# On the quantum mechanical derivation of the Wallis formula for $\pi$

## TL;DR

This paper examines the quantum mechanical derivation of the Wallis formula for pi, showing that different trial functions can lead to the same result and analyzing the integral structures involved.

## Contribution

It demonstrates that both Gaussian and Lorentz trial functions can derive the Wallis formula, expanding the understanding of its quantum mechanical origins.

## Key findings

- Both Gaussian and Lorentz trial functions yield the Wallis formula
- The integral structure leading to the Wallis ratio is explicitly analyzed
- The derivation is not limited to a specific trial function

## Abstract

We comment on the Friedmann and Hagen's quantum mechanical derivation of the Wallis formula for $\pi$. In particular, we demonstrate that not only the Gaussian trial function, used by Friedmann and Hagen, but also the Lorentz trial function can be used to get the Wallis formula. The anatomy of the integrals leading to the appearance of the Wallis ratio is carefully revealed.