= Hydrogen radial second moment
{title2=$\langle r^2\rangle_{nl}$}
For a nonrelativistic point-nucleus <hydrogen atom>, $\langle r^2\rangle_{nl}=a_0^2n^2[5n^2+1-3l(l+1)]/2$. Integrating the normalized radial <wavefunction> squared with measure $r^4dr$ 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.
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