Hydrogen ground-state probability beyond the classical turning point ($P(r>2a_0)=13e^{-4}$)

The nonrelativistic hydrogen ground energy is $-e^2/(8\pi {ϵ}_0{a}_0)$. Requiring nonnegative classical kinetic energy restricts radius to $r\le 2a_0$. Integrating the quantum ground-state radial probability density beyond that radius instead gives $13e^{-4}\simeq 0.238$. A bound-state tail can therefore have substantial integrated probability outside the classically allowed region without violating energy conservation.