Hydrogen

The Hydrogen atom has a classical hamiltonian of,

\[ \begin{equation} \boxed{\hat H = \frac{-\hbar^2}{2m}\hat\nabla^2 - \frac{e^2}{4\pi\epsilon_0}\frac{1}{r} } \end{equation} \]

With an eigenvalue, $\( \begin{align} E_n &= -\frac{m_e}{2\hbar^2}\left(\frac{e^2}{4\pi\epsilon_0}\right)^2\frac{1}{n^2} \\ &= \frac{m\alpha^2 c^2}{2}\frac{1}{n^2}\\ &\approx \frac{13.6 \text{ eV}}{n^2} \end{align} \)$