= Classical grand potential in a harmonic trap
{title2=$\Phi=-k_BT e^{\beta\mu}[1-e^{-\beta\hbar\omega}]^{-3}$}
For noninteracting particles in a three-dimensional isotropic <quantum harmonic oscillator>, measure the one-particle ground energy as zero. The single-particle <partition function> is $Z_1=(1-e^{-\beta\hbar\omega})^{-3}$, from the product of three <geometric series>. In the dilute <Maxwell–Boltzmann statistics> regime, $\ln\Xi=e^{\beta\mu}Z_1$, so the <grand potential> is the displayed expression and $N=e^{\beta\mu}Z_1$. For $k_BT\gg\hbar\omega$,
$$
\Phi\simeq-\frac{(k_BT)^4}{(\hbar\omega)^3}e^{\beta\mu},\qquad\mu\simeq k_BT\ln\left[N\left(\frac{\hbar\omega}{k_BT}\right)^3\right].
$$
The same approximation follows from integrating the <density of states> $g(E)\simeq E^2/[2(\hbar\omega)^3]$ against $e^{-\beta E}$. Extrapolating the classical formula until the fugacity is one gives the scale of <Bose condensation in a three-dimensional harmonic trap>, but the quantitative transition temperature requires the Bose occupation law rather than Maxwell-Boltzmann statistics.
Back to article page
