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Suppose we have two free electrons described by two exactly the same space wavefunctions (which are the space part of the combined wavefunction, the other part being the spin part).

Now two wavefunctions can never be in the same position state. How does this space part of the wavefunction contributes to Pauli's exclusion principle? Do the space wavefunctions only have to overlap (in which case they are not in the same state) for the spins to be in different (opposite) states?

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The real condition is that the overall wave function must be antisymmetric under exchange of any two electrons. One implication of this is that two electrons cannot be in exactly the same state (spatial and spinor), since the antisymmetric requirement would forbid it.

Spatial states can and do overlap without issue even when the spinor states are the same, as long as the states are orthogonal. For instance, all of the hydrogen orbitals overlap with all the other hydrogen orbitals to some degree.

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  • $\begingroup$ But when will the spatial overlap be such that the spins must be opposite? $\endgroup$ – descheleschilder Dec 8 '17 at 9:43

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