What is the expected distance of the electron from the nucleus in the hydrogen atom? Specifically, I would like to know the general formula, in terms of $n$ and $l$, assuming the electron is in an orbital (i.e. simultaneous eigenstate of $H$, $L^2$, and $L_z$).
I understand that it involves integrating an associated Laguerre polynomial, but I wasn't able to find the formula for the integral. I looked on Wikipedia and in Abramowitz & Stegun, but no luck.
 A: $$\langle r\rangle_{n,l,m}=\frac{a_0n^2}{Z}\left[1+\frac{1}{2}\left(1-\frac{l(l+1)}{n^2}\right)\right].$$
Source: McQuarrie, Quantum Chemistry.
A: There exists something called Kramers's recursion rule and I think it is what are you looking for.
$$\frac{k+1}{n^2} \left\langle r^k \right\rangle - \frac{a_0}{Z} \left(2k+1\right)\left\langle r^{k-1} \right\rangle + \frac{k a_0^2}{4Z^2} \left( \left(2l+1\right)^2 - k^2 \right) \left\langle r^{k-2} \right\rangle,$$
where $k$ is integer and $a_0$ Bohr radius. For deriving $\left\langle r \right\rangle$ you have to calculate $\left\langle r^{-1} \right\rangle$ at first by setting $k=0$ and then you can set $k=1$ and calculate $\left\langle r \right\rangle$. And of course you know $\left\langle r^{0} \right\rangle = 1$.
The result is
$$\left\langle r \right\rangle = \frac{a_0}{2Z}\left(3n^2-l\left(l+1\right)\right).$$
A: Bethe and Salpeter, in Quantum Mechanics of One- and Two-Electron Atoms (Academic Press, 1957) derive a general expression for definite integrals of associated Laguerre functions times decaying exponentials times powers of the Laguerre argument on pages 13 and 14.  They use this result to produce the result given above by Wagner, which appears as equation 3.20.
A: 
I would like to know the general formula ...

The electron’s orbital distance, ionization energy and shape can be modeled based on classical mechanics when the recently-discovered pentaquark structure is used as the model of the proton.
General algorithm and calculation from Helium to Calcium inside paper: "Atomic Orbitals: Explained and Derived by Energy Wave Equations.":
https://vixra.org/abs/1708.0146
According to linked calculation an expected distance electron for hydrogen is:
$r_{1s} = 5.2918*10^{-11}$ [m]
or
$r_{1s} = 52.9$ [pm]
