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

Keywords

Quantum oscillator
Tunnelling
Asymptotic expansion

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Paris_AsymptoticEvaluationOfAnIntegralArisingInQuantumHarmonic_Published_2015Final published version, 201 KBLicence: CC BY

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title = "Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities",

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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DO - 10.12988/ams.2015.53200

M3 - Article

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VL - 9

SP - 3561

EP - 3567

JO - Applied Mathematical Sciences

JF - Applied Mathematical Sciences

IS - 72

ER -

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

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