXiv:quant-ph/0009085Physical Review ANov 3, 200061 citations

Analytic treatment of interacting Fermi gas in an arbitrary dimensional harmonic trap

Hiroyuki Yoshimoto, Susumu Kurihara

arXiv:cond-mat/0210198Journal of Physics A Mathematical and GeneralOct 8, 20036 citations

Some exact results for a trapped quantum gas at finite temperature

Brandon P. van Zyl, Rajat K. Bhaduri, Akira Suzuki +1 more

arXiv:cond-mat/0209460Physical Review AFeb 24, 200336 citations

Exact coherent states of a noninteracting Fermi gas in a harmonic trap

Dae-Yup Song

arXiv:cond-mat/0609253Physical Review ANov 16, 20066 citations

Journal of Physics A Mathematical and General

Exact particle and energy densities for a noninteracting Fermi gas in a harmonic trap

Xian Zhi Wang

Published Oct 28, 2002• 5 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/35/45/308

Related papers

Simple Analytical Particle and Kinetic Energy Densities for a Dilute Fermionic Gas in a d -Dimensional Harmonic Trap

M. Brack, Brandon P. van Zyl

We derive simple analytical expressions for the particle density $\ensuremath{\rho}(r)$ and the kinetic energy density $\ensuremath{\tau}(r)$ for a system of noninteracting fermions in a $d$ -dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\ensuremath{\tau}[\ensuremath{\rho}]$ using the exact $\ensuremath{\rho}(r)$ and show that it locally reproduces the exact kinetic energy density $\ensuremath{\tau}(r)$, including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of ${\ensuremath{\tau}}_{\mathrm{TF}}[\ensuremath{\rho}(r)]$ yields the exact total kinetic energy.We derive simple analytical expressions for the particle density $\ensuremath{\rho}(r)$ and the kinetic energy density $\ensuremath{\tau}(r)$ for a system of noninteracting fermions in a $d$-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\ensuremath{\tau}[\ensuremath{\rho}]$ using the exact $\ensuremath{\rho}(r)$ and show that it locally reproduces the exact kinetic energy density $\ensuremath{\tau}(r)$, including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of ${\ensuremath{\tau}}_{\mathrm{TF}}[\ensuremath{\rho}(r)]$ yields the exact total kinetic energy.

arXiv:cond-mat/0010201Physical Review LettersFeb 19, 200186 citations

Exact finite-temperature particle and energy densities for noninteracting Bose and Fermi gases in d -dimensional anisotropic harmonic traps

Xian Zhi Wang

Physical Review AApr 4, 200212 citations

Noninteracting fermions in a one-dimensional harmonic atom trap: Exact one-particle properties at zero temperature

F. Gleisberg, W. Wonneberger, U. Schlöder +1 more

One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution ${n}_{0}(z,N).$ For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near ${2k}_{F},$ with ${k}_{F}$ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution ${n}_{0}(z,N).$ For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near ${2k}_{F},$ with ${k}_{F}$ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.

arXiv:quant-ph/0009085Physical Review ANov 3, 200061 citations

Analytic treatment of interacting Fermi gas in an arbitrary dimensional harmonic trap

Hiroyuki Yoshimoto, Susumu Kurihara

We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.

arXiv:cond-mat/0210198Journal of Physics A Mathematical and GeneralOct 8, 20036 citations

Some exact results for a trapped quantum gas at finite temperature

Brandon P. van Zyl, Rajat K. Bhaduri, Akira Suzuki +1 more

We present closed analytical expressions for the particle and kinetic-energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic-oscillator potential. For $d=2$ and $3,$ exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l degeneracy in a harmonic-oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.We present closed analytical expressions for the particle and kinetic-energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic-oscillator potential. For $d=2$ and $3,$ exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l degeneracy in a harmonic-oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.

arXiv:cond-mat/0209460Physical Review AFeb 24, 200336 citations

Exact coherent states of a noninteracting Fermi gas in a harmonic trap

Dae-Yup Song

Exact and closed-form expressions of the particle density and the momentum distribution are derived for a coherent state of a noninteracting Fermi gas, while such a state can be obtained from the ground state in a $d$ -dimensional isotropic harmonic trap by modulating the trap frequency and shifting the trap center. The profile of the momentum distribution turns out to be identical in shape with that of the particle density, however, as a manifestation of the Heisenberg uncertainty principle, the dispersion of the distribution increases (decreases) when that of the particle density is decreased (increased). The expressions are also applicable for a sudden and total opening of the trap, and it is shown that, after the opening, the gas has a stationary momentum distribution whose dispersion could be arbitrarily large or small.Exact and closed-form expressions of the particle density and the momentum distribution are derived for a coherent state of a noninteracting Fermi gas, while such a state can be obtained from the ground state in a $d$-dimensional isotropic harmonic trap by modulating the trap frequency and shifting the trap center. The profile of the momentum distribution turns out to be identical in shape with that of the particle density, however, as a manifestation of the Heisenberg uncertainty principle, the dispersion of the distribution increases (decreases) when that of the particle density is decreased (increased). The expressions are also applicable for a sudden and total opening of the trap, and it is shown that, after the opening, the gas has a stationary momentum distribution whose dispersion could be arbitrarily large or small.

arXiv:cond-mat/0609253Physical Review ANov 16, 20066 citations

We derive simple analytical expressions for the particle density $\ensuremath{\rho}(r)$ and the kinetic energy density $\ensuremath{\tau}(r)$ for a system of noninteracting fermions in a

d

-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\ensuremath{\tau}[\ensuremath{\rho}]$ using the exact $\ensuremath{\rho}(r)$ and show that it locally reproduces the exact kinetic energy density $\ensuremath{\tau}(r)$, including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of ${\ensuremath{\tau}}_{\mathrm{TF}}[\ensuremath{\rho}(r)]$ yields the exact total kinetic energy.

{n}_{0}(z,N).

For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near

{2k}_{F},

with

{k}_{F}

being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.

d=2

and

3,

exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l degeneracy in a harmonic-oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.

d

-dimensional isotropic harmonic trap by modulating the trap frequency and shifting the trap center. The profile of the momentum distribution turns out to be identical in shape with that of the particle density, however, as a manifestation of the Heisenberg uncertainty principle, the dispersion of the distribution increases (decreases) when that of the particle density is decreased (increased). The expressions are also applicable for a sudden and total opening of the trap, and it is shown that, after the opening, the gas has a stationary momentum distribution whose dispersion could be arbitrarily large or small.