 Hydrogenic ground-state radius scaling

For the one-electron Coulomb potential $-KZ/r$ with fixed nucleus, the normalized spherically symmetric ground state is ${\psi }_Z=({\alpha }_Z^3/\pi {)}^{1/2}{e}^{-{\alpha }_{Z}r}$, where ${\alpha }_Z=Z/a_{\ast }$ and $a_{\ast }={\hslash }^2/(mK)$. Its radial probability density is $p_Z(r)=4{\alpha }_Z^3{r}^2{e}^{-2{\alpha }_{Z}r}$. The factorial integral givesMore generally $⟨r^q{⟩}_Z=\Gamma (q+3)/[2(2{\alpha }_Z{)}^q]$ for real $q>-3$, by the gamma function integral. The density transformation $p_Z(r)=Z{p}_1(Zr)$ 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.