Two-dimensional Thomas–Fermi condensate ($n(r)=(\mu -m{\omega }^2{r}^2/2{)}_{+}/g$)

In a two-dimensional isotropic harmonic trap with repulsive effective coupling $g>0$, neglecting density-gradient kinetic energy gives the displayed parabolic number density. Normalization to $N$ particles yields $\mu =\sqrt{Ngm{\omega }^2/\pi }$, radius $R^2=2\mu /(m{\omega }^2)$ and central density $n_c=\mu /g$. The bulk approximation requires $\mu \gg \hslash \omega$, equivalently $mgN/(\pi {\hslash }^2)\gg 1$. It fails near the outer edge and inside quantum vortex cores. The coupling must have two-dimensional units; an unreduced three-dimensional contact coupling cannot be inserted without a normalization convention.