Quantum Realization of the Wallis Formula
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Quantum Physics arXiv:2604.03662 (quant-ph) [Submitted on 4 Apr 2026] Title:Quantum Realization of the Wallis Formula Authors:Bin Ye, Ruitao Chen, Lei Yin View a PDF of the paper titled Quantum Realization of the Wallis Formula, by Bin Ye and 1 other authors View PDF HTML (experimental) Abstract:We present a unified quantum-mechanical derivation of the Wallis formula from two solvable radial systems: the circular states of the three-dimensional isotropic harmonic oscillator and the lowest-radial-branch states of the planar Fock--Darwin problem, including the lowest Landau level sector. In both cases, the radial probability density has the exact form $P(r)\propto r^\nu e^{-\lambda r^2}$, which yields the scale-independent reciprocal observable $Q=\langle r\rangle\langle r^{-1}\rangle$. The two systems realize the even and odd half-integer Gamma-function branches of the same moment formula, so that the associated finite Wallis partial products are determined by $Q$ in one case and by $Q^{-1}$ in the other. In the large-angular-momentum regime, the corresponding states become localized on a thin spherical shell or a narrow annulus, with vanishing relative radial width, so that $Q\to1$ and both finite-product representations reduce to the Wallis formula for $\pi$. Comments: Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th) Cite as: arXiv:2604.03662 [quant-ph] (or arXiv:2604.03662v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.03662 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Lei Yin [view email] [v1] Sat, 4 Apr 2026 09:29:20 UTC (57 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum Realization of the Wallis Formula, by Bin Ye and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 Change to browse by: hep-th References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)