Write down the expressions for the classical energy and angular momentum for an electron in a hydrogen atom. In the Bohr model the angular momentum $L$ is quantised so that

$$L=n \hbar,$$

for integer $n$. Assuming circular orbits, show that the radius of the $n$ 'th orbit is

$$r_{n}=n^{2} a$$

and determine $a$. Show that the corresponding energy is then

$$E_{n}=-\frac{e^{2}}{8 \pi \epsilon_{0} r_{n}}$$