= Solution
{c}
Measure single-particle <energy> from the <quantum harmonic oscillator> <ground state> <energy> $\hbar\omega$, and write $\widetilde\mu=\mu_{\mathrm{physical}}-\hbar\omega\leq0$. The <Bose-Einstein distribution> and the supplied <density of states> give
$$
N_{\mathrm{ex}}(T,\widetilde\mu)=\int_0^\infty\frac{E\,dE}{(\hbar\omega)^2[e^{(E-\widetilde\mu)/(k_BT)}-1]}.
$$
This excited-state population increases with the <chemical potential> and has the finite saturation value
$$
N_{\mathrm{ex}}^{\max}(T)=\left(\frac{k_BT}{\hbar\omega}\right)^2\int_0^\infty\frac{u\,du}{e^u-1}
=\frac{\pi^2}{6}\left(\frac{k_BT}{\hbar\omega}\right)^2.
$$
Any additional particles must occupy the <ground state>: this is <two-dimensional harmonic-trap Bose condensation>. The <critical temperature> and the <condensate fraction> are therefore
$$
\boxed{N\omega^2=\frac{\pi^2}{6}\left(\frac{k_BT_c}{\hbar}\right)^2,\qquad
k_BT_c=\frac{\hbar\omega\sqrt{6N}}\pi,\qquad
N_0(T)=\begin{cases}N[1-(T/T_c)^2],&T\leq T_c,\\0,&T>T_c.\end{cases}}
$$
Here $N$ is the total <particle number>, the integral of the inhomogeneous <number density> over the trap; there is no fixed uniform area in which $N$ itself is a <number density>. Equivalently $N\kappa=\pi^2m(k_BT_c)^2/(6\hbar^2)$ relates the trap population to the <critical temperature>. Harmonic confinement changes the low-<energy> <density of states> and permits this saturation in two spatial dimensions.
These formulas are leading large-$N$ results with $k_BT_c\gg\hbar\omega$. A sharp transition belongs to the trapped-gas <thermodynamic limit> $N\to\infty$, $\omega\to0$ at fixed $N\omega^2$; a finite trap has <finite-size rounding>. In particular, finite $N_0$ requires $\widetilde\mu=-k_BT\log(1+1/N_0)<0$, approaching zero in the condensation limit.
Back to article page
