Two-spin system 1. Consider a system with two spins S₁ and 52, both with spin 1/2. Solve the eigenvalues and the eigenstates for the Hamiltonian Ĥ = Ŝ₁. Ŝ₂. (14) Hint: Use the basis S, S₂, S1, S₂). 2. For each of the eigenstates, what are its total S and total S₂ quantum numbers? 3. The eigenstates can be classified to one singlet state and three triplet states. Which one is singlet and which three are triplets? 4. Proof that all the eigenstates of $2 are also eigenstates of the total Ŝ, operator, where S₂ = S1 + Ŝ2z.

Two-spin system 1. Consider a system with two spins S₁ and 52, both with spin 1/2. Solve the eigenvalues and the eigenstates for the Hamiltonian Ĥ = Ŝ₁. Ŝ₂. (14) Hint: Use the basis S, S₂, S1, S₂). 2. For each of the eigenstates, what are its total S and total S₂ quantum numbers? 3. The eigenstates can be classified to one singlet state and three triplet states. Which one is singlet and which three are triplets? 4. Proof that all the eigenstates of $2 are also eigenstates of the total Ŝ, operator, where S₂ = S1 + Ŝ2z.