In quantum mechanics, measuring the position of a particle causes the wave function to collapse, fixing the particle at a measured position. Given this collapse, how can it be claimed that as the observation time tends to zero, the uncertainty in position becomes infinitely large? Once the position is measured, the wave function collapses to the measured location, and it is not possible to take successive measurements of the position on the original wave function. Therefore, how can we define the uncertainty in position measurements for the original wave function if we are restricted to observing a single value?
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2 Answers
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From Griffiths' Introduction to Quantum Mechanics:
The expectation value is the average of measurements on an ensemble of identically-prepared systems, not the average of repeated measurements on one and the same system.
The uncertainty is defined in terms of expectation values: $$\Delta A = \sqrt{\langle \hat{A}^2\rangle - \langle \hat{A}\rangle^2}$$ so it is the standard deviation of the measurement on a large number of systems in the same state.
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$\begingroup$ But that is the expected value (the average value), instead of the uncertainty delta_x I meant $\endgroup$CuriousMind– CuriousMind2024-05-30 20:02:30 +00:00Commented May 30, 2024 at 20:02
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3$\begingroup$ @CuriousMind Read the last line again. $\endgroup$Vincent Thacker– Vincent Thacker2024-05-30 20:09:03 +00:00Commented May 30, 2024 at 20:09
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In quantum mechanics, measuring the position of a particle causes the wave function to collapse, fixing the particle at a measured position.
No it doesn't. The equations of motion of quantum systems do not imply wave function collapse. There are some variants of quantum theory, such as spontaneous collapse theory that feature wave function collapse
https://arxiv.org/abs/2310.14969
But such theories haven't reproduced the predictions of relativistic quantum field theories, i.e.- almost all predictions of quantum theory in the real world:
https://arxiv.org/abs/2205.00568
A real measurement is an interaction that produces a copyable record of some property of a system. Such interactions suppress quantum interference between versions of a system that are macroscopically different from one another:
https://arxiv.org/abs/quant-ph/0306072
This leads to a reality that looks like a collection of approximately independent histories of the same system on the scales of everyday life:
https://arxiv.org/abs/0903.1802
https://arxiv.org/abs/1111.2189
Once the position is measured, the wave function collapses to the measured location, and it is not possible to take successive measurements of the position on the original wave function. Therefore, how can we define the uncertainty in position measurements for the original wave function if we are restricted to observing a single value?
You prepare multiple copies of the same kind of system under the same conditions so that they end up in the same state. You then measure different observables, get the probability distribution over the possible outcomes and use that information to work out the state: this is called quantum tomography
