Consider a two slit interference experiment. It is a well known fact that when a detector is placed in front of one of the slits that the pattern disappears. Quantum mechanically the wave function of the photon collapses into one of the slits (in which the detector is placed).
Now consider the following scenario: We consider a 3 slit interference experiment in which in front of one of the slits a detector is placed. How will it affect the pattern? How will it be different if the detector had not been placed?
Now let us generalize the results to a n slit interference experiment in which at most there are n-2 detectors placed randomly. How will the pattern change again?
A mathematical description would be most welcome.
They're called "path integrals" for a reason.
You don't need a mathematical description if you involve a detector at any slit in the "classic" set-up (more recently experimental designs have allowed us to realize Wheeler's delayed-choice thought experiment and we've had Schrödinger cats and kittens for years; in the former instance various designs allow us to e.g., switch a system continuously from displaying "particle-like" to "wave-like" dynamics or to determine whether the result of an observation/measurement of a quantum system will be wave-like or particle-like AFTER the observation/,measurement has already been made.
Classically, however, any detector will cause decoherence/projection/collapse. In fact, the reason we see interference patterns with two slits, three slits, etc., as we do is because the screen is already causing a kind of collapse. Hence path integrals. Theoretically, the photon or electron or whatever takes all possible paths if it isn't interfered with. Having any slits prevents this. Putting a detector at any slit ensures a point-like measurement and no interference patterns.
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