 Ground-state oscillator probability beyond the turning points

The ground-state turning points are $\pm \sqrt{\hslash /(m\omega )}$. In dimensionless position $y=x\sqrt{m\omega /\hslash }$, the probability outside them is $(2/\sqrt{\pi }){\int }_1^{\infty }{e}^{-y^2}dy=\mathrm{erfc}(1)\simeq 0.15730$. This tail demonstrates that a bound quantum state need not be confined to its classically allowed region.