This concerns the famous two-slit experiment. Electrons or photons or your favorite particle, doesn't matter. As we all know, the attempt to detect which slit the quanta pass through leads to loss of the diffraction pattern.
The interesting part of the experiment is describing what happens when neither slit has a particle detector. It is commonly said that each particle goes through both slits at the same time. At the popular level this makes QM seem weird and mystical, and at the professional level we might cite the Copenhagen Interpretation which boils down to: Don't ask questions about things you can't observe, and so we don't speak of such things, just do the math.
How sound this reasoning is at a fundamental level? Can we really conclude each quantum goes through both slits yet as a whole entity? Can we define a "it went through both slits" observable? Is there a proper Hermitian operator with (I suppose) eigenvalues 0 and 1, that can distinguish a quantum going through both slits vs. only one slit but without saying which one?
Perhaps it would make more sense to think about an N-slit experiment, and ask about a Hermitian operator that can report n, the number of slits a quantum takes?
I suspect I'm not asking this question quite right, but have an intuition there's something yet to be dug up from this age old gedankenexperiment.
Answer
It's a very good question but the answer is No, there is nothing such as "it went through both slits" observable (i.e. no linear operator that would correspond to this Yes/No question). The reason is that such "information" cannot be observed, not even in principle and not even statistically.
Much more generally, there don't exist any observables that would say "the physical system did X or Y before it was observed". It's the very reason why the term "observable" was chosen because things that are, by definition, unobservable because they only occur in the middle of the experiment are simply not observables.
All these statements are very manifest e.g. in Feynman's path integral formulation of quantum mechanics. One always has to sum the probability amplitudes over all conceivable histories. In this sense, using the double-slit example, the particle always goes through both slits. For some particular places where the particle may be detected, the first or second slit may contribute zero to the probability density. But the whole probability distribution is always affected by both slits.
I wrote that this question is good because these issues are misunderstood by most laymen - and not just those who admit that they are laymen but also many authors of philosophical books about quantum mechanics etc. They usually want to simplify their life and imagine that the particle, after some moment, is guaranteed to "behave as a particle". When the experimenter does something, he may forget that the particle has quantum and wave-like properties, and he may imagine it has only gone through one of the slits.
But as the "delayed choice quantum eraser" experiments show, this reasoning is always invalid. The particle may always "recall" that it has wave-like properties, and by modifying the future portions of its journey, the other slit - and the interference between both slits - may always become important once again. There's no reason why Nature should have answers to questions that can't be measured - such as "what was happening before the measurement" - and indeed, whenever we are in the quantum regime and the classical approximation is inapplicable, Nature chooses not to have answers to any of these questions.
Observables are quantities that can be shown as values at actual measuring apparatuses and quantum mechanics can only predict the odds that the value of an observable will be one number or another. Everything else that some people may want to imagine as the "detailed history" that "preceded" the measurement is unphysical.
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