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

2015/02/11 by R B Paris, Paris, R B · 1 citation
Mathematics · #33C45 #34B05 #41A30 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C45 #msc:34B05 #msc:41A30 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1502.03382

6 pages, 0 figures

arxiv created 2015/02/11 · arxiv updated 2015/02/12

Abstract

We obtain an asymptotic evaluation of the integral ∫_√(2n+1)^∞ e^-x2 H_n²(x) dx for n→∞, where H_n(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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