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When you measure $\vec p$ the wave function collapses to an eigenstate of the momentum operator. These eigenfunctions are always plane waves, correct? Does it mean that momentum always collapses into a plane wave? What if the particle is confined in space? How can the wave function collapse into a function that is non-zero at infinity if the particle has zero probability of being found outside of the confined location?

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    $\begingroup$ A minor addition to the good answer by @Emilio Pisanty: the reason we need to let the wavefunction expand before we can get a good momentum-measurement is because interactions are local in the spatial domain, not in the momentum domain. $\endgroup$ – Chiral Anomaly Mar 1 at 18:55
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When you measure $\vec p$ the wave function collapses to an eigenstate of the momentum operator. These eigenfunctions are always plane waves, correct? Does it mean that momentum always collapses into a plane wave?

Yes, and yes.

What if the particle is confined in space? How can the wave function collapse into a function that is non-zero at infinity if the particle has zero probability of being found outside of the confined location?

To perform an accurate momentum measurement, regardless of what technique you use, you need to let the wavefunction expand to a given extent $L$, with the resolution of your momentum measurement getting fixed by the uncertainty principle at $\Delta p \sim h/L$. Performing a measurement of momentum to infinite precision is unphysical (exactly as with an infinite-precision measurement of position), but in the ideal case you would need to let the wavefunction expand to an unbounded size.

Measurements are not just abstract applications of projection operators: if you want them to happen in the real world, then they need to be physical processes in which things change. If you don't let your wavefunction expand, then you won't be able to measure momentum.

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