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I was reading this web page about interchange symmetry for states with identical particles here:

http://quantummechanics.ucsd.edu/ph130a/130_notes/node317.html

The article states that the highest total angular momentum state will always be symmetric under interchange and the next highest state is antisymmetric.

So my questions is:

Why does the symmetry of the angular momentum wave function alternate? How to prove it?

Thank you in advance.

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    $\begingroup$ Relevant things to google or otherwise investigate: Permutation group, young tableaux, representation theory and young tableaux, young tableaux and angular momentum. You might find the following useful: hep.caltech.edu/~fcp/math/groupTheory/young.pdf $\endgroup$ – joshphysics Jun 10 '14 at 6:53
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Let there be two identical particles with angular momentum operators $J_1, J_2$ having the value $j$.
A state in the joined angular momentum basis will be $|J, m\rangle$
A state in the separated angular momentum basis will be $|m1, m2\rangle$

We shall observe the state with the highest azimuthal component $m = j + j = 2j$
$$ |J=2j,m=2j\rangle = |m_1=j,m_2=j\rangle $$

It is easy to see that the state is symmetrical under interchange. The $J_-$ operator is commuting with the permutation operator, hence every $|J=2j,m\rangle$ state is also symmetrical under interchange.

Next, we know that $$ |J=2j-1,m=2j-1\rangle = const * (|m_1=j,m_2=j-1\rangle - |m_1=j-1,m_2=j\rangle) $$ We can see that this next state with lower total angular is anti-symmetrical, and because of the previous statement every $|2j-1,m\rangle$ state is also symmetrical under interchange.

We can repeat this process down to the desired value of $J$

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