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

Matthias Brack·Brandon P. van Zyl
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Abstract

We derive simple analytical expressions for the particle density $\rho(r)$ and the kinetic energy density $\tau(r)$ 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 $\tau[\rho]$ using the exact $\rho(r)$ and show that it locally reproduces the exact kinetic energy density $\tau(r)$, {\it 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 $\tau_{TF}[\rho(r)]$ yields the {\it exact} total kinetic energy.

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Matthias Brack, Brandon P. van Zyl. 2000-10-15. Simple Analytical Particle and Kinetic Energy Densities for a Dilute Fermionic Gas in a d-Dimensional Harmonic Trap. https://doi.org/10.1103/physrevlett.86.1574

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