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xkcd #3027: Exclusion Principle
(xkcd.com)
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this post was submitted on 21 Dec 2024
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But it's an outlier right? I guess that's why it's joked as an additional force in the comic, because even by "maths just describes reality" standards, it's weird. Like, it feels like a particle "knows" if the state is already occupied, and that's why it can't occupy it. But that implies some communication - the conveying of some force - else how does it "know"?
It’s not really communication. They ‘know’ because they become part of the same wave function. The wave function of the system is
|psi1 psi2> +- |psi2 psi1>
Note that if the +- is a plus, then exchanging psi1 and psi2 yields the exact same equation. If it’s a minus, you get a negative sign out front. Electron systems have a negative sign because of the spin statistics theorem (I don’t understand that part, so you can look it up if you want—it involves field theory iirc) Now, if electrons are exactly the same (indistinguishable), then exchanging them will yield the exact same wave function, leading to
|psi1 psi2> - |psi2 psi1> = |psi2 psi1> - |psi1 psi2>
The only solution here is |psi1 psi2> - |psi2 psi1> = 0
But recall that |psi1 psi2> - |psi2 psi1> describes the system as a whole. So this system is prohibited by quantum mechanics, and there’s no way for two electrons to have indistinguishable states (be in the same place at the same time).