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Could somebody please explain what is going on here?

We have a system of two indistinguishable spin-1 bosons. We shall adopt the center of mass frame.

Let $S$ = total spin
$L$ = relative orbital angular momentum
$J$ = $L+S$ = total angular momentum

Prove that $J = 2m$ where $m$ is an integer. If given that $J=1$, what are the permissible $(L,S)$ pairs?

I am lost with this. I have managed to show that for the states $S=0, S=2$ interchanging particles $1,2$ is symmetric, whereas it is antisymmetric for $S=1$. However, I don't know how to use this here. What is the significance of the CoM frame choice?

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You're probably halfway there. How do even or odd $L$ orbital wavefunctions behave under particle exchange? How should the total wavefunction behave? – Emilio Pisanty Nov 18 '12 at 16:24
@EmilioPisanty, thanks for commenting! :) I can answer the second question: since they are bosons the total wavefunction should be symmetric under the exchange. However, I have no idea how the $L$ orbital wavefunctions work... – Glen Nov 18 '12 at 16:30
Well, the exchange symmetry will be exactly equivalent to parity, right? All you need to see then is how the orbital wavefunctions behave under that, which should be easy. – Emilio Pisanty Nov 18 '12 at 19:54
@EmilioPisanty: Thanks! – Glen Nov 19 '12 at 8:01
Glad to help. Consider writing it up as an answer below. – Emilio Pisanty Nov 19 '12 at 10:44

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