Spherical Thomas–Fermi condensate ($R=a_0(15Na/a_0{)}^{1/5}$)

For repulsive contact interactions in a three-dimensional isotropic harmonic trap, neglecting the kinetic term in the Gross–Pitaevskii equation gives $n(r)=(\mu -m{\omega }^2{r}^2/2)/U_0$ inside $R=\sqrt{2\mu /(m{\omega }^2)}$ and zero outside. Normalization yields $N=8\pi \mu R^3/(15U_0)$, hence $\mu =(\hslash \omega /2)(15Na/a_0{)}^{2/5}$ and $R=a_0(15Na/a_0{)}^{1/5}$, with oscillator length $a_0=\sqrt{\hslash /(m\omega )}$. The density is parabolic but the wavefunction is its square root. Neglected kinetic energy rounds the edge and supplies a tail.

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 ${\hslash }^2/(2m{d}^2)$ against the local potential change $Fd$ gives $d\sim [{\hslash }^2/(2mF){]}^{1/3}$. 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.