Corrections to Thomas-Fermi Densities at Turning Points and Beyond

Uniform semiclassical approximations for the number and kinetic-energy densities are derived for many noninteracting fermions in one-dimensional potentials with two turning points. The resulting simple, closed-form expressions contain the leading corrections to Thomas-Fermi theory, involve neither sums nor derivatives, are spatially uniform approximations, and are exceedingly accurate.

Figure 1

Thomas-Fermi (dashed) and semiclassical (dotted) approximations to the density (solid) of 2 particles in a Morse potential, $v(x)=15(e^{-x/2}-2e^{-x/4})$.

Figure 2

Error in semiclassical density for $N=2$ (solid), and $N=8$ (dashed) in the Morse potential of Fig. 1.

Figure 3

Integrated measure of error [Eq. (26)] in TF density multiplied by 0.1 (top) and semiclassical uniform approximation (bottom) for the Morse potential of Fig. 1.

Figure 4

Thomas-Fermi (dashed), uniform semiclassical (dotted), and exact (solid) kinetic energy density for 2 particles in the Morse potential of Fig. 1. The value of ${\pi }^2[n^{\mathrm{sc}}(x){]}^3/6$ is also shown (dot-dashed).

Figure 5

Error (see text) in kinetic energy densities in the Morse potential of Fig. 1 with the semiclassical uniform approximation (squares), Thomas-Fermi theory (dots), and $t^{\mathrm{loc}}[n^{\mathrm{sc}}]$ (rhombs).