What Are the Permissible (L,S) Pairs for J=1 in a System of Two Spin-1 Bosons?

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Homework Statement



We have a system of 2 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?

Homework Equations


Bose-Einstein Stats?

The Attempt at a Solution


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?
Also, due to Bose-Einstein stats, the overall wavefunction should be symmetric under the interchange of the 2 bosons. However, I don't know how even and odd wavefunctions for orbital angular momentum behave. Anyone please?
 
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No worries. Problem resolved. Thanks for reading anyways.
 
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