= Solution
{c}
The <Hartree product state> underlying the <Gross–Pitaevskii equation> has the factorized <one-body coherence of a Hartree condensate>. Choosing its nodeless <ground state> <orbital> real gives
$$
\boxed{G_{\mathrm{GP}}(r)=N\phi^*(0)\phi(r)=\sqrt{n(0)n(r)}.}
$$
In particular, the <Thomas–Fermi approximation for a condensate> from part (d) predicts
$$
\boxed{G_{\mathrm{TF}}(r)=\begin{cases}\displaystyle\frac\mu g\sqrt{1-r^2/R^2},&0\leq r<R,\\0,&r\geq R.\end{cases}}
$$
The curve starts at $\mu/g$, decreases with zero slope at the origin, and reaches zero with an infinite slope at $R$. Restoring the <kinetic energy> in the full <Gross–Pitaevskii equation> rounds that edge and produces a nonzero tail. This corrects the <Thomas–Fermi approximation for a condensate>, but the resulting <one-body density matrix> still factorizes.
\b[The additional missing physics is quantum depletion.] Interactions scatter pairs of <bosons> into <orbitals> outside the <Bose-Einstein condensate> even at zero <temperature>. In terms of the true <Bose-Einstein condensate> <orbital> $\varphi_0$ the exact <one-body density matrix> decomposes as
$$
G(r)=N_0\varphi_0^*(0)\varphi_0(r)+\rho_{\mathrm{ex}}(0,\mathbf r),\qquad N_0<N,
\qquad
G(0)=N_0|\varphi_0(0)|^2+n_{\mathrm{ex}}(0).
$$
The single-<orbital> <Hartree product state> sets $N_0=N$ and omits $\rho_{\mathrm{ex}}$ altogether. In a <weakly interacting Bose gas>, <Bogoliubov theory> adds a comparatively short-range noncondensed contribution near the origin, together with reduced weight in the broad <Bose-Einstein condensate> component. The central value remains the total <number density>, not just the <condensate density>. Its numerical shift relative to the original <Gross–Pitaevskii equation> solution also depends on the interaction-induced redistribution of <number density>; it is not fixed by the <condensate fraction> alone.
\Image[/past-exam-of-the-quantum-matter-course-of-the-university-of-oxford/original2025-interacting-correlations.png]
{title=Edge smoothing and quantum depletion give different corrections to the correlation}
{description=Left: a numerical radial Gross–Pitaevskii solution at $gN/(\hbar\omega\ell^2)=100\pi$ is compared with the Thomas–Fermi curve, each at the same $N$. Right: an illustrative depletion correction reduces the broad component and adds a narrow contribution, with the central density held fixed for comparison. Its weight and width are schematic, not a calculation of the exact interacting gas. Solving the full Gross–Pitaevskii equation accounts for the left correction only.}
Back to article page
