= Two-dimensional Thomas–Fermi condensate
{title2=$n(r)=(\mu-m\omega^2r^2/2)_+/g$}
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 $\mu\gg\hbar\omega$, equivalently $m gN/(\pi\hbar^2)\gg1$. 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.
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