A harmonic oscillator of frequency $\omega$ is in thermal equilibrium with a heat bath at temperature $T$. Show that the mean number of quanta $n$ in the oscillator is

$$n=\frac{1}{e^{\hbar \omega / k T}-1} .$$

Use this result to show that the density of photons of frequency $\omega$ for cavity radiation at temperature $T$ is

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

By considering this system in thermal equilibrium with a set of distinguishable atoms, derive formulae for the Einstein $A$ and $B$ coefficients.

Give a brief description of the operation of a laser.