Quantum Realization of the Wallis Formula

Quantum Physics [Submitted on 4 Apr 2026] Quantum Realization of the Wallis Formula Bin Ye, Ruitao Chen, Lei Yin 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: 8 pages, 1 figure 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 Submission history From: Lei Yin [view email] [v1] Sat, 4 Apr 2026 09:29:20 UTC (57 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: hep-th References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)