= Solution
Measure energies from the single-particle <ground state>, and assume the \b[three-dimensional ideal Bose gas in the large-particle-number limit]. The <harmonic-trap chemical-potential pinning> discussed here uses a <chemical potential> satisfying $\mu_{\rm ch}\leq0$. On cooling, it approaches zero; below the condensation <temperature> it remains at zero to leading thermodynamic order. Above that <temperature> it is negative. The detailed negative-chemical-potential branch shape is determined by the <density of states>; the figure uses the harmonic trap analysed below. The <finite-size condensate chemical potential> stays slightly below zero rather than being exactly pinned there.
A \b[<Bose-Einstein condensate> has macroscopic occupation of a single quantum state]. A dilute atomic gas needs low <temperature> and high phase-space <density> while remaining metastable against ordinary liquid or solid formation. <Laser cooling> followed by <evaporative cooling of atoms> in a weak trap can achieve the required regime. For the trapped ideal gas the quantitative threshold is derived below; a uniform ideal gas has $n\lambda_T^3=\zeta(3/2)$ at onset. An experimental signature is a narrow low-momentum component appearing within the broader thermal distribution after time-of-flight expansion, as observed in magnetically trapped rubidium-87.<Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor by M. H. Anderson and others>[1995 rubidium experiment: low-velocity condensate component and nonthermal anisotropic expansion]{ref}
\b[<Spinless cubic-box wave-vector counting>.] Impose <periodic boundary conditions> on a cube of side $L$. The allowed vectors are $\mathbf k=(2\pi/L)(n_x,n_y,n_z)$ with integer components. Each state occupies $ (2\pi/L)^3$ in wave-vector space. For a spinless particle, counting the spherical shell gives
$$
d\mathcal N=\frac{V}{(2\pi)^3}4\pi k^2dk,\qquad\boxed{g_V(k)=\frac1V\frac{d\mathcal N}{dk}=\frac{k^2}{2\pi^2}.}
$$
There is no <spin> multiplicity. This is continuum shell counting; a finite box has discrete levels. Using $\epsilon=\hbar^2k^2/(2m)$ gives the free-particle <density of states> per unit <energy> and <volume>
$$
g_V(\epsilon)=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt\epsilon.
$$
\b[<Semiclassical density of states in a central potential>.] At position $\mathbf r$, the available <kinetic energy> is $\epsilon-U(r)$. The local-density approximation integrates the preceding free-particle result over the classically allowed region:
$$
g(\epsilon)=\int_{U<\epsilon}\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{\epsilon-U(r)}\,d^3r=\boxed{\frac{(2m)^{3/2}}{\pi\hbar^3}\int_0^{r_\epsilon}r^2\sqrt{\epsilon-U(r)}\,dr.}
$$
This form assumes the allowed radial region is the connected interval $0\leq r\leq r_\epsilon$. A nonmonotonic potential would require integrating all allowed shells.
For the isotropic harmonic trap, $r_\epsilon=\sqrt{2\epsilon/(m\omega^2)}$. Substituting $r=r_\epsilon\sin\vartheta$ gives
$$
\int_0^{r_\epsilon}r^2\sqrt{\epsilon-\tfrac12m\omega^2r^2}\,dr=r_\epsilon^3\sqrt\epsilon\int_0^{\pi/2}\sin^2\vartheta\cos^2\vartheta\,d\vartheta=\frac\pi{16}r_\epsilon^3\sqrt\epsilon.
$$
Consequently
$$
\boxed{g(\epsilon)=\frac{\epsilon^2}{2(\hbar\omega)^3}.}
$$
The exact oscillator ground <energy> is $3\hbar\omega/2$ above the potential minimum. The <harmonic-trap energy-zero convention> subtracts that constant, while the displayed <density of states> is the leading semiclassical continuum expression, neglecting the finite-level corrections produced by that shift. The discrete <ground state> must be counted separately.
\b[Condensate population.] Below the transition, use $\mu_{\rm ch}\simeq0$ for the excited states. The <saturation of harmonic-trap excited states> follows from the supplied integral:
$$
N_{\rm ex}=\int_0^\infty\frac{g(\epsilon)\,d\epsilon}{e^{\epsilon/(k_BT)}-1}=\frac12\left(\frac{k_BT}{\hbar\omega}\right)^3\int_0^\infty\frac{x^2dx}{e^x-1}=\zeta(3)\left(\frac{k_BT}{\hbar\omega}\right)^3,
$$
where $\zeta$ is the <Riemann zeta function> and $\zeta(3)\simeq1.202$. Setting $N_{\rm ex}(T_0)=N$ gives the <harmonic-trap critical-temperature prefactor>:
$$
\boxed{T_0=\frac{\hbar\omega}{k_B}\left(\frac N{\zeta(3)}\right)^{1/3}\simeq0.9405\frac{\hbar\omega}{k_B}N^{1/3},\qquad \frac{N_0(T)}N=1-\left(\frac T{T_0}\right)^3\quad(T<T_0).}
$$
Above $T_0$ there is no macroscopic ground-state fraction in this approximation. For completeness, above the transition the <quantum gas fugacity> $z=e^{\mu_{\rm ch}/(k_BT)}<1$ satisfies $N=(k_BT/\hbar\omega)^3\operatorname{Li}_3(z)$; the <polylogarithm> $\operatorname{Li}_3$ therefore defines the plotted negative branch. A finite trapped gas has rounded behaviour, weak interactions shift the threshold, and the continuum thermal population is inaccurate when $k_BT$ is comparable to or below $\hbar\omega$. The power law is intended for the large-$N$ condensed regime, not an exact discrete-spectrum formula at arbitrarily small <temperature>.<Quantum Gases by David Tong>[Section 3.5: Bose distribution, excited-state saturation and separate condensate population; trap density-of-states calculation derived here]{ref}
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2016/paper-1-trapped-condensate.png]
{title=Leading ideal harmonic-trap chemical potential and condensate fraction versus temperature, with energy measured from the ground state}
{source=paper-1-trapped-condensate.py}
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