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)∝ r^ν e^-λ r², which yields the scale-independent reciprocal observable Q= r r⁻¹. 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→1 and both finite-product representations reduce to the Wallis formula for π.