Quantum Realization of the Wallis Formula
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 , which yields the scale-independent reciprocal observable . 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 in one case and by 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 and both finite-product representations reduce to the Wallis formula for .
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