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Two particles with spins S1 and S2 are combined to make a system of spin S, expressed by: Sˆ2 =  Sˆ 1 + Sˆ 2 2 =  Sˆ2 1 + Sˆ 1.Sˆ 2 + Sˆ 2.Sˆ 1 + Sˆ2 2  (1) where, Sˆ 1.Sˆ 2 = Sˆ 2.Sˆ 1 = Sˆ 1xSˆ 2x + Sˆ 1ySˆ 2y + Sˆ 1zSˆ 2z (2) a) Find the product of Sˆ 1.Sˆ 2 in terms of Sˆ 1±, Sˆ 2±, Sˆ 1z, Sˆ 2z. b) Assume that S1 = 1 and S2 = 2. What is the form of combined state |3, 3 > ? Calculate Sˆ z|3, 3 > and Sˆ2 |3, 3 >. Construct |3, 1 > .

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