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 $P(r)\propto r^νe^{-λr^2}$, 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 $π$.

• The paper derives the Wallis formula for π by linking quantum radial probability densities from the isotropic harmonic oscillator and Fock–Darwin models.
• The methodology exploits closed-form Gamma function identities and scale-invariant moment ratios to reveal the quantum-classical connection.
• The results offer practical insights into semiclassical localization and pave the way for extending these techniques to other solvable quantum systems.

Quantum Realization of the Wallis Formula: An Analytical Perspective

Overview of Objectives and Context

This paper develops a unified, quantum-mechanical derivation of the Wallis formula for $\pi$ by exhibiting it as a direct consequence of the radial probability structure in two canonical solvable systems: the circular states of the three-dimensional isotropic harmonic oscillator and the lowest-radial-branch states of the planar Fock–Darwin (and Landau level) problem. The analysis pivots on the realization that in both models, the radial probability density is of the form $P(r) \propto r^\nu e^{-\lambda r^2}$, which gives rise to a scale-invariant, dimensionless observable $Q = \langle r \rangle \langle r^{-1} \rangle$. The Wallis product appears not as a mathematical curiosity but as a finite partial product directly linked to $Q$ (or its inverse), converging to the Wallis formula in the semiclassical limit.

Historically, Friedmann and Hagen first elucidated a quantum-mechanical connection to the Wallis formula via a variational treatment of the hydrogen atom, with follow-ups demonstrating the phenomenon's independence from model-specific dualities or trial functions (Friedmann et al., 2015, Chashchina et al., 2017, Cortese et al., 2017). The present approach departs from those routes by relying exclusively on exact properties of fully solvable models and is not predicated on energy comparison, but rather on the explicit structure of radial probability densities and Gamma function ratios.

The Common Radial-Gaussian Framework and Wallis Product Connection

At the core of the analysis is the family of probability distributions

$$P_{\nu,\lambda}(r) = N_{\nu,\lambda} \, r^\nu e^{-\lambda r^2}, \qquad r>0,\, \nu > -1,\, \lambda > 0,$$

which encompasses both the oscillator and Fock–Darwin/Landau regimes through appropriate choices of $\nu$ and $\lambda$. The essential observable is

$$Q_\nu := \langle r \rangle_{\nu,\lambda} \langle r^{-1} \rangle_{\nu,\lambda} = \frac{\Gamma\left(\frac{\nu+2}{2}\right)\Gamma\left(\frac{\nu}{2}\right)}{\Gamma\left(\frac{\nu+1}{2}\right)^2}.$$

Here, all scale dependence cancels, and only the shape parameter $\nu$ enters. The positivity and Cauchy-Schwarz bounds give $Q_\nu \geq 1$, with equality only for delta-distributions (perfect radial rigidity). The connection to the Wallis formula emerges in the structure of half-integer Gamma-function products, where the even and odd branches correspond to the oscillator and magnetic models, respectively.

Application to the Three-Dimensional Isotropic Harmonic Oscillator

For the three-dimensional isotropic harmonic oscillator, the circular (node-less) states are characterized by angular momentum quantum number $P(r) \propto r^\nu e^{-\lambda r^2}$0 and radial probability densities

$P(r) \propto r^\nu e^{-\lambda r^2}$1

where $P(r) \propto r^\nu e^{-\lambda r^2}$2 is the characteristic scale. Calculation of the moments yields

$P(r) \propto r^\nu e^{-\lambda r^2}$3

Via explicit evaluation, this ratio is written in terms of the finite Wallis product $P(r) \propto r^\nu e^{-\lambda r^2}$4,

$P(r) \propto r^\nu e^{-\lambda r^2}$5

In the large-$P(r) \propto r^\nu e^{-\lambda r^2}$6 regime, these states develop into thin spherical shells (semiclassical circular orbits), leading to $P(r) \propto r^\nu e^{-\lambda r^2}$7, and hence $P(r) \propto r^\nu e^{-\lambda r^2}$8, manifesting the Wallis formula in the limit.

Application to the Planar Fock–Darwin and Landau Systems

The planar Fock–Darwin system, describing a charged particle in a uniform magnetic field with optional harmonic confinement, similarly yields, for the lowest-radial-branch (lowest Landau level-like) states,

$P(r) \propto r^\nu e^{-\lambda r^2}$9

with $Q = \langle r \rangle \langle r^{-1} \rangle$0, and quantum number $Q = \langle r \rangle \langle r^{-1} \rangle$1. The corresponding observable is

$Q = \langle r \rangle \langle r^{-1} \rangle$2

Inverting, the Wallis partial product appears as

$Q = \langle r \rangle \langle r^{-1} \rangle$3

Structurally, the inversion arises from the difference in measure factors between the three- and two-dimensional cases. In the large-$Q = \langle r \rangle \langle r^{-1} \rangle$4 limit, the states localize on narrow annuli (guiding-center orbits), again yielding $Q = \langle r \rangle \langle r^{-1} \rangle$5 and thus $Q = \langle r \rangle \langle r^{-1} \rangle$6, reproducing the Wallis product.

Theoretical and Practical Implications

This analysis demonstrates that the appearance of the Wallis formula in quantum mechanics is not tied to a specific system or to an ad hoc variational approach, but is underpinned by a structural pattern: any solvable radial quantum system with explicit Gaussian weighting and integer/half-integer radial exponents realizes a simple, closed-form, scale-invariant moment ratio. When the angular quantum numbers become large (the semiclassical regime), the quantum radial distributions become sharply localized, and the reciprocal radial observable approaches unity—a signature of correspondence with classical motion. The Wallis product thereby emerges as a finite quantum correction to the classical limit, readable directly from closed-form expressions.

On a practical level, this offers insight into the universality and structural recurrence of the Wallis formula in quantum models, suggesting that any system furnishing an exact, half-integer parameter Gaussian radial density will yield analogous results. On a theoretical level, the identification of a shared reciprocal-radial mechanism extends the interpretative reach of semiclassical analysis and may guide the search for new quantum systems and observables where nontrivial mathematical identities manifest physically.

Outlook and Possible Extensions

The methodology outlined in this paper can be directly generalized to other radial quantum problems with similar symmetry and exactly solvable spectral properties, and may inform analyses of deformation, perturbation, or mapping procedures in integrable models. Additionally, the bridge between semiclassical localization and closed-form moment-ratio identities provides a platform for re-evaluating the appearance of special function products (beyond Wallis-type) in quantum expectation values.

One possible direction involves exploring families of solvable potentials with alternative weightings, or extending to higher-dimensional isotropic systems with nontrivial measure contributions. There is also potential in using the observable $Q = \langle r \rangle \langle r^{-1} \rangle$7 or its relatives as benchmarks for quantum-classical transition metrics in quantum chaos or for quantifying rigidity in non-Gaussian radial distributions.

Conclusion

This work rigorously establishes the Wallis product formula for $Q = \langle r \rangle \langle r^{-1} \rangle$8 as a quantum-mechanical phenomenon arising from the explicit form of the radial probability density in both the three-dimensional harmonic oscillator and planar Fock–Darwin (Landau-level) systems. By leveraging closed-form Gamma function identities, it unifies and demystifies previous observations by showing that both models instantiate a common, physically transparent semiclassical mechanism encoded in the scale-invariant reciprocal radial observable. The results clarify when and how classical analytic constants such as $Q = \langle r \rangle \langle r^{-1} \rangle$9 emerge from quantum structure and point to broader classes of solvable systems where similar connections must exist.

Reference: "Quantum Realization of the Wallis Formula" (2604.03662)