Suppose one measures the position, then momentum, then position of a particle, and that all these measurements are done in quick succession of one another (ie. arbitrarily close to zero-time as possible). After the momentum measurement, the particles position is very uncertain. Therefore, the second measurement of the particles position can be quite different from the first. But does this not violate causality, since the particle can move a great distance in practically no time?
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$\begingroup$ Either you can't make measurements infinitely quickly, or you can't measure the momentum very certainly, which decreases the uncertainty of position brought by moment measurement. $\endgroup$– LuessiawCommented Jul 26, 2023 at 6:27
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1$\begingroup$ This question and answers might help you see how complicated causality is when quantum mechanics is involved ( the HUP) physics.stackexchange.com/questions/346780/… $\endgroup$– anna vCommented Jul 26, 2023 at 9:11
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$\begingroup$ This may be related to the Mott problem - (how do a trajectory emerges from wave functions?). The culprit is that the whole detection context must be considered. See the original paper. $\endgroup$– Stéphane RollandinCommented Jul 26, 2023 at 14:15
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$\begingroup$ Related: physics.stackexchange.com/q/762398/226902 and physics.stackexchange.com/q/561760/226902 $\endgroup$– QuilloCommented Jul 26, 2023 at 15:42
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The HUP applies to simultaneous measurements (of $x$ and $p$). What you propose are consecutive measurements.
A nice discussion of the distinction is given in
Raymer, M. G. "Uncertainty principle for joint measurement of noncommuting variables." American Journal of Physics 62.11 (1994): 986-993.
answered Jul 26, 2023 at 15:19