  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.
