Thomas–Fermi approximation for a condensate

Neglecting density-gradient kinetic energy in the stationary Gross–Pitaevskii equation gives $n=(\mu -V{)}_{+}/g$ for repulsive interactions. For a radial harmonic trap $V=m{\omega }_r^2{r}^2/2$, this is $n=n_0(1-r^2/R_r^2{)}_{+}$ with $R_r=\sqrt{2g{n}_0/(m{\omega }_r^2)}$. Hard boundaries can truncate this natural support. The approximation fails in vortex cores and boundary healing regions and requires a cloud much larger than its healing length.