= Solution
{c}
Write $G(r)=\langle\psi^\dagger(0)\psi(\mathbf r)\rangle$, $r=|\mathbf r|$, and $\ell^2=\hbar/(m\omega)$. At zero <temperature> all $N$ particles occupy $\phi_0(\mathbf r)=(\ell\sqrt\pi)^{-1}e^{-r^2/(2\ell^2)}$. Expanding the <bosonic annihilation operator> in single-particle <orbitals> gives
$$
\boxed{G(r)=N\phi_0^*(0)\phi_0(\mathbf r)=\frac{N}{\pi\ell^2}\exp\!\left(-\frac{r^2}{2\ell^2}\right)\quad(T=0).}
$$
This <one-body density matrix> is a <Gaussian function> of width $\ell$; it contains one <wavefunction> evaluated at $r$, so its exponent is half that of the spatial <number density> $N|\phi_0(r)|^2$. In particular,
$$
\boxed{G(0)=\langle\psi^\dagger(0)\psi(0)\rangle=n(0),}
$$
the total <number density> at the trap centre, including condensed and noncondensed particles.
At positive <temperature>, with occupations $\overline n_a$ of the trap <energy eigenstates>, the exact ideal-gas decomposition is
$$
G(r)=\frac{N_0}{\pi\ell^2}e^{-r^2/(2\ell^2)}+\sum_{a\ne0}\overline n_a\phi_a^*(0)\phi_a(\mathbf r).
$$
For $\hbar\omega\ll k_BT<k_BT_c$, the sketch has a broad <Bose-Einstein condensate> component on scale $\ell$ and a narrower thermal peak near the origin. On approaching $T_c$ the broad component loses its weight $N_0$, while the thermal part remains. Above $T_c$ there is no macroscopic broad <Bose-Einstein condensate> component; close to $T_c$, low <energy levels> can still give a shoulder, so a single narrow <Gaussian function> is only a high-<temperature> description. At $k_BT\lesssim\hbar\omega$ individual excited <energy levels> matter and the two-scale description need not apply.
For $T\gg T_c$ the <Maxwell-Boltzmann distribution> applies. The leading off-diagonal <one-body density matrix> is
$$
G(r)\simeq\frac{N\kappa}{2\pi k_BT}\exp\!\left(-\frac{mk_BT}{2\hbar^2}r^2\right),\qquad
\boxed{\sigma_r=\frac{\hbar}{\sqrt{mk_BT}}=\frac{\lambda_T}{\sqrt{2\pi}},\quad
\lambda_T=\sqrt{\frac{2\pi\hbar^2}{mk_BT}}.}
$$
Here $\sigma_r$ is the <Gaussian function> width defined by $e^{-r^2/(2\sigma_r^2)}$, and $\lambda_T$ is the <thermal de Broglie wavelength>. The thermal cloud itself extends over the much larger spatial scale $\sqrt{k_BT/\kappa}$; this is a <correlation> width, not the cloud radius.
\Image[/past-exam-of-the-quantum-matter-course-of-the-university-of-oxford/original2025-ideal-correlations.png]
{title=Ideal trapped-gas one-body correlations across condensation}
{description=Curves for $N=10^4$ use the discrete oscillator spectrum and its thermal kernel, with the chemical potential chosen to fix $N$. Each vertical axis shows $G(r)/(N/(\pi\ell^2))$; the last panel enlarges the vertical scale. The dotted curve below $T_c$ is the condensate contribution. The value of $T_c$ used in the labels is the leading large-$N$ expression.}
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