20. If j 1 ≥ j 2 show that j 1 + j 2 X j = j 1 − j 2 (2 j +1)=(2 j 1 +1)(2 j 2 +1) . 21. Consider three particles of spin 1 2 , and let ~ S 1 , ~ S 2 , and ~ S 3 denote the corresponding spin operators. In the combined spin space associatedw iththetota lsp inoperator ~ S = ~ S 1 + ~ S 2 + ~ S 3 , what irreducible invariant subspaces S ( τ ,s ) occur, and how many spaces occur for each value of s .[Hint: f rst determine the invariant subspaces obtained by combining any two of the spins together, and then combine the third spin with each of the subspaces of the two previously coupled spins. Make sure the total number of f nal basis states | τ ,s,m i is the same as the total number of initial basis states | m 1 ,m 2 ,m 3 i . Note: you do not actually have to produce any of the vectors | τ ,s,m i to solve this problem.] 22. Consider three particles of spin 1 2
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