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Firstly, I don't seem to fully understand the idea of an "observation." From my own intuition, I would guess that a measurement of position would be very difficult to make without constraining the particle to wherever you wanted it to be. The simplest approach seems to be "shooting" out millions of equally spaced deep potential wells at the wave function at the same time to "trap" the particle between two adjacent wells. However, even then the initial state of the quantum system would change, and so the resulting wave equation would still not contain exact numbers, and would have completely no relevance to the initial wave equation.

So the potential wells would need to be "gradually" grown up around the particle. But in that case, when would the wave equation ever need to "choose"? Wouldn't everything would be defined by solving the wave equation as a function of time?

Sorry for not being more specific, but any insight would be welcome.