Say I have 1 electron in some quantum state Defined by some wave function, and it's doing its thing fluctuating the probabilities of where it might be.

What if I put a measuring device in an area where there's consistently a 1% chance of the particle being and then quickly remove the measuring device 1 milisecond later vs what if I leave the measuring device there for 1 second

Each case wouldn't both have a 1% chance of finding the particle, right? I feel like the one left there for 1 second made 1thousand more measurements than the one there for 1 milisecond, so it should be 1thousand times more likely to be discovered by the 1 second detector? What are the details here?

Bonus part 2: If after our measurement we see absolutely nothing (my electron wasn't there). Does the fact we know it isn't there affect the wave function? (It can't still have a 1% chance of being there if it's not there, right?)

Actually I happen to know the answer to this is yes. Looking and seeing nothing does affect the particle's wave function, because in the double slit experiment if you measure slit A and find nothing, you'll find the particle behind slit B with its wave function collapsed as if you detected it at B. But, how exactly does this work mathematically? I've never heard anything about how the wave function gets modified by detecting nothing in some area. Only how it completely collapses when the particle is detected.

  • $\begingroup$ There is no correlation between the probability of finding a system in a certain state and the strength of the coupling of the measurement apparatus that makes that measurement. Both are independent properties. One is a property of the system, the other is a property of the measurement apparatus. What we know or do not know about a system doesn't affect the system. The act of measurement is not an act of knowledge. It's an act that introduces a physical coupling that otherwise wouldn't be there. What happens to the result of that coupling later is irrelevant. $\endgroup$ – CuriousOne Sep 11 '15 at 4:04
  • $\begingroup$ @CuriousOne That's a confusing comment. I'm asking if you leave the measuring device there 1 thousand times longer, how much does that increase the chance of the particle being detected during that time. You're saying that's somehow irrelevant? $\endgroup$ – Farzher Sep 11 '15 at 4:20
  • $\begingroup$ The measurement device either makes a detection or it doesn't. If it doesn't couple strongly enough to make the detection in the time that you leave it there, then you are simply getting a measurement error rather than the true value for the probability. Just because it's a quantum measurement doesn't mean that you can't make a huge measurement error. $\endgroup$ – CuriousOne Sep 11 '15 at 4:23
  • $\begingroup$ Ah, I see. I'm assuming the measurement is 100% accurate (or 99+% accurate if you'd like). Therefore if nothing is detected, it really wasn't there. $\endgroup$ – Farzher Sep 11 '15 at 4:28
  • $\begingroup$ Whether the measurement is accurate or not depends on the coupling of the measurement device. Some detectors couple only very weakly (e.g. a neutrino detector), so that most of the quanta are lost and only very few are detected. The detector efficiency has to be calibrated out in the data analysis. In your case one would have to measure in the volume element and calibrate the detector sensitivity by measuring in the entire volume (where the integral over the probability adds up to 100%). Even then one would have to have a sufficient number of detections in both to avoid statistical errors. $\endgroup$ – CuriousOne Sep 11 '15 at 4:36

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