We obtain an asymptotic evaluation of the integral ʃ ∞ √2n+1 e−x2 H2 n (x) dx for n → ∞, where Hn(x) is the Hermite polynomial. This integral is used to determine the probability for the quantum harmonic oscillator in the nth energy eigenstate to tunnel into the classically forbidden region. Numerical results are given to illustrate the accuracy of the expansion.
Paris, R. B. (2015). Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities. Applied Mathematical Sciences, 9(72), 3561-3567. https://doi.org/10.12988/ams.2015.53200