{"paper":{"title":"Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R B Paris","submitted_at":"2015-02-11T17:15:09Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03382","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}