The Expectation of ${1\over r}$ in the Ground State

$$\begin{eqnarray*} R_{10} &=& 2\left({Z\over a_0}\right)^{3\over 2}\; e^{ - Z r/ ... ...t)^{3} \left({a_0\over 2Z}\right)^2\; 1! \\ &=&{Z\over a_0} \\ \end{eqnarray*}$$

We can compute the expectation value of the potential energy.

$$\bgroup\color{black}\langle \psi_{100}\vert-{Ze^2\over r}\ver... ...=Z^2e^2{\alpha mc\over\hbar} =-Z^2\alpha^2mc^2=2E_{100} \egroup$$