Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another ia the Hamiltonian H = Eo [S(1) · S(2)] + E₁ [S₂(1) + S:(2)], ħ here Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the article on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?

Consider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another ia the Hamiltonian H = Eo [S(1) · S(2)] + E₁ [S₂(1) + S:(2)], ħ here Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the article on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?