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I was reading this and it mentions in the 3-electron section, that for a spacial wave function to be symmetric under fermion swapping, it must be a function of even parity. Similarly for anti-symmetry under fermion swapping, it must be a function of odd parity.

It is not immediately obvious to me why parity symmetry $(-1)^l$ has anything to do with the bosonic or fermionic properties of the spacial wave function. So I suppose my questions are:

Is it true that spacial inversion is the same symmetry as swapping?

And if so, why?

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  • $\begingroup$ parity is what happens when +x is turned to -x, mirror. when you swap you are mirroring. $\endgroup$ – anna v Jun 21 '19 at 4:32
  • $\begingroup$ think of conservation of parity $\endgroup$ – anna v Jun 21 '19 at 4:53
  • $\begingroup$ I can see that being the case for say, 2 electrons in an $s$ shell. but if I am swapping 2 electrons, one in $s$ and one in $p$, I see no reason for swapping to be a negation of coordinates, just a trading of coordinates, which may or may not be negating. $\endgroup$ – Craig Jun 21 '19 at 14:34
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    $\begingroup$ well, I do not know whether swapping is allowed in this case, so cannot go further into discussion $\endgroup$ – anna v Jun 21 '19 at 14:50
  • $\begingroup$ @Craig I read the site you're quoting and couldn't find the statement for a spacial wave function to be symmetric under fermion swapping, it must be a function of even parity. Would you please tell us where exactly did you find it? $\endgroup$ – Elio Fabri Jun 24 '19 at 9:29

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