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In a beam splitter experiment with a single photon, if I place a detector on one path and after some time it never clicks, why does the wavefunction still collapse to the other path even though I haven’t interacted with the photon? And how much time it takes for the wavefunction to collapse to the other path?

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    $\begingroup$ See so-called negative result measurements and interaction free measurements. $\endgroup$ Commented Sep 8 at 12:43
  • $\begingroup$ No this is not a negative measurement and interaction free measurement in my view. In all those cases, at the end of paths there is a detector and these detectors exhaust all possibilities. The interaction free effect occurs in an intermediate part of the paths. We can attest it only when one of the final mutually exclusive possibilities took place, and here an explicit interaction happens. This also fixes when the collapse occurs. In the raised issue instead one paths has no final detector. We cannot say that the particle stays on the other path. $\endgroup$ Commented Sep 9 at 7:25
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    $\begingroup$ @ValterMoretti even after enough time ? Also what would be that enough time? $\endgroup$ Commented Sep 9 at 7:28
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    $\begingroup$ What time? There is no time observable in QM. You should need a precise choice of some “arrival time observable”, but also in that case, to measure it you need an explicit interaction with a detector. $\endgroup$ Commented Sep 9 at 7:30
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    $\begingroup$ The subject has a link with : en.wikipedia.org/wiki/Wheeler%27s_delayed-choice_experiment $\endgroup$ Commented Sep 9 at 9:51

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In quantum mechanics, non-direct interaction with the system can itself be interpreted as measurement.

This is known as interaction-free measurement. Here, the absence of a detection on one path provides information. i.e., the photon must have taken the other path, and it is this information that causes wavefunction collapse.

According to the Copenhagen interpretation, the wavefunction evolves unitarily in a superposition until a measurement occurs. When the systems state becomes definite (eigenstate), we say measurement has occurred. So, a "measurement" includes any interaction or "interaction like" inference, such as the observation of no clicks on one path.

The absence of physical detection updates information about the state and therefore collapses the wave function. i.e., the photon is now known to take the other path.

See the Renninger negative-result experiment$^1$ and counterfactual quantum communication that has been demonstrated$^2$.

And how much time it takes for the wavefunction to collapse to the other path?

In most standard interpretations of quantum mechanics, the wavefunction collapse is considered instantaneous (though this notion comes with important subtleties).

$^1$ From source

In quantum mechanics, the Renninger negative-result experiment is a thought experiment that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The statement is that a particle need not be detected in order for a quantum measurement to occur, and that the lack of a particle detection can also constitute a measurement. The thought experiment was first posed in 1953 by Mauritius Renninger. The non-detection of a particle in one arm of an interferometer implies that the particle must be in the other arm. It can be understood to be a refinement of the paradox presented in the Mott problem.

$^2$ The first experimental verification of counterfactual quantum communication was completed in 2012 by Liu, Ju, Liang, Tang, Peng, Pan, et. al. They implemented a (counterfactual) quantum cryptography protocol, achieving high visibility (over $98\%$) in their interferometers

Even more recently (this month of this year, 09/2025), researchers presented an on-chip quantum photonic implementation of counterfactual communication using the quantum Zeno effect (with probabilities of about $74\%$ for transmitting a $0$-bit and $85\%$ for a $1$‑bit, while detecting only $\approx 0.17\%$ of photon leakage).

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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 operator called an observable. The eigenvalues of the observable are the possible results of measuring the relevant quantity and quantum theory predicts the probability of each of the possible outcomes.

In general the probabilities of the outcomes depend on what happens to all of the possible values. This is called quantum interference. For an example see Section 2 of

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

When you walk through a doorway you don't have to consider all of the possible ways you might walk through. When you do a measurement you don't experience seeing more than one result. This appears inconsistent with describing systems in terms of observables.

Collapse was suggested as a solution to this problem: it is an alleged process that eliminates all but one of the possible outcomes of a measurement. Many accounts in textbooks state that collapse happens but provide no explanation of how it happens. Also collapse is inconsistent with the equations of motion of quantum theory so this lack of an explicit theory created problems. Some physicists have modified the equations of motion of quantum theory to include collapse, see

https://arxiv.org/abs/2310.14969

I don't know what such theories imply about exactly how to describe the measurement you described but you could try working it out.

Other physicists actually worked out what quantum theory implies about what happens after a measurement. A measurement is an interaction that produces a record of some property of a system. When information is copied out of a quantum system interference is suppressed: this is called decoherence

https://arxiv.org/abs/1911.06282

For systems you see in everyday life information is copied out of them on scales of space and time much smaller than those over which they change significantly. As a result they are decohered very effectively and don't undergo interference. An electron in an atom doesn't have the same issue and so interference is important for electrons. There have been experiments that increase and decrease such interactions to test what happens and those experiments have matched the predictions of decoherence theory.

Decoherence doesn't eliminate the other possible outcomes of a measurement, it just prevents interference. As a result on the scales of space and time in everyday life it predicts that reality looks a bit like a collection of parallel classical universes

https://arxiv.org/abs/1111.2189

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

This is often called the many worlds interpretation but it is just the result of applying quantum theory as one would apply any other theory.

In the MWI when you do a measurement of whether a photon is present or absent at a location, there are two versions of the measurement result one for presence and the other for absence. The results of measurements in different branches of the interferometer only match up when they are compared.

There are many proposals for measurements where you can detect some property of a state without interacting with it: interaction free measurements, see

https://arxiv.org/abs/quant-ph/0103081

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We know in advance in this type of experiment that the photon has a corpuscle aspect, why speak of the collapse of its wave?

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  • $\begingroup$ Do we know its path or trajectory especially after the beam splitter? $\endgroup$ Commented Sep 9 at 13:18
  • $\begingroup$ even if we don't know its trajectory, we know in advance that it is a particle $\endgroup$ Commented Sep 9 at 13:23
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    $\begingroup$ There is no particles or waves there is wave particle duality where we describe the particles with wavefunctions $\endgroup$ Commented Sep 9 at 13:29
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    $\begingroup$ The particle is what we detect while the wavefunction tells us how it behaves before we detect it $\endgroup$ Commented Sep 9 at 13:37
  • $\begingroup$ So the 2022 Nobel Prize winner Alain Aspect says anything on the board: ""first experiment particle like behavior: goes either to one side or the other , not both" (see video) $\endgroup$ Commented Sep 9 at 13:41

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