Perturbation of a Three-Level System

Consider a spin-1 system (like a Nitrogen nucleus) that is placed in a constant magnetic field $\mathbf{B}_0 = B_0\hat{\mathbf{z}}$ has a Larmor frequency of $\omega_0$. We perturb this system by turning on another magnetic field $\mathbf{B}’ = B_1\hat{\mathbf{z}}$. This perturbation Hamiltonian is characterized by a different Larmor frequency, $\omega_1$.

  1. Construct the original Hamiltonian, $H_0$, and the perturbation Hamiltonian, $H’$. Use the matrix representation.
  2. What are the original energy eigenvalues?
  3. Use Perturbation Theory to determine the first-order energy corrections.
  4. Draw a sketch of the energy levels and how they are affected by increasing $B_1$. What do you notice about the shape of the curves?