# Null measurement in the which-path experiments

• A

## Main Question or Discussion Point

Say we set up a which-path experiment in which there is a detector at only one of the two slits ("Slit A"). In the case of a null measurement, where an electron arrives at the ultimate screen without triggering the detector at Slit A, is there any physical interaction between that electron and the detector?

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Suppose there is a coupling constant of a set up $0\le g \le 1$. If g=1 then the electron interacts very strongly and no interference is visible. For g=0 there is no interaction. For values in between the contrast between the light and dark bands greys out as g decreases.

This is an analogy and I don't know of any experimental support.

So are you saying g=1 in the case where the electron arrives at the screen without triggering the detector (since an ensemble of similar cases will not produce an interference pattern)?

So are you saying g=1 in the case where the electron arrives at the screen without triggering the detector
That would be g=0 (no coupling)
(since an ensemble of similar cases will not produce an interference pattern)?
The maximum interaction, g=1 will wash out the interference pattern completely.

So I guess you are wondering something like 'If the electron passes trough slit B, how can the measurement possibly influence the electron if there is no direct interaction between the detector and the electron?'

I think the issue becomes clearer when being more precise with the semantics. I think the following way of thinking is the most appropriate:

A detector at slit A does not measure 'the position of a given electron'. It measures the occupation $N_A$ of electrons in mode A. This means that the number of electrons in A always collapses to an eigenstate of the particle number operator: $N_A=$ either 0 or 1. Because the total electron number $1=N=N_A+N_B$ is conserved this automatically causes $N_B$ to be 1 or 0 as well as a consequence.

I was more wondering about the argument that the electron's wave function is a physical thing and so there is a physical interaction in the case of null measurement between the detector at Slit A and the electron's wave function.

Strilanc
is there any physical interaction between that electron and the detector?
You need to define what you mean by "physical" and "physical interaction". And not in a loose way using english prose, but in the form of an equation.

Personally I would say that the answer to your question is interpretation dependent. In some interpretations,the wavefunction collapses into either "no detection, electron at left slit" or "yes detection, electron at righ slit" as soon as the electron's wavefunction reaches the detector. But you can also make interpretations where the detector marks the parts of the wavefunction passing through it, and it's only as the two cases recombine at the screen that the marked and unmarked parts get appropriately separated or interfered. There is quite a lot of flexibility in terms of where and when the "important stuff" happens.

By physical interaction I am thinking of an interaction that can be observed, for example through an observable amplification of the phenomenon. So if this wave function is a physical thing and it is interacting physically with the detector then there should be some way of detecting that phenomenon?

By physical interaction I am thinking of an interaction that can be observed, for example through an observable amplification of the phenomenon. So if this wave function is a physical thing and it is interacting physically with the detector then there should be some way of detecting that phenomenon?
There aren't any experiments that can tell if the WF is real. It is the electron that interacts with the detector.

I notice this thread is marked 'A' for advanced. So far its been Basic.

Strilanc
By physical interaction I am thinking of an interaction that can be observed [...] if this wave function is a physical thing and it is interacting physically with the detector then there should be some way of detecting that phenomenon?
Like, if it made an interference pattern go away? Seems pretty observable. But clearly that's not what you have in mind.

Boing3000
Gold Member
is there any physical interaction between that electron and the detector?
Obviously, because the interference disappear.

Like, if it made an interference pattern go away? Seems pretty observable. But clearly that's not what you have in mind.
The disappearance of the interference behind Slit B when a detector is introduced at Slit A struck me as a phenomenon that some seek to explain by positing an interaction between the electron's wave function and the detector. But that phenomenon of lost interference did not strike me as independent evidence of the posited interaction.

Is there any way to generate independent evidence of the posited interaction in this case?

PeterDonis
Mentor
2019 Award
The disappearance of the interference behind Slit B when a detector is introduced at Slit A
This is not a correct description. The correct description is "the disappearance of interference at the original detector when a detector is introduced at Slit A. The interference at the original detector is not attributable to either slit by itself.

that phenomenon of lost interference did not strike me as independent evidence of the posited interaction.
The independent evidence of interaction between the electron and the detector at Slit A is that the detector at Slit A detects an electron. More precisely, that the detector at Slit A is certain to detect an electron if an electron passes through Slit A; we establish this by testing the detector in a simpler environment where the trajectory of the electrons is controlled so that they all pass through the detector. That is what justifies the explanation of the lost interference as being due to the presence of the detector at Slit A, even if on some runs of the experiment the detector at Slit A does not register a detection.