Uncertainty and particle in a box

[hokhani]

TL;DR

More confinement doesn't result in more uncertainty!

As far as I know about the position -momentum uncertainty if a quantum particle is more confined, we expect its momentum to be more uncertain. However, I think, as a counterexample one may take the particle in a box. Each sin wave function (which is the solution of particle in the box) is always a combination of the two opposite momenta no matter how much the particle is confined. So, it seems that the uncertainty in momentum doesn't change by confinement. I appreciate any explanation.

 

[anuttarasammyak]

TL;DR: More confinement doesn't result in more uncertainty!

Each sin wave function (which is the solution of particle in the box) is always a combination of the two opposite momenta no matter how much the particle is confined.

These solution wave functions must be zero outside the box. They are not simple sinusoids functions. Their momentum wave functions do not give simple two opposite values. Ref. https://en.wikipedia.org/wiki/Particle_in_a_box

 

[Morbert]

The wavefunction is zero outside the box, so you don't strictly have two exact but opposite momenta. Also, these two momenta are further from the mean the smaller the box is, increasing uncertainty.