= Hydrogen ground-state nuclear-volume probability
{title2=$P(r<r_p)\simeq4r_p^3/(3a_0^3)$}
For the point-Coulomb <ground state>, radial <probability density> is $4r^2e^{-2r/a_0}/a_0^3$. Integrating to $r_p$ gives $1-e^{-2u}(1+2u+2u^2)$, with $u=r_p/a_0$, whose small-$u$ expansion is the displayed result. A finite <proton> charge distribution changes the interior potential, and a <proton> radius is not literally a sharp material boundary; the expression is an estimate of a small overlap region.
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