= Hydrogenic ground-state radius scaling
{title2=$\langle r\rangle_Z=3a_*/(2Z),\quad a_*=\hbar^2/(mK)$}
For the one-electron <Coulomb potential> $-KZ/r$ with fixed <nucleus>, the normalized spherically symmetric <ground state> is $\psi_Z=(\alpha_Z^3/\pi)^{1/2}e^{-\alpha_Zr}$, where $\alpha_Z=Z/a_*$ and $a_*=\hbar^2/(mK)$. Its <radial probability density> is $p_Z(r)=4\alpha_Z^3r^2e^{-2\alpha_Zr}$. The <factorial> integral gives
$$
\langle r\rangle_Z=4\alpha_Z^3\frac{3!}{(2\alpha_Z)^4}=\frac3{2\alpha_Z}=\frac{3a_*}{2Z}.
$$
More generally $\langle r^q\rangle_Z=\Gamma(q+3)/[2(2\alpha_Z)^q]$ for real $q>-3$, by the <gamma function> integral. The density transformation $p_Z(r)=Zp_1(Zr)$ expresses the full scaling, not just its mean. It applies to a <hydrogen-like ion> with one <electron>; the exact neutral <helium atom> radius additionally depends on the two-electron <Coulomb interaction>.
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