English

Quantum Realization of the Wallis Formula

Quantum Physics 2026-04-07 v1 High Energy Physics - Theory

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)rνeλr2P(r)\propto r^\nu e^{-\lambda r^2}, which yields the scale-independent reciprocal observable Q=rr1Q=\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 QQ in one case and by Q1Q^{-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 Q1Q\to1 and both finite-product representations reduce to the Wallis formula for π\pi.

Keywords

Cite

@article{arxiv.2604.03662,
  title  = {Quantum Realization of the Wallis Formula},
  author = {Bin Ye and Ruitao Chen and Lei Yin},
  journal= {arXiv preprint arXiv:2604.03662},
  year   = {2026}
}

Comments

8 pages, 1 figure

R2 v1 2026-07-01T11:53:47.294Z