Hydrogen#

The Hydrogen atom has a classical hamiltonian of,

$$\boxed{{\displaystyle \hat{H}=\frac{-{\hslash }^2}{2m}{\hat{\mathrm{\nabla }}}^2-\frac{e^2}{4\pi {ϵ}_0}\frac{1}{r}}}$$

With an eigenvalue, $$\begin{array}{rl}{E}_n& =-\frac{m_e}{2{\hslash }^2}{\left(\frac{e^2}{4\pi {ϵ}_0}\right)}^2\frac{1}{n^2}\\ & =\frac{m{\alpha }^2{c}^2}{2}\frac{1}{n^2}\\ & \approx \frac{13.6\text{ eV}}{n^2}\end{array}$$