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Suppose we want to measure a modulus square of a wave function $\psi(\vec{r},t)$ for a single particle. It is important that the wave function depends on time. Are there any fundamental limitations to such measurement? I would expect we cannot resolve the details of $|\psi(\vec{r},t)|^2$ below Planck length and Planck time, but are there any other restrictions?

Edit: In the case of the quantum field theory, we would be looking at the expectation value of the operator of the number of particles $\hat{n}(r)$ in different regions of space, and the question would be whether there are restrictions on the size of the region of space we would be able to probe.

Pavlo. B.
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    You're asking about fundamental restrictions, but you're referring to a kind of model that already assumes a crude approximation (strictly non-relativistic single-particle quantum mechanics). It's like asking if there are any limitations to how well we can measure the exact radius of the earth, as if the earth were a perfect sphere. Our current understanding of matter is based on relativistic quantum field theory, which doesn't have strict particle-position-observables in the usual sense (see here). Is that what you're asking? – Chiral Anomaly Apr 30 '21 at 19:29
  • You have a great answer there, but I am not sure how to apply it here. I could adopt my question to the quantum field theory, and instead of the wave function look for the expectation value of the operator $ \hat{n}(r)$ in different regions of space (here $\hat{n}(r)$ is the particle number operator of the interacting field). Would I be restricted in the choice of the region? – Pavlo. B. Apr 30 '21 at 19:57
  • Relativistic QFT has observables that are strictly localized in arbitrarily small regions of space, and it has observables that count particles with perfect reliability (if the particle is perfectly stable), but it can't have observables that do both of those things. So no, you're not restricted in the choice of the region -- unless you want the detector to be perfectly reliable. (...) – Chiral Anomaly Apr 30 '21 at 20:05
  • (...) Relativistic QFT doesn't limit how well a given observable can be measured, but it does (unavoidably) limit the types of observables that exist at all. You can perfectly measure an observable that does not detect particles with perfect reliability, but then you can't interpret the result strictly as the position of a particle. That interpretation is only approximately valid. It's not a measurement-precision issue, it's a what-kinds-of-observables-exist issue. – Chiral Anomaly Apr 30 '21 at 20:05
  • To be honest, I am not sure I understand how one can actually measure QFT observables. Since all particles and all field excitations are actually "dressed", is it even possible to probe the original "observables" from which we start our QFT? Aren't the original observables inaccessible? Or we actually can probe them? – Pavlo. B. Apr 30 '21 at 20:32
  • Those questions lead into some interesting territory, and I don't think I can address them in a short comment. Would you be interested in continuing this in chat? (If so, then we can post a few dummy comments to make the system give us the option to move to chat.) – Chiral Anomaly Apr 30 '21 at 21:09
  • Yeah, that would be great. (1st dummy comment) – Pavlo. B. Apr 30 '21 at 21:18
  • Let us continue this discussion in chat. – Pavlo. B. Apr 30 '21 at 21:19

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