Consider two electron at $|x\rangle$ and $|y\rangle$ respectively. Is it possible to anti-symmetrize the total state of the the system just with the above information? If I make $$|\psi\rangle = [|x\rangle|y\rangle-|y\rangle|x\rangle]\frac{1}{\sqrt{2}}$$ is it not zero as tensor product vectors $|x\rangle|y\rangle $and $|y\rangle|x\rangle$ are the same? Can someone explain when one needs to do antisymmetrization for electron wave functions and with respect what?
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