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Consider the following scenario:

You have a particle detector that clicks when a quantum particle is near its vicinity. This is a "position measurement." Therefore, under the Copenhagen interepretation, this "collapses" the wavefunction to a particular position. The new wavefunction is now some localized position of the particle. This wavefunction propagates over time, and you could mathematically calculate the probability density for where you are likely to measure it next.

If your detector was perfect, it would collapse the wavefunction to a Dirac Delta. But, your detector wasn't perfect, and so you don't know the actual location of the particle perfectly. There seems to be two choices here:

  1. Say that the collapsed wavefunction is actually something like a sharply peaked Gaussian (or a top-hat function, or anything sharply localized).
  2. Say that the collapses wavefunction collapses to a position ket (Dirac Delta), but you don't know which one, so you describe the system as a density matrix over the possible positions. This density matrix can have an identical probability mass as the wavefunction squared described in (1) above.

Now, these two approaches will lead to different predictions. So which is correct? In this idealized setup the "right" answer is probably (1), but the boundary between the two gets more difficult to precisely delineate in real world scenarios where the density matrix is actually used. For example, for a quantum many body system, where you describe it with some density matrix, will this density matrix approach give different predictions than if you had just assumed the system was in some pure state, and you put all of your "uncertainty" into the wavefunction itself?

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    $\begingroup$ Good question, +1. See the notion of "coarse grained measurements", "non-selective measurements" and the notion of "quantum instruments" in general. The overall long story short is that the post-measurement state depends on the measurement scheme, and there are generally infinitely many different measurement schemes associated to the same observable. $\endgroup$ Commented Dec 17, 2025 at 23:00
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    $\begingroup$ See also this excellent post. $\endgroup$ Commented Dec 17, 2025 at 23:43
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    $\begingroup$ @TobiasFünke Thank you for the references, and that is indeed a good post. $\endgroup$ Commented Dec 18, 2025 at 1:48

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A single slit is an example of such a detector for photons.

A photon is prepared in a state so that it has a uniform probability of hitting anywhere on the screen or slit. If it hits the screen, it is absorbed by some atom in the screen. The wave function collapses and vanishes.

If it doesn't hit the screen, it passes through the slit. It now is in a state that uniformly fills the slit. The wave function has collapsed without interacting with any atom.

You can calculate the wave function as it moves forward. If there is a second screen, you find that it gives a single slit diffraction pattern. This matches the calculation.

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A delta function of position is a coherent superposition of plane waves that are in phase at one time at this position. Clearly the position will spread out with high velocities and in an instant the particle position will be unknown.

Your second option appears to be a more moderate superposition of momentum states. The same argument holds. After a characteristic time the wave function will have spread over a large volume.

The way to know where a particle is to trap it in a potential well, such as a Penning trap.

Btw a density matrix describes incoherent superposition of wave functions. I don’t think the concept is needed here.

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In classical physics, the evolution of a measurable quantity such as the $x$ position of a particle is described by a function $x(t)$ such that if you measure $x$ at time $t$ you get the result $x(t)$.

In quantum physics the evolution of a measurable quantity is described by an observable whose value at any given time is a Hermitian operator. The possible results of measuring that quantity are the eigenvalues of that operator and quantum physics predicts the probability of each of the possible values.

In general the probabilities depend on what happens to all of the possible values of the observable: quantum interference. For an example see Section 2 of

https://arxiv.org/abs/math/9911150

But when information is copied out of a quantum system interference is suppressed: this is called decoherence

https://arxiv.org/abs/1911.06282

The objects you see in everyday life have information copied out of them on scales of space and time much smaller than those over which they change significantly so on those scales interference is very effectively suppressed.

Decoherence selects a set of pointer states that don't interfere much with one another but doesn't eliminate any of the pointer states. So reality as described by quantum equations of motion involves multiple versions of all of the systems you see around you in everyday life this is often called the many worlds interpretation

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

Some people don't like this and say that all but one of the possible states disappears. When they are vague about how and why this happens or indeed whether quantum theory describes reality at all this tends to be called the Copenhagen interpretation or statistical interpretation and the answer to your question is that you're not supposed to ask that question. Some physicists have proposed theories of how collapse happens, such as saying that states get squashed with Gaussians with some probability

https://arxiv.org/abs/2310.14969

Decoherence is fine with imperfect measurements, repeated measurements, unsharp measurements and so on, but such measurements don't really fit into collapse theories

https://arxiv.org/abs/1604.05973

So the answer to your question is that quantum theory can cope with any kind of measurement that can be done in reality, but if you want collapse you're going to have a much harder time.

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    $\begingroup$ I did not downvote, but I'm not really sure if this answers the question; it just reads like polemics. The question should be answerable within the Copenhagen interpretation; if not, this would imply that the Copenhagen interpretation makes some testable prediction that differs from the Many Worlds interpretation, which would be quite a discovery. Perhaps the links you provided do answer the question though. $\endgroup$ Commented Dec 18, 2025 at 7:01
  • $\begingroup$ @SorenJ You're not sure whether an answer with many references explaining how the spread of information is described in quantum theory and how it selects sets of states that can be regarded as measurement results answers a question about what kind of measurement results occur in quantum theory? $\endgroup$ Commented Dec 18, 2025 at 16:37
  • $\begingroup$ I typically use SE when I am hoping to get an answer to a question in a relatively short period of time. I am aware that my initial question probably has answers somewhere in the literature, but I have not taken the time to go through the ~180 pages you provided. I don’t come to SE hoping to, for example, understand 2D quantum materials from just one question, I would just start with textbooks instead. It would always be fair, though, to tell me something like, “this is a complicated question that can’t be answered simply. Here are the relevant materials.” $\endgroup$ Commented Dec 18, 2025 at 17:43

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