= Solution
Use the <condensate wavefunction> convention $\int |\psi|^2\,d^3r=N$, so $n(\mathbf r)=|\psi(\mathbf r)|^2$ is the <number density> and $\psi/\sqrt N$ is a normalized one-particle orbital. The <contact interaction> strength is $U_0=4\pi\hbar^2a/m$ for dilute atoms with <s-wave scattering length> $a$. The <chemical potential> $\mu=\partial E/\partial N$ fixes the stationary time dependence $e^{-i\mu t/\hbar}$ and enforces normalization when minimizing the <energy>. The <Gross–Pitaevskii equation> follows from an <energy functional> that contains the pair-counting factor $1/2$:
$$
E[\psi]=\int\left[\frac{\hbar^2}{2m}|\nabla\psi|^2+V|\psi|^2+\frac{U_0}{2}|\psi|^4\right]d^3r.
$$
Varying $E-\mu\int|\psi|^2d^3r$ gives the stationary <Gross–Pitaevskii equation>; the interaction contribution to that equation is $U_0 n\psi$, without the factor $1/2$.
For the broadening discussed here assume \b[repulsive interactions], $a>0$. The <Thomas–Fermi approximation for a condensate> neglects the <gradient> <kinetic energy> compared with the trapping and interaction <energies>. At fixed spatial scale, increasing $N U_0$ increases the mean-field potential. Expansion then reduces the <kinetic energy> further: for a cloud radius $R$, its scale per atom is $\hbar^2/(mR^2)$, whereas the trap <energy> scale is $m\omega^2R^2$. Their ratio decreases as $(a_0/R)^4$. In the weak-interaction limit the <kinetic energy> instead balances confinement, giving the <Gaussian wave packet> <ground state> of the <quantum harmonic oscillator>.
The <spherical Thomas–Fermi condensate> has
$$
n_{\rm TF}(r)=\frac{\mu-\tfrac12m\omega^2r^2}{U_0}\quad(0\leq r\leq R),\qquad n_{\rm TF}(r)=0\quad(r>R),\qquad \mu=\tfrac12m\omega^2R^2.
$$
Integrating this <density> gives $N=8\pi\mu R^3/(15U_0)$. Thus
$$
\boxed{R=a_0(15Na/a_0)^{1/5},\qquad \mu_{\rm TF}=\frac{\hbar\omega}{2}(15Na/a_0)^{2/5}.}
$$
At fixed $N$, stronger repulsion produces a wider, lower <wavefunction>: the <Gaussian wave packet> crosses over towards the <square root> of an inverted <parabola>, with an almost flat central region and a steeper outer drop. The \i[<density>] is the inverted <parabola>; the \i[<wavefunction>] is its <square root>. The exact <wavefunction> remains smooth and has a tail, since <kinetic energy> cannot be neglected near the <Thomas–Fermi condensate edge rounding> region. The <healing length> must be small compared with the cloud size in the bulk. Attractive interactions would not follow this large-$N$ expansion argument.
The original plot compares the noninteracting endpoint with two normalized repulsive <spherical Thomas–Fermi condensate> profiles. The latter are asymptotic shapes, rather than exact intermediate solutions of the <Gross–Pitaevskii equation>. Each has the same integrated atom number; $\chi=Na/a_0$ labels the illustrative interaction strengths.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2012/paper-1-condensate-profiles.png]
{title=Condensate broadening}
Choose $A$ real and positive. The <Gaussian integral> gives
$$
N=A^2\pi^{3/2}a_0^3,\qquad
\boxed{A=\frac{\sqrt N}{\pi^{3/4}a_0^{3/2}},\quad \mu_0=\frac32\hbar\omega,\quad E_0=\frac32N\hbar\omega.}
$$
The three <zero-point energies> add for each atom. The ideal-gas <chemical potential> is a one-atom <energy>; the total <energy> includes all $N$ atoms.
For weak interactions, the first-order correction in <perturbation theory> is the <energy functional> of the <contact interaction> evaluated on the unperturbed <Gaussian wave packet>. The <integral> needed is
$$
\int |\psi_0|^4d^3r=\frac{N^2}{\pi^3a_0^6}\left(\frac{\pi a_0^2}{2}\right)^{3/2}=\frac{N^2}{(2\pi)^{3/2}a_0^3}.
$$
Consequently the <Gaussian interaction energy of a trapped condensate> is
$$
\boxed{E_I=\frac{N^2U_0}{2(2\pi)^{3/2}a_0^3},\qquad \frac{E_I}{\hbar\omega}=\frac{N^2a}{\sqrt{2\pi}\,a_0}.}
$$
Here $\hbar\omega=\hbar^2/(ma_0^2)$. These are the mean-field expressions consistent with the supplied equation and $\int|\psi|^2=N$. Exact first-order <perturbation theory> for precisely $N$ atoms in one orbital replaces $N^2$ by $N(N-1)$ through the number of distinct pairs; this distinction is negligible in the large-$N$ regime of the crossover estimate.
To judge the ideal-gas approximation, compare the interaction <energy> with the \i[total] ideal <energy>, or compare the interaction <energy> per atom with $\hbar\omega$:
$$
\frac{E_I}{E_0}=\frac{2}{3\sqrt{2\pi}}\frac{Na}{a_0}.
$$
Thus the <crossover from Gaussian to Thomas–Fermi condensate> is governed by $\chi=Na/a_0$: <perturbation theory> is reliable for $\chi\ll1$, while the regime of the <Thomas–Fermi approximation for a condensate> requires $\chi\gg1$. \b[The characteristic crossover number is $N_c\sim a_0/a$; there is no sharp critical atom number here.] If one specifically defines crossover by $E_I=E_0$ using the weak-interaction expression, it gives $N_c=(3\sqrt{2\pi}/2)a_0/a$. That prefactor is a convention-dependent estimate extrapolated to the edge of <perturbation theory>, not a phase transition. Comparing $E_I$ itself to a single $\hbar\omega$ would incorrectly omit a factor of $N$.
For the numerical estimate use $m\simeq87\times1.66054\times10^{-27}\,\mathrm{kg}$ and $\hbar\simeq1.05457\times10^{-34}\,\mathrm{J\,s}$. Then
$$
a_0=\sqrt{\frac{\hbar}{m(2\pi\times50\,\mathrm{s^{-1}})}}\simeq1.52\,\mathrm{\mu m},\qquad
\boxed{N_c\sim\frac{a_0}{a}\simeq3.0\times10^2\ \text{atoms}.}
$$
The particular equality $E_I=E_0$ would instead give about $1.15\times10^3$ atoms. These quantify different choices within a smooth crossover; a well-developed <spherical Thomas–Fermi condensate> profile requires going substantially beyond the characteristic scale.
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