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.
Highlights
We obtain an asymptotic evaluation of the integral2n+1 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
The quantum harmonic oscillator in dimensionless variables has the Hamiltonian given by H = (p2+ x2), where x, pare the position and momentum operators satisfying [x, p] = i
It has been pointed out recently by Jadczyk [1] that the tunnelling probabilities Pn,tun are rarely discussed in the literature on quantum mechanics
Summary
2n+1 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
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