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In the double-slit experiment why does the light hitting the back screen not result in a measurement? How far away from the slit can the measurement take place and still cause a wave collapse and is there a consequence to moving the measurement further away from the slits)? What qualifies as a measurement?

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In the double-slit experiment why does the light hitting the back screen not result in a measurement?

Light hitting the back screen does result in a measurement, which is why the photon only makes a single localized spot on the screen. The interference pattern becomes evident only after multiple photons have hit the screen, so we can see the distribution of spots.

...is there a consequence to moving the measurement further away from the slits...?

If the back screen is moved closer and closer to the slits, the distribution of spots accumulated at that new screen-position will look less like an interference pattern and more like an image of the two slits, depending on how close the screen is to the slits.

What qualifies as a measurement?

This is a very broad but important question. In a nutshell: any time the quantity-of-interest (the location of a photon, in this case) influences its surroundings in a prolific and practically irreversible way, as long as the effect is sensitive to the value of that quantity, then that quantity has effectively been measured.

(This principle doesn't solve the infamous "measurement problem", because it doesn't tell us how to calculate the distribution of outcomes. We still need Born's rule for that. But for all practical purposes, this principle does tell us when we can safely apply Born's rule.)

In the case of the double-slit experiment, the effect on the interference pattern depends on how far downstream we measure the (transverse) location of the photon, because the farther downstream the measurement is, the less sensitive it is to which slit the photon may have gone through.

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    $\begingroup$ Re, "a measurement that detects the passage of a photon through an individual slit (preferably without destroying the photon," Really? How would it be possible for an instrument to register the passage of a photon, but not change the photon in any way? How "close" would the photon have to pass (assuming that "close" even has any meaning in this context)? $\endgroup$ – Solomon Slow Dec 15 '18 at 16:20
  • $\begingroup$ @SolomonSlow Yes, you're right. That was careless of me. A better way to word the answer would be to consider what happens to the interference pattern when the back screen is moved closer to the slits. I'll edit the answer. $\endgroup$ – Chiral Anomaly Dec 15 '18 at 16:38
  • $\begingroup$ I don't agree that the screen is the observation point, the wave function continues until absorbed by an eye or detector or something. But maybe this is just semantics. $\endgroup$ – PhysicsDave Dec 16 '18 at 2:51
  • $\begingroup$ Thank you very much your answer is very helpful! Just to make completely sure I’m following what you are saying. If some measurement is performed right by the slits then we get no interference pattern. As we move the measurement further away from the slits then an interference pattern begins to emerge? So its not either or, it’s gradual. Also would collisions (and thus measurements) not occur when hitting air molecules or is air not dense enough? $\endgroup$ – Sparky Dec 16 '18 at 3:25
  • $\begingroup$ @Sparky Right -- it's gradual, not either or, just like in classical optics, except that the pattern accumulates one photon at a time. However, no matter where we put the screen, the occurrence of measurement at that location is a relatively sudden event. The physical consequences of the photon being absorbed propagate quickly throughout the surroundings, making the event practically irreversible. Collisions with air molecules don't necessarily have this effect if air is transparent, because then the photon propagates through the air without significantly changing the behavior of the air. $\endgroup$ – Chiral Anomaly Dec 16 '18 at 15:04
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The measurement is occurring in your eye and that is where collapse happens. You continue to see the screen because the light source is continuous. So if a 1000 photons go thru the slit maybe you observe 100 and your buddy sees 100 and the blackboard sees 500 and 300 slip out the window into outer space ( they don't collapse ! ).

Observation is a measurement.

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    $\begingroup$ observation is interaction - the screen, the material of the slit borders, the air molecules between the slits and the screen, all are capable of “observing” photons. Replace the eyeballs in your answer with rocks and you get the same thing. $\endgroup$ – JPattarini Dec 15 '18 at 15:45
  • $\begingroup$ A photon interaction like diffraction or refraction is still not an observation until we detect it with our eyes or cameras, etc. Furthermore photons are reflected and scattered off a screen and the wave function has not yet collapsed. $\endgroup$ – PhysicsDave Dec 16 '18 at 2:47

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