How particles become localized

[dsdsuster]

I know when u take a measurement of an observable, the state vector collapses into one of the eigenvectors of the observable's operator.

So what happens if I try to localize the particle within some delta_x, using a projection operator for example. Let's say the particle exists previously in some known state with some position probability distribution. Since any probability distribution that is localized within the delta_x is an eigenvector of this projection operator how would we decompose the existing state vector in terms of allowed position distributions eigenvectors? There are clearly infinite allowed eigenvectors of the projection operator all localized within delta_x.

 

[Nugatory]

The "eigenfunctions" (scare-quotes because there are some mathematical subtleties here - google for "rigged Hilbert space") of the position operator are not physically realizable, so your position measurement does not collapse the wave function to one of them.

Even though they're not physically realizable we still use them in our calculations because the physically realizable states can be written as superpositions of them. This Before you measure, the wave function is one such superposition; the measurement collapses it to a different superposition, one in which all the amplitudes outside of the region between $x$ and $\Delta{x}$ cancel to zero.

And yes, we do need to add an infinite number of these "eigenfunctions" to get the right superposition; that's because the position observable is continuous not discrete, and it's why use an integral instead of a summation here.