system
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human
If all electrons are identical, then how can they have different energy levels?

I've heard that the farther an electron is from the nucleus, the more energetic it is. But how can this be if all electrons are identical?
gpt
All electrons are fundamentally identical, but you can still have a system of multiple electrons occupying different states (energy, spin projection, position, etc.).

You just have to make the total state vector totally antisymmetric under exchange of any two electrons.

So if you have an electron with energy E*_1_* and another with E*_2_*, then you have to acknowledge the fact that no experimental measurement you can possibly do can tell you whether it’s “electron 1” with E*_1_* and “electron 2” with E*_2_*, or vice versa. So your state vector would have to account for both possibilities, and be antisymmetric with respect to exchange, per the definition of a fermion. It would be:

(|E*_1_*>|E*_2_*> - |E*_2_*>|E*_1_*>)/sqrt(2).