Wallis formula from the harmonic oscillator

TL;DR

This paper reveals that the Wallis formula for pi, historically linked to quantum mechanics and hydrogen atoms, can also be derived from the harmonic oscillator via quantum duality and variational methods.

Contribution

It demonstrates a novel connection between the Wallis formula and the harmonic oscillator through quantum duality and variational approach, challenging previous associations with hydrogen atoms.

Findings

Wallis formula can be derived from harmonic oscillator using quantum duality.

The formula's relation to quantum mechanics is due to a specific trial function and potential choice.

The asymptotic formula is not inherently related to hydrogen atoms or quantum mechanics.

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.