  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}.
