Asymptotic formula for quantum harmonic oscillator tunneling probabilities

TL;DR

This paper derives an asymptotic formula showing that the tunneling probability of a quantum harmonic oscillator in the n-th energy state decreases proportionally to the inverse cube root of n, revealing a simple relationship in the high-energy limit.

Contribution

It introduces a straightforward asymptotic analysis method to determine tunneling probabilities for quantum harmonic oscillators in high-energy states.

Findings

Tunneling probability scales as n^(-1/3) for large n.

The asymptotic formula simplifies understanding of quantum tunneling in harmonic oscillators.

The result is derived using elementary asymptotic analysis techniques.

Abstract

Using simple methods of asymptotic analysis it is shown that for a quantum harmonic oscillator in n-th energy eigenstate the probability of tunneling into the classically forbidden region obeys an unexpected but simple asymptotic formula: the leading term is inversely proportional to the cube root of n.