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Am I misunderstanding the significant of the double slit experiment?

I thought the takeaway wasn’t that the particle comes out both sides, the implication is that the behavior of a single particle is the same as the behavior of multiple particles - that is to say, it appears to be an interference pattern, even when there should be no other particles to interferes with the single one.

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No you're understanding correctly (I think), the behaviour of a single detected particle depends on all possible paths it could take to get to the detection.

This is fundamental to 100 years of quantum mechanics and underlies most of physics including all semiconductors, materials science, chemistry, lasers, etc. The double slit experiment is just a very good illustration of the principle boiled down to its essentials, which is why it's everywhere in pop-sci. It makes for more accessible story than describing how a hydrogen atom works.

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