Answer Happy

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please answer a,b,c
Consider two interacting spins. The operators of the spin projections in one-spin Hilbert space are Ŝ₁ = ¹ħơ, Ŝy = ½ħơy, Ŝ₂ = ¹⁄ħơ₂, where σ, σy and σ₂ are Pauli matrices. a.) The quantum basis for two spins consists of 4 states, which can be chosen as the projections of the spins on the z-axis | 11), | ↑↓), | ↓↑), | ↓↓). Find the matrix repre- sentation of spin operators S₁, S2x, Sly, S2y, S1z, S2z in the above four-dimensional Hilbert space. Obtain explicitly the Hamiltonians as a 4 × 4 matrix in the above basis and find the eigenvectors and eigenvalues of the following models: b.) Ising model: Ĥ₁ = JŜ1zŜ2z c.) Heisenberg model: µ = J(Ŝ₁xŜ2x + Ŝ1yS2y + Ŝ1zŜ2z). Here J is the interaction constant. Hint: In part (b), the solution may simplify if you use the spin rising operator ŜÅ and the spin lowering operator S. They are defined as follows: Ŝ+ = Ŝz+iSy (1) Ŝ= Ŝz - iSy (2)