this post was submitted on 21 Dec 2024
103 points (99.0% liked)
xkcd
8968 readers
107 users here now
A community for a webcomic of romance, sarcasm, math, and language.
founded 2 years ago
MODERATORS
you are viewing a single comment's thread
view the rest of the comments
view the rest of the comments
It's a lot more complicated than that even.
Pauli Exclusion Principal is that two or more identical particles of half integer spin cannot occupy the same quantum state. So two electrons in an orbital must be made of a +1/2 and -1/2 spin. This is evidenced by observation, but the prediction was made long before that.
This is because the total wave function for fermions is antisymmetric (bosons, like the photon, are symmetric). It's sort of hard to describe how this works without paper and pen, but essentially there is different formula of solving a wave function. A symmetric wave function is a sum, and an antisymmetric wave function is a difference. The issue arises when you have two identical particles - symmetric functions can be any state as it results in a solution >0. If you have an assymetric function of two identical particles, the result is 0, which isn't a valid state.
The very uncomfortable part of physics is here: when we ask "why" the answer based on the math and the observation is quite literally "because that is the way math works." It's fundamental - just like x * 0 = 0.
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).
So the way I "understood" the spin-statistics theorem is that it's basically this:
A given particle with a given intrinsic spin has a direct relationship to a collection of the same particles as a consequence of quantum math. Yeah. Just "it's related."
Proving that math is really freaking difficult and you need to use relativistic quantum field theory. I think it was Richard Feynman who said "We apologize for the fact that we cannot give you an elementary explanation."
Actually when I graduated there was another professor (can't remember his name) who was discussing his frustration with how they still can't explain it without all of QFT steps.
Basically, this is where the shared attitude of "the more you know about quantum physics, the more confusing it becomes."