In case it's relevant, the context of my question is finding the allowed states of an atom. For example, given a nitrogen atom with (1s)(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}(2s)^{2}2p^{3}, how do we find the possible states in terms of total orbital angular momentum L, total spin S, and total angular momentum J = L + S. It seems that the solutions to problems like this rely on determining whether given pairs of (L,S) lead to completely antisymmetric states or not. Only the antisymmetric states are allowed because of the symmetrization postulate.

Now, my question: say we add together angular momenta ## J = J_1 + \cdots + J_n ## and get a resulting basis ## \left| J, M \right\rangle ##. Now, is it true that the exchange symmetry of ## \left| J, M \right\rangle ## dependsonlyon J? For example, if I know that one state with J = 4 is completely antisymmetric, can I be sure thatallstates with J = 4 are completely antisymmetric with respect to interchange? (I don't know if it makes a difference, but if it does, I'm only really interested in cases where ## j_1, \ldots, j_n ## are either all integers or all half-integers.)

It seems that this is true for the case of adding two angular momenta (though I don't really know how to prove it). The states with ## J = J_{max} ## will all be symmetric, the states with ## J = J_{max} - 1 ## will be antisymmetric, etc. The key point though is that states with the same ## J ## will have the same exchange symmetry, even if ## M ## is different. Is there a good justification for this, and does it hold for addition of three or more angular momenta?

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# Exchange symmetry and addition of angular momentum

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