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

R. B. Paris

doi:10.12988/ams.2015.53200

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 language	English
Pages (from-to)	3561-3567
Number of pages	7
Journal	Applied Mathematical Sciences
Volume	9
Issue number	72
DOIs	https://doi.org/10.12988/ams.2015.53200
Publication status	Published - 27 Apr 2015

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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

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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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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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