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The wave function in Quantum Mechanics tells us, among other things, the probability of finding a particle at somme location. According to the Copenhagen interpretation of QM, when we look for and find the particle somewhere, we say the wave function has collapsed.

If the wave function is defined in a particular region and we try to measure the particle at one particular sub-region of that region but don't find it there, does the wave function still collapse such that the particle is localized somewhere else after the measurement?

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Measurements are represented by operators, and collapse is described as producing an eigenstate of that operator. The measurement you're doing is represented by a projection operator. In the situation you describe, the wavefunction after collapse will have had one region projected down to zero, but the other region will still all be there. The particle is not localized to one point within that region.

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