What is the expected distance of the electron from the nucleus in the hydrogen atom?

Question

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.

Jakub Wagner · Accepted Answer

There exists something called Kramers's recursion rule and I think it is what are you looking for.
$$frac{k+1}{n^2} leftlangle r^k rightrangle - frac{a_0}{Z} left(2k+1right)leftlangle r^{k-1} rightrangle + frac{k a_0^2}{4Z^2} left( left(2l+1right)^2 - k^2 right) leftlangle r^{k-2} rightrangle,$$
where $k$ is integer and $a_0$ Bohr radius. For deriving $leftlangle r rightrangle$ you have to calculate $leftlangle r^{-1} rightrangle$ at first by setting $k=0$ and then you can set $k=1$ and calculate $leftlangle r rightrangle$. And of course you know $leftlangle r^{0} rightrangle = 1$.
The result is
$$leftlangle r rightrangle = frac{a_0}{2Z}left(3n^2-lleft(l+1right)right).$$

user41298 · Answer

$$langle rrangle_{n,l,m}=frac{a_0n^2}{Z}left[1+frac{1}{2}left(1-frac{l(l+1)}{n^2}right)right].$$
Source: McQuarrie, Quantum Chemistry.

user1785960 · Answer

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]

What is the expected distance of the electron from the nucleus in the hydrogen atom?

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