The penetration probability of the quantum harmonic oscillator into the classically forbidden region is examined in the limit of large quantum numbers using analytical methods. An expression is derived for the asymptotic form of the penetration probability, and is shown to be in good agreement with an earlier result obtained by numerical methods. The asymptotic form of the probability density in the region between the classical turning points is also presented and found to have a simple physical interpretation.
Atems DE & Wadehra JM. Asymptotic form of the penetration probability of the quantum harmonic oscillator into the classically forbidden region. Am. J. Phys. 1995;63(5):443-6. doi: 10.1119/1.17908