Mean-square orbital radius ($⟨r^2⟩$)

For a normalized spatial wavefunction, $⟨r^2⟩=\int r^2|\psi (\mathbf{r}){|}^2{d}^{3}r$. Its square root is the root-mean-square radius, which differs from the exponential decay length. A spherical hydrogenic donor state with $\psi \propto e^{-r/a}$ has $⟨r^2⟩=3a^2$, obtained by dividing the radial integrals of $r^4{e}^{-2r/a}$ and $r^2{e}^{-2r/a}$.