A quantum mechanical well and a derivation of a $\pi^2 $ formula

TL;DR

This paper derives two mathematical formulas involving pi from the energy spectrum of a quantum particle in an infinite potential well, revealing specific momentum measurement outcomes and their relation to classical values.

Contribution

It introduces novel derivations of pi-related formulas within quantum mechanics, connecting spectral properties to classical pi series and integrals.

Findings

Derived a pi series from quantum momentum spectra

Identified momentum peaks at classically allowed values for even quantum states

Connected quantum spectral analysis with classical pi formulas

Abstract

Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a $\pi$ series. In this work, two "$\pi$ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, $\frac{\pi^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}$, and the integral $\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =\pi$. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of $\pi$ formula in \cite{wallys}.