At temperature $T$ and chemical potential $\mu$, the number density of a non-relativistic particle species with mass $m \gg k_{B} T / c^{2}$ is given by

$$n=g\left(\frac{m k_{B} T}{2 \pi \hbar^{2}}\right)^{3 / 2} e^{-\left(m c^{2}-\mu\right) / k_{B} T},$$

where $g$ is the number of degrees of freedom of this particle.

At recombination, electrons and protons combine to form hydrogen. Use the result above to derive the Saha equation

$$n_{H} \approx n_{e}^{2}\left(\frac{2 \pi \hbar^{2}}{m_{e} k_{B} T}\right)^{3 / 2} e^{E_{\mathrm{bind}} / k_{B} T}$$

where $n_{H}$ is the number density of hydrogen atoms, $n_{e}$ the number density of electrons, $m_{e}$ the mass of the electron and $E_{\text {bind }}$ the binding energy of hydrogen. State any assumptions that you use in this derivation.