= Solution
{c}
In the large-$N$ <Gross–Pitaevskii equation>, dropping the <kinetic energy> gives $[\kappa r^2/2+gn(r)-\mu]\Psi(r)=0$. The physical nonnegative <Thomas–Fermi approximation for a condensate> is therefore
$$
\boxed{n(r)=\frac1g\left(\mu-\frac\kappa2r^2\right)_+,\qquad R=\sqrt{\frac{2\mu}\kappa},}
$$
where $(z)_+=\max(z,0)$ and $R$ is the cloud radius. Its <particle number> is
$$
N=2\pi\int_0^Rr\,n(r)\,dr=\frac{2\pi}{g}\left(\frac{\mu R^2}2-\frac{\kappa R^4}8\right)
=\frac{\pi\mu^2}{g\kappa}.
$$
Thus the <two-dimensional Thomas–Fermi condensate> obeys
$$
\boxed{\mu=\sqrt{\frac{Ng\kappa}\pi},\qquad R=\left(\frac{4Ng}{\pi\kappa}\right)^{1/4},\qquad n(0)=\frac\mu g.}
$$
Here $N=\int n\,d^2r$ is a total <particle number>. Keeping the exact pair count in the fixed-$N$ <Hartree product state> would replace $g$ by $g(1-1/N)$ in these formulas expressed in terms of $n=N|\phi|^2$.
The approximation is \b[best at the centre, $r=0$, and worst near the edge, $r=R$]. For example its omitted <quantum pressure of a condensate>, evaluated on the interior profile, is
$$
Q(r)=-\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{n}}{\sqrt n}
=\frac{\hbar^2}{2mR^2}\frac{2-r^2/R^2}{(1-r^2/R^2)^2}.
$$
It is smallest at the centre and diverges at the artificial sharp edge. The central <healing length> $\xi_0=\hbar/\sqrt{2m\mu}$ is much smaller than $R$ when $\mu\gg\hbar\omega$, since $R/\xi_0=2\mu/(\hbar\omega)$. This is the bulk validity condition; the full <Gross–Pitaevskii equation> smooths the edge and gives a decaying tail outside $R$.
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