From the classical point of view, the electron in a hydrogen atom moves under the influence of a central attractive force \(F=-e^{2} / r^{2}\) caused by the proton nucleus. According to "old quantum theory" the phase integrals over \(r\) and \(\theta\) are given by

\[\oint p_{r} d r=n_{1} h \quad \text { and } \quad \oint p_{\theta} d \theta=n_{2} h\]

where \(p_{r}\) and \(p_{\theta}\) are the classical canonical momenta, \(h\) is Planck's constant and \(n_{1}\) and \(n_{2}\) are positive integers. Show that according to old quantum theory there are only a discrete set of possible energy levels, given by

\[E_{n}=-\frac{m e^{4}}{2 n^{2} \hbar^{2}}\]

where \(\hbar \equiv h / 2 \pi\) and \(n=n_{1}+n_{2}\).