Hydrogen radial second moment ($⟨r^2{⟩}_{nl}$)

For a nonrelativistic point-nucleus hydrogen atom, $⟨r^2{⟩}_{nl}=a_0^2{n}^2[5n^2+1-3l(l+1)]/2$. Integrating the normalized radial wavefunction squared with measure $r^{4}dr$ gives $3a_0^2,42a_0^2,30a_0^2$ for $1s,2s,2p$. For a hydrogen-like ion, replace $a_0$ by $a_0/Z$ in the infinite-nuclear-mass model. These moments determine first-order shifts under quadratic potential energy perturbations.