Infinite-well ground-state uncertainty product ($\Delta x\Delta p=\hslash \sqrt{{\pi }^2/12-1/2}$)

For the normalized ground state $\sqrt{2/L}\sin (\pi x/L)$ on $0<x<L$, direct position integrals give $\Delta x^2=L^2(1/12-1/(2{\pi }^2))$. The kinetic momentum second moment gives $\Delta p^2={\pi }^2{\hslash }^2/L^2$. Their product is about $0.568\hslash$, strictly above the Heisenberg uncertainty principle bound. Infinite walls do not make this non-Gaussian state a minimum-uncertainty wave packet.