Spin Combination of Two Particles

Question:
Two particles with spins S1 and S2 are combined to make a system of spin S, expressed by:

where,

a) Find the product of Ŝ12 in terms of Ŝ, Ŝ, Ŝ1z, Ŝ2z.

b) Assume that S1 = 1 and S2 = 2. What is the form of combined state |3, 3 > in terms of eignstates of Ŝ12 and Ŝ22 (or Ŝ1z and Ŝ2z )?

Calculate Ŝz |3, 3 > and Ŝ2|3, 3 >.
Construct |3, 1 > .

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