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.
Wallis formula from the harmonic oscillator
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We show that the asymptotic formula for $\pi$, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in \cite{HF}, can also be related to the harmonic oscillator using a quantum duality between these two systems. As a corollary we show that this very interesting asymptotic formula is not related with the hydrogen atom or quantum mechanics itself but with a clever choice of a trial function and a potential in the Schroedinger equation when we use the variational approach to calculate the ground state energy associated with the given potential function.
fields
quant-ph 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Variational minimization on the 4D singular harmonic oscillator with quartic trial functions yields the Wallis product for odd effective angular momenta and its reciprocal for even ones in the large-nu limit via Gamma-function ratios.
Quantum states in the 3D harmonic oscillator and planar Fock-Darwin systems realize the Wallis formula for pi through the scale-independent observable Q = <r><r^{-1}> that approaches 1 at high angular momentum.
citing papers explorer
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Emergence of $\pi$ from Equatorial Quantum Localization
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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Wallis Products from the Four-Dimensional Singular Harmonic Oscillator
Variational minimization on the 4D singular harmonic oscillator with quartic trial functions yields the Wallis product for odd effective angular momenta and its reciprocal for even ones in the large-nu limit via Gamma-function ratios.
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Quantum Realization of the Wallis Formula
Quantum states in the 3D harmonic oscillator and planar Fock-Darwin systems realize the Wallis formula for pi through the scale-independent observable Q = <r><r^{-1}> that approaches 1 at high angular momentum.