7. Consider a system of two spin 1/2 particles. The particles interact with one another via the Hamiltonian Ho AS(1)S(2), = where A is a constant, S is the spin 1/2 operator and the labels (1) and (2) indicate the particle on which it operates. The system is immersed in a magnetic field pointing in the z-direction, which introduces a second term to the Hamiltonian: H₁ = E(S₂(1) + S₂(2)), where the constant ε = (-eB/m). The overall Hamiltonian is then given by H = H0 + H₁. (a) Find the spectrum of this system, i.e. the eigenvalues of H. (b) Find the normalized eigenstates of H corresponding to the eigenvalues of part (a)? (c) How is the energy gap between the ground state and the first excited state behave as (ε/A) → 0? and as (ε/A) → ∞0?

7. Consider a system of two spin 1/2 particles. The particles interact with one another via the Hamiltonian Ho AS(1)S(2), = where A is a constant, S is the spin 1/2 operator and the labels (1) and (2) indicate the particle on which it operates. The system is immersed in a magnetic field pointing in the z-direction, which introduces a second term to the Hamiltonian: H₁ = E(S₂(1) + S₂(2)), where the constant ε = (-eB/m). The overall Hamiltonian is then given by H = H0 + H₁. (a) Find the spectrum of this system, i.e. the eigenvalues of H. (b) Find the normalized eigenstates of H corresponding to the eigenvalues of part (a)? (c) How is the energy gap between the ground state and the first excited state behave as (ε/A) → 0? and as (ε/A) → ∞0?