# How should get the expectation value of $1/r$ in the hydrogen atom? [duplicate]

I have a question of quantum mechanics. I want to calculate \begin{align}\left\langle \frac{1}{r} \right\rangle \end{align} in the state $$n, l$$ of the hydrogen atom.

In the textbook I have, the wavefunction is given as \begin{align} \psi_{nlm} = \sqrt{\left(\frac{2Z}{na_0}\right)^3\frac{(n-l-1)!}{2n[(n+l)!]^3}} e^{-Zr/na_{0}}\left(\frac{Zr}{na_{0}}\right)^{l}L^{2l+1}_{n+l}\left(\frac{2Z}{na_{0}}r\right)Y^{m}_{l}(\theta, \phi)\end{align}

Though I referred this link, the index of the associated Laguerre polynomial and the normalization coefficient are quite different hence I cannot get an idea how to simplify the result. How can I calculate the expectation of \begin{align}\left\langle \frac{1}{r}\right\rangle?\end{align}

As the potential in the hydrogen hamiltonian, is $$V(r)= - \frac{e^2}{4\pi \epsilon_0 r}$$ The easiest way is to use the formula for the energy $$E_{n,l}= \frac{m_e e^4} {2(4\pi \epsilon_0)^2 \hbar^2 n^2},$$ in conjuction with the Feynman-Hellmann formula $$\frac{\partial E_{n,l}}{\partial \lambda} = \langle n,m| \frac{\partial H}{\partial \lambda}| n,l\rangle$$ and just differentiate $$E_{nl}$$ with respect to $$e$$.