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 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.
Original languageEnglish
Pages (from-to)3561-3567
Number of pages7
JournalApplied Mathematical Sciences
Volume9
Issue number72
DOIs
Publication statusPublished - 27 Apr 2015

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Harmonic Oscillator
Hermite Polynomials
Evaluation
Tunnel
Numerical Results
Energy

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abstract = "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.",
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Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities. / Paris, R. B.

In: Applied Mathematical Sciences, Vol. 9, No. 72, 27.04.2015, p. 3561-3567.

Research output: Contribution to journalArticle

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AB - 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.

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