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>What happens if Hitachi's experimental setup had only one path for the electron to follow rather than two? You're claiming that the frogs are a good model for Hitachi's setup, so it should be the same result that you claim for the frog model with one door. You're claiming that the frog model with one door produces a pattern of alternating wet and dry. What happens in the lab when you try it with electrons? Electrons following a single path don't produce an alternating pattern; they produce a pattern very close to a Gaussian.
The electrons, photons, and frogs (jumping over the wall when probability below some threshold) would produce an alternating pattern like this one:
http://www.a-levelphysicstutor.com/images/waves/ys-1s-2s-gra...
Because it is result of the same geometry as on the image that i posted before : where 2 greens touch the wall simultaneously - deep trough, zero intensity, and where 2 reds - peak. The meaning of the green/red isn't important - be it probability of a frog's legs touching ground or the probability density of position of electron - the resulting pattern is the same. The image can be interpreted as either while it is just an image of concentric circles originating from 2 points. Gaussian outline comes to play only because it describes the normal deviation of a frog/electron from the preferred direction they jump/fired along.
> If you let electrons go through either just one path or just the other path, you'll get nice smooth Gaussians from both (translated a bit from each other).
this is true only in classical mechanics where position of electron or a bullet isn't quantized - you'll get a smooth Gaussian. In QM, i.e. real electrons, photons (or jumping frogs having the property of going through/over the wall where ground touch probability is close to 0) single slit produces dim/light pattern like on the image above. Like the double-slit pattern, this single-slit pattern is also result of position quantization.
> But open both paths up, you'll see the distribution dim at places. That's very unexpected! Somehow letting more electrons through has decreased the electrons hitting certain parts of the screen.
This - double slit is more frequent than single-slit or dimming at some places - happens only for stream of photons as photons interfere with one another and the photon's wavelength is the same as its position quantization period (note - for electron DeBroglie and positional quantization are different wavelengths), so the photon_A-photon_B interference is visible on the scale of the pattern produced as a result of positional quantization. In case of Hitachi, we have single electrons, so no electron_A-to-electron_B interference, so the double-slit pattern is less frequent than single-slit, ie. where was light on single slit - there will continue to be the same or brighter light on double-slit, it is just some (not all) troughs/dims will disappear.
> You "see" the effect in the probability distribution for where that electron appears
The quantized position probability distribution already explains the pattern. I don't see how adding the interference or superposition changes it.
>If you still think there's some effect being transmitted between different electrons
no, the many particles, i.e. many samples is just results in better visualization of underlying probability distribution. I think here we agree.
> You would get exactly the same effects by setting up a statistically significant number of screens all spatially separated from each other and running the experiments without signals being able to be communicated between screens and then overlaying the resultant observed positions.
the same here. We agree about independent events and their cumulative statistics. Let frogs jump in separate setups, and mark their results on a separate screen - the pattern will emerge as if they all jumped in the same yard.
>However, the Mach–Zehnder Interferometer is, I believe, an better physical model to witness a single photon interfering with itself.
Will definitely spend a time on it.
> Young's double-slit experiment was invented first, so for really that reason alone, it's taught first, but because of its continuous nature, it's easier to get bogged down in certain irrelevancies.
it only looks somewhat "continuous" with light. That is the beauty of the Hitachi experiment as it took away a lot of "continuity" effect as well as "photon A interfering with photon B" effect.