Wallis formula from the harmonic oscillator

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.

v1:8 pages; v2: reference added, typos corrected

References in corpus (3)

• Quantum Mechanical Derivation of the Wallis Formula for $\pi$
• On the quantum mechanical derivation of the Wallis formula for $\pi$
• From $e$ to $\pi$: Derivation of the Wallis formula for $\pi$ from $e$

Cited by in corpus (2)

• Relativistic Runge-Lenz vector: from $\mathcal{N}=4$ SYM to SO(4) scalar field theory
• Euler's Reflection Formula, Infinite Product Formulas, and the Correspondence Principle of Quantum Mechanics