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Quantum Realization of the Wallis Formula

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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 $\pi$.

fields

quant-ph 1

years

2026 1

verdicts

UNVERDICTED 1

representative citing papers

Emergence of $\pi$ from Equatorial Quantum Localization

quant-ph · 2026-04-29 · unverdicted · novelty 6.0

A natural geometric rigidity index for equatorial localization on the sphere in highest-weight spherical harmonics is exactly a Wallis partial product, recovering π via the correspondence principle in the large-quantum-number limit.

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  • Emergence of $\pi$ from Equatorial Quantum Localization quant-ph · 2026-04-29 · unverdicted · none · ref 5 · internal anchor

    A natural geometric rigidity index for equatorial localization on the sphere in highest-weight spherical harmonics is exactly a Wallis partial product, recovering π via the correspondence principle in the large-quantum-number limit.