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The 1-D infinite square well is a classic problem in introductory QM. We find that the position-space eigenfunctions of the Hamiltonian (the "allowed wave functions") are sine waves with wavelengths such that they vanish at the boundaries of the well. It seems to me that, because these wave functions are pure sine waves, they are eigenstates of momentum, so the momentum uncertainty for these states is 0.

At the same time, the position uncertainty is finite, since the particle is confined inside the well. Then the product of the position and momentum uncertainties is 0, which seems to violate the Uncertainty Principle. If this argument is correct so far, it implies that a perfect infinite square well cannot be constructed in principle. Then one of the assumptions behind the problem is unphysical. There are at least two possibilities:

1. The assumption that the walls are infinitely high.

2. The assumption that the "corners" on the ends are infinitely sharp.

What I am wondering is, which of those two assumptions, or both, is to blame for the paradox? And how can we prove within quantum mechanics that that assumption is unphysical? This isn't homework, I just thought it was interesting. Thanks for any thoughts you have!