(i) A system of $N$ identical non-interacting bosons has energy levels $E_{i}$ with degeneracy $g_{i}, 1 \leq i<\infty$, for each particle. Show that in thermal equilibrium the number of particles $N_{i}$ with energy $E_{i}$ is given by

$$N_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}-1}$$

where $\beta$ and $\mu$ are parameters whose physical significance should be briefly explained.

(ii) A photon moves in a cubical box of side $L$. Assuming periodic boundary conditions, show that, for large $L$, the number of photon states lying in the frequency range $\omega \rightarrow \omega+d \omega$ is $\rho(\omega) d \omega$ where

$$\rho(\omega)=L^{3}\left(\frac{\omega^{2}}{\pi^{2} c^{3}}\right)$$

If the box is filled with thermal radiation at temperature $T$, show that the number of photons per unit volume in the frequency range $\omega \rightarrow \omega+d \omega$ is $n(\omega) d \omega$ where

$$n(\omega)=\left(\frac{\omega^{2}}{\pi^{2} c^{3}}\right) \frac{1}{e^{\hbar \omega / k T}-1} .$$

Calculate the energy density $W$ of the thermal radiation. Show that the pressure $P$ exerted on the surface of the box satisfies

$$P=\frac{1}{3} W$$

[You may use the result $\int_{0}^{\infty} \frac{x^{3} d x}{e^{x}-1}=\frac{\pi^{4}}{15}$.]