Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities

TL;DR

This paper derives an asymptotic approximation for an integral involving Hermite polynomials, which models tunneling probabilities in quantum harmonic oscillators, providing insights into quantum behavior at large quantum numbers.

Contribution

It introduces a novel asymptotic evaluation of a key integral related to quantum tunneling probabilities in harmonic oscillators for large quantum numbers.

Findings

Asymptotic formula accurately estimates tunneling probabilities.

Numerical results confirm the expansion's precision.

Provides a mathematical tool for quantum physics applications.

Abstract

We obtain an asymptotic evaluation of the integral \[\int_{\sqrt{2n+1}}^\infty e^{-x^2} H_n^2(x)\,dx\] for $n\rightarrow\infty$, where $H_n(x)$ is the Hermite polynomial. This integral is used to determine the probability for the quantum harmonic oscillator in the $n$th energy eigenstate to tunnel into the classically forbidden region. Numerical results are given to illustrate the accuracy of the expansion.