Mean-square orbital radius
= Mean-square orbital radius
{title2=$\langle r^2\rangle$}
For a normalized spatial <wavefunction>, $\langle r^2\rangle=\int r^2|\psi(\mathbf r)|^2d^3r$. 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 $\langle r^2\rangle=3a^2$, obtained by dividing the radial integrals of $r^4e^{-2r/a}$ and $r^2e^{-2r/a}$.