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## Main Question or Discussion Point

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.

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.