Let $\sigma_x$ and $\sigma_p$ be the standard deviation of position and momentum of a particle. The ordinary uncertainty relation tells us that in general we have
Now, consider a particle in an interval of length $L$, contrained by an infinite potential well. Since the wave function of that particle is zero at the boundary, the product $\sigma_x\,\sigma_p$ should be striktly greater that $\hbar/2$ because the particle can not be in the (gaussian) state of minimum uncertainty.
Moreover, it is not to expect that the ground state of a particle in the box is the state of minimum uncertainty. So, my question is:
What is the minimum value of the product $\sigma_x\,\sigma_p$ which a particle in the box can reach?