The Hamiltonian for two interacting spins (both s=1/2 ) in a magneticfield B directed along the z axis is H=B\left(a_{1} \sigma_{z}^{(1)}+a_{2} \sigma_{z}^{(2)}\right)+K \sigma^{(1)} \cdot \sigma^{(2)} where an and az are the negatives of the magnetic moments (assumedto be unequal to avoid degeneracy), and K is the interaction strength. Use second order perturbation theory to calculate the energy eigen-values, assuming that B is small. Use second order perturbation theory to calculate the energy eigen values, under the opposite assumption that K is small. Find the exact energy eigenvalues for this Hamiltonian, and verifythe correctness of your answers in parts (a) and (b).

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