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

R B Paris

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arXiv · 1502.03382

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

R B Paris

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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.

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R B Paris. 2015-02-11. Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities. https://arxiv.org/abs/1502.03382

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