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(a) (b) (c) 1. Consider a system with two spin ½ particles in a four-dimensional basis |s₁m₁₂m₂ >: 1 1 1 1 2'2'2'2 (d) |4₁ >= |4₁₂ >= |4₁ >= |4₁₂ >= >= xi xi 1 1 1 -1 1 -1 1 1 2' 2 '2'2 >= x+xz where x and x are the eigenspinors of the operator Ŝ₂. The indices 1 and 2 refer to particle 1 and particle 2, respectively. In addition, 1 -1 1-1 2' 2 '2 Xi x > = X₁ X₂ S₁ is the spin operator of particle 1. $₂ is the spin operator of particle 2. Ŝ is the total spin operator: S = S₁ + S₂. Find the matrix representations of S1z, Szz, S2, and $2 in the four-dimensional basis |Yn >. Hint: make a table. Find the matrix representations of the total spin operators S² and S₂ in the basis |Yn >. Find the normalized eigenspinors and eigenvalues of $² and Ŝ₂. Congratulations: you just derived the Clebsch Gordon coefficients in a way that is different from the method used in Griffiths. Explain! Indicate degeneracies. Using the eigenspinors of Ŝ² and $₂ representations of Ŝ² and S₂ as your new basis, write down the matrix

(a) (b) (c) 1. Consider a system with two spin ½ particles in a four-dimensional basis |s₁m₁₂m₂ >: 1 1 1 1 2'2'2'2 (d) |4₁ >= |4₁₂ >= |4₁ >= |4₁₂ >= >= xi xi 1 1 1 -1 1 -1 1 1 2' 2 '2'2 >= x+xz where x and x are the eigenspinors of the operator Ŝ₂. The indices 1 and 2 refer to particle 1 and particle 2, respectively. In addition, 1 -1 1-1 2' 2 '2 Xi x > = X₁ X₂ S₁ is the spin operator of particle 1. $₂ is the spin operator of particle 2. Ŝ is the total spin operator: S = S₁ + S₂. Find the matrix representations of S1z, Szz, S2, and $2 in the four-dimensional basis |Yn >. Hint: make a table. Find the matrix representations of the total spin operators S² and S₂ in the basis |Yn >. Find the normalized eigenspinors and eigenvalues of $² and Ŝ₂. Congratulations: you just derived the Clebsch Gordon coefficients in a way that is different from the method used in Griffiths. Explain! Indicate degeneracies. Using the eigenspinors of Ŝ² and $₂ representations of Ŝ² and S₂ as your new basis, write down the matrix

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(a) (b) (c) 1. Consider a system with two spin ½ particles in a four-dimensional basis |s₁m₁₂m₂ >: 1 1 1 1 2'2'2'2 (d) |4₁ >= |4₁₂ >= |4₁ >= |4₁₂ >= >= xi xi 1 1 1 -1 1 -1 1 1 2' 2 '2'2 >= x+xz where x and x are the eigenspinors of the operator Ŝ₂. The indices 1 and 2 refer to particle 1 and particle 2, respectively. In addition, 1 -1 1-1 2' 2 '2 Xi x > = X₁ X₂ S₁ is the spin operator of particle 1. $₂ is the spin operator of particle 2. Ŝ is the total spin operator: S = S₁ + S₂. Find the matrix representations of S1z, Szz, S2, and $2 in the four-dimensional basis |Yn >. Hint: make a table. Find the matrix representations of the total spin operators S² and S₂ in the basis |Yn >. Find the normalized eigenspinors and eigenvalues of $² and Ŝ₂. Congratulations: you just derived the Clebsch Gordon coefficients in a way that is different from the method used in Griffiths. Explain! Indicate degeneracies. Using the eigenspinors of Ŝ² and $₂ representations of Ŝ² and S₂ as your new basis, write down the matrix
Transcribed Image Text:(a) (b) (c) 1. Consider a system with two spin ½ particles in a four-dimensional basis |s₁m₁₂m₂ >: 1 1 1 1 2'2'2'2 (d) |4₁ >= |4₁₂ >= |4₁ >= |4₁₂ >= >= xi xi 1 1 1 -1 1 -1 1 1 2' 2 '2'2 >= x+xz where x and x are the eigenspinors of the operator Ŝ₂. The indices 1 and 2 refer to particle 1 and particle 2, respectively. In addition, 1 -1 1-1 2' 2 '2 Xi x > = X₁ X₂ S₁ is the spin operator of particle 1. $₂ is the spin operator of particle 2. Ŝ is the total spin operator: S = S₁ + S₂. Find the matrix representations of S1z, Szz, S2, and $2 in the four-dimensional basis |Yn >. Hint: make a table. Find the matrix representations of the total spin operators S² and S₂ in the basis |Yn >. Find the normalized eigenspinors and eigenvalues of $² and Ŝ₂. Congratulations: you just derived the Clebsch Gordon coefficients in a way that is different from the method used in Griffiths. Explain! Indicate degeneracies. Using the eigenspinors of Ŝ² and $₂ representations of Ŝ² and S₂ as your new basis, write down the matrix
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