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arXiv:cond-mat/0209460 · Physical Review A

Some exact results for a trapped quantum gas at finite temperature

Brandon P. van Zyl, Rajat K. Bhaduri, Akira Suzuki, M. Brack

Published Feb 24, 2003• 36 citations
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Abstract

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.

DOI
10.1103/physreva.67.023609

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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=2d=2d=2 and 3,3,3, 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 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 ddd-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 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 investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas–Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation τ TF [ρ] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities τ( r ) and ρ( r ) including their leading-order oscillating terms.
We present a new, simple closed form expression for the exact Wigner function of an ideal gas of harmonically trapped fermions or bosons at arbitrary temperature and dimensionality. One of the primary advantages of our expression is that it provides a clean separation of the spatial and momentum variables, which can significantly simplify analytical calculations involving the Wigner function. Our analysis is based on the explicit Fourier transform of the exact first-order density matrix, which affords an exceedingly simple derivation of the Wigner function. We demonstrate the utility of our approach by presenting several analytical results which may be trivially derived from the Wigner function, but otherwise require more involved calculations. We also examine the temperature and dimensionality dependence of the Fermi Wigner function in detail, thereby pointing out some subtle issues regarding the applicability of the Thomas–Fermi approximation to the Wigner function in the limit of a 'macroscopic' number of particles, and finite-temperatures.
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation tau(TF)[rho] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities tau(r) and rho(r) including their leading-order oscillating terms.