  Quantum corrections to Thomas-Fermi (TF) theory are investigated for
noninteracting one-dimensional fermions with known uniform semiclassical
approximations to the density and kinetic energy. Their structure is analyzed,
and contributions from distinct phase space regions (classically-allowed versus
forbidden at the Fermi energy) are derived analytically. Universal formulas are
derived for both particle numbers and energy components in each region. For
example, in the semiclassical limit, exactly 1/(6\pi3^{1/2}) of a particle
leaks into the evanescent region beyond a turning point. The correct
normalization of semiclassical densities is proven analytically in the
semiclassical limit. Energies and densities are tested numerically in a variety
of one-dimensional potentials, especially in the limit where TF theory becomes
exact. The subtle relation between the pointwise accuracy of the semiclassical
approximation and integrated expectation values is explored. The limitations of
the semiclassical formulas are also investigated when the potential varies too
rapidly. The approximations are shown to work for multiple wells, except right
at the mid-phase point of the evanescent regions. The implications for density
functional approximations are discussed.
