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The overall wavefunction must be antisymmetric with respect to particle exchange and so for para-hydrogen (an antisymmetric spin state) the spatial part of the wavefunction must be symmetric with respect to particle exchange.

My notes say that because the parity of a state of angular momentum quantum number l is (-1)^l then for para-hydrogen l may only take even values. I'm struggling to see the relationship between parity and particle exchange in this case, can anyone help?

Thanks