And I'm assuming if you instead chose to make ∣𝜓⟩ a superposition of 5 basis states. And you looked at a 5 electron system. When you do the slater determinant math you would similarly have only one possible solution wavefunction. Which involves each electron in a different basis state?

Also, it looks like you are expressing the system wavefunction as a linear combination of a product of the individual electron wavefunctions. From what I've read I think this is only allowed if there is negligible electron-electron interaction. If you don't do that, will you still get the Pauli Exclusion Principle result to arise from the anti-symmetry conditions?

@roobee What do you mean by 5 basis states? The spin of a single electron can only be spin up or spin down

for example, if ∣𝜓⟩=a ∣1⟩+b ∣2⟩+c ∣3⟩+d ∣4⟩+e ∣5⟩ where ∣1⟩ is the spin up in ground state, ∣2⟩ is the spin down in ground state, ∣3⟩ is spin up in state above ground state, ∣4⟩ is spin down in state above ground state, etc.

@roobee Ah ok I guess I got mixed up with your linked question. Yeah if the electrons only differed by those quantum numbers you would get an appropriate state that was anti-symmetric under particle exchange. If you want to add more interactions and quantum numbers you can, but I am just considering when the electrons just differ in one observable, i.e. spin.

Ok. But even when looking at only one observable, spin, if there is significant electron electron interaction you can't express the system wavefunction as a linear combination of the products of the individual wavefunctions right? Oh, but in that case than you wouldn't even have individual wavefunctions to examine. All the electrons would be in the same hybrid orbital. So the Pauli Exclusion Principle wouldn't even be used right?

@roobee Right, then you will just have more quantum numbers associated with the system, right?

well, quantum numbers wouldn't exist anymore because there would only be one system wavefunction. and there would be no wavefunction associated with individual electrons anymore. I think?

@roobee Well there is always one system wavefunction either way. I guess I am not fully understanding what you are getting at

hi i'm moving this to chat since the website suggested it

Good idea :)

So when we express a system wavefunction as a product of individual electron wavefunctions. Ie: ∣↑↓⟩, which means electron 1 looks like the single electron wavefunction ∣↑⟩. where ∣↑⟩ means spin up in the ground state. similarly for electron 2. then the pauli exclusion principle makes sense.

But if you can't express the system wavefunction as a product of individual wavefunctions. than there isn't even a wavefunction associated with each electron, only with the system. so you can't invoke Pauli's Exclusion principle to say that the 2 electrons can't be in the same state, because the electrons do not occupy a state individually. they only occupy a state as a system.

is that understanding right? adding this just in case you don't get automatically pinged, @AaronStevens

Yeah sorry I am doing some online tutoring as well right now, so my replies will be sparse at the moment