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When talking about the double-slit experiment, most physics books consider the wall to be a infinitely high potential, so that the photon is either reflected or transmitted through one of the slits.

(Side-question: is it reasonably correct to imagine the wall as a mirror with two small slits? It either reflects or transmits the photon, with maybe some edge cases like partial reflection).

Now let's replace the mirror by a black body, so that the photon is either absorbed or transmitted through one of the slits (furthermore, let's assume that the screen behind the wall is very far away). Quantum mechanics predicts (I'm talking about Copenhagen interpretation only) that there is a certain probability $P_W$ that the photon is absorbed by the wall, and also a probability $P_T = 1 - P_W$ that it is transmitted.

If that is correct, then this seems to me like some kind of half-measurement. The wave function collapses only if the photon is absorbed, which means that the photon "has to decide" where it hits the wall. If the photon is transmitted, the wave function cannot completely collapse (if it did, there would not be any interference pattern behind the wall).

However, the wave function is not left unchanged too. Those eigenstates of the position operator that correspond to an absorption have been "filtered out", because after the wave has passed the wall, the photon can no longer be detected to be absorbed by the wall. Furthermore, while the photon passed the wall, the wave function has been changed in a way that (as far as I understand it) does not seem to satisfy the Schrödinger equation - it is a partial collapse, not an evolution over time.

If that is correct, what process describes such a partial collapse? In the mathematical formulation of quantum mechanics, there's no mention of such a partial measurement - it's just measurement or no measurement, and it says that a measurement let's the wave function collapse to an eigenstate of the measurement operator. Clearly, if the photon does not hit the wall, it does not collapse to an eigenstate of the position operator.

Related question

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  • $\begingroup$ There are more than one question here. That's always bad because it means that for someone to write an answer, that person has to be able to answer multiple questions. I recommend splitting this post into as many individual posts as there are question marks in the test you've written. $\endgroup$ – DanielSank Aug 10 '15 at 8:07
  • $\begingroup$ There are exactly three questions: 1) The title, which is just the title. You're right, maybe I should not have placed a question mark there. 2) The side-question about the wall as a mirror. I edited the question to explicitly mark it as such. 3) The main question, which I marked in bold to emphasize it. You removed this formatting in your edit out of some reason. Do you mind if I restore it, so that the main question sticks out, as I intented it at first? $\endgroup$ – Bass Aug 10 '15 at 9:39
  • $\begingroup$ I removed the bold because I was unnecessary. Think about it like this: if you're asking so many things that you think you need bold to help the reader focus on what's important, then really what you should do is focus the question instead. It's totally find to post multiple related questions, and in fact this can really help you get better answers. $\endgroup$ – DanielSank Aug 10 '15 at 16:20
  • $\begingroup$ I understand your objection, but if you read my question attentively, you can see that it focuses on one real question. I cannot formulate this question in just one sentence. It needs some introduction to make the context clear, and when the introduction is finished, the real question is marked in bold. There are many SE sites where this is common. It's one way to do a TL;DR. I'm sorry if I don't yet have the physics language to formulate it in less words. Never mind, I got the answer I was looking for. Thanks for your efforts. $\endgroup$ – Bass Aug 11 '15 at 18:31
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For a measurement event to occur, it's not necessary that the particle is detected.

This is outlined in detail in the Renninger negative-result experiment, which consists of a radioactive atom surrounded by two hemispherical particle detectors, which are assumed to be 100% efficient so they detect every particle coming out from the atom. Generally one of the two detectors has a larger radius than the other.

This experiment has a similar structure as the one you described.

When the atom decays by emitting an alpha ray, the wave function of the outgoing alpha ray is spherical at the beginning. When it reaches the first detector (say, the upper hemisphere), there's a 50% chance that it is detected (this corresponds to the wall in your setup). If not, then the wave function must have partially collapsed to the bottom hemisphere (like in the double-slit experiment after the wall).

At this point, we have the situation that the particle has not yet been detected. Nonetheless, a measurement has been made, because we know with 100% certainty that the alpha ray is going downwards. Its wavefunction is mostly spherical, but vanishes in the upper hemisphere, and shows diffraction patterns at the "equator".

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    $\begingroup$ This looks more like a comment. Answers should elaborate more and address the actual questions. You are probably familiar with Stack Exchange sites, so you should know this. $\endgroup$ – rmhleo Aug 10 '15 at 9:38
  • $\begingroup$ Yep, corrected that. $\endgroup$ – cheeesus Aug 10 '15 at 11:18

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