(Solution) - Consider three weakly interacting identical spin 1 particles a -(2025 Original AI-Free Solution)

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Consider three weakly interacting, identical spin 1 particles.
a. Suppose the space part of the state vector is known to be symmetric under interchange of any pair. Using notation | + > | 0 > | + > for particle 1 in ms = + 1, particle 2 in ms = 0, particle 3 in ms = + 1, and so on, construct the normalized spin states in the following three cases:
(i) All three of them in | + >.
(ii) Two of them in | + > one in | 0 >.
(iii) All three in different spin states.
What is the total spin in each case?
b. Attempt to do the same problem when the space part is anti symmetric under interchange of any pair.