For the reduced quantum harmonic oscillator, and the ground-state energy is . Thus the classical turning points satisfy , giving . Normalizing the wavefunction gives , because . The Born rule therefore gives the harmonic-oscillator forbidden-region probability
The factor two includes both tails. Using the exact complementary error function gives . A stationary real wavefunction has zero probability current, so this result describes forbidden-region occupancy, not continuous loss of the bound particle.
Figure 1.
Normalized ground-state probability with both classically forbidden tails shaded
. Details.