More on QM Probability vs Frequency

2020 February 28
by Daniel Lakeland

So in my previous post, I mention the possibility to use Bohmian deterministic mechanics to discover the probability that a photon goes through a given slit if you observe it flash at a certain spot on your detector. In the setup there’s a shutter that opens and closes the second slit randomly according to radioactive decay or the like. Let’s see how we can use this information.

The frequency distribution for photons with both slits open is an “interference pattern” which has an oscillatory nature. For example from the Wikipedia article on the double slit experiment:

Single vs Double Slits
A single slit has no fast-oscillating pattern but still develops a diffraction-spreading pattern. The double slit pattern shows the fast oscillations with regions of “dropout”, dark regions even in the center of the pattern.

So, suppose we observe a photon in the general brightest central region. Suppose that it flashes within one of those “dark bands” that the double-slit pattern shows. Obviously psi_double^2 is very small in this region whereas psi_single^2 is large. Therefore the posterior probability that the particle went through the first slit because the second slit is closed… is very high. On the other hand, if we see the flash in one of the regions that is bright in both the diffraction and the interference pattern, then we have a harder time knowing whether the second slit was open or closed, though if the brightnesses are slightly different, then we infer that one vs the other was more probable.

What about the situation where we know the second slit is open, and we see a flash at a particular spot. Consider the 2 slit picture from above. If the flash comes from say far to the right where the diffraction pattern is quite dark, but the interference pattern has more light… Then when we run the Bohmian mechanics we will probably find that the photon came from one or the other slit with higher probability. Not having done the calculations I just don’t know, but let’s suppose for example it has a 80% chance of coming through the second slit. What does this matter? In particular, the pattern is the way it is because the apparatus is the way it is… in other words the second slit is open, there is nothing interfering with the passage of particles through that slit, there is no special magnetic fields or electron clouds or glass pieces or anything in the way, and so the physical scenario is such that the wave function does a particular thing, and voila… If we did something to perturb the apparatus, a different wave function would get set up, and a different path would be taken by particles initially coming from the same place as the original particle, and so it wouldn’t hit in the same place and might go through a different slit. What physical consequence can our knowledge that the particle had a high probability that it came through a given slit have? It’s a fact about the past, so we can’t act on it to change the past. It might matter if it would help us decide whether the particle might have interacted with some apparatus along the way, but if there were an apparatus along the way, the wave function would have been different and we’d have a different probability that the particle went along the path.

It seems to me this is one of the essential features of the problem of “whether a particle went through one or the other slit or both”. People whose interpretation of QM is that the particle doesn’t exist until it hits our detector are interpreting “there is no physically observable consequence of inferring the path that a particle took in the past” as evidence that “the path that the particle took in the past doesn’t exist”. This is rather odd. The fact is, by coupling our knowledge that the path might more likely have been X to knowledge of what things might have affected that path (such as the shutter) we can potentially infer that stuff we don’t know was more likely to be one way or another… For example, perhaps we can infer that there was unlikely to have been radioactive decay in the shutter mechanism.

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