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For a system of two coupled spin 21 particles, one can write the composite state as s1s2m1m2 where s1=s2=21 and m1=21,m2=21. The following relations are true: S1^2s1s2m1m2=s1(s1+1)2s1s2m1m2S2^2s1s2m1m2=s2(s2+1)2s1s2m1m2S1z^s1s2m1m2=m1s1s2m1m2S2z^s1s2m1m2=m2s1s2m1m2 The spins can be arranged in 4 possible ways (in the notation of s1s2m1m2 ) as 21212121, 21212121,21212121,21212121 In order to measure the total spin of the system one can define the operator S^=S^1+S^2 and the coupled basis {s,s1,s2,m}.S^ is the total spin of the system and an angular momentum. Therefore, s must be a positive integer or half-integer, and m varies by oneunit jumps between s and +s. Hence, the problem is to find what values s and m can have, and to express the basis vectors {s,s1,s2,m} in terms of those of the known uncoupled basis. As a starting point you should consider which combination of m1 and m2 gives the maximum value for m. In this case the wavefunction of s=1,s1=21,s2=21,m=1 can only be the state where both spins have m1=m2=21. So, s=1,s1=21,s2=21,m=1=m1=21,m2=21. You now need to find an expression for the other states s=1,s1=21,s2=21,m=0, s=1,s1=21,s2=21,m=1 and s=0,s1=21,s2=21,m=0. Hint: Think about how you can use the ladder operator S^=S1^+S2^ on the states s,s1,s2,m to come up with expression for the other wavefunctions Compare your results with the Clebsch-Gordan coefficients shown in Table 4.8 of Griffiths. Do your results match with those from the table? Table 4.8: Clebsch-Gordan coefficients. (A square root sign is understood for every entry; the minus sign, if present, goes outside the radical.)

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