= Thomas–Fermi condensate edge rounding
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The <Thomas–Fermi approximation for a condensate> neglects density-gradient <kinetic energy>, predicting a sharp edge where the trapping <potential energy> equals the <chemical potential>. Near that edge, the density becomes small and its fractional spatial variation grows, so the neglected gradient term can no longer be treated as small.
Let $F=|\nabla V|$ at a smooth edge and let $d$ be the rounding width. Balancing $\hbar^2/(2md^2)$ against the local potential change $Fd$ gives $d\sim[\hbar^2/(2mF)]^{1/3}$. The full stationary <Gross–Pitaevskii equation> then smooths the density and allows a tail beyond the nominal Thomas–Fermi radius. This boundary layer can be narrow compared with the cloud radius while still being essential for edge-sensitive quantities.
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