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

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