Corrections to Thomas-Fermi densities at turning points and beyond.

The semiclassical limit of quantum mechanics has long guided the construction of powerful approximations to quantum problems [1–3]. Both dynamical evolution [4– 6] and spectral properties of quantum systems [7, 8] have been extensively studied with semiclassical approximations. But the asymptotic expansion in powers of ~ is in general singular. Consider the eigenfunctions of a onedimensional potential v(x). The standard semiclassical approach is to find WKB wave functions[9–12] separately in the classically allowed and evanescent regions, linearize the potential near each turning point and match the solutions with connection formulae [1]. Langer[13] found a more elegant solution in the form of a single expression involving the Airy function (Eq. 4). Langer’s approximation is spatially uniform. Its error vanishes in the semiclassical limit for all points in space [14]. A more demanding challenge is to find the number density, n(x), of the lowest N occupied levels. The classical limit may be obtained by replacing the discrete sum by an integral over classical states [1, 15]. The result is the Thomas-Fermi (TF) density [16, 17]:

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