Day 37 In-Class Activity

Perturbing 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 ${\mathbf{B}}^{\prime }=B_2\hat{\mathbf{x}}$. This perturbation Hamiltonian is characterized by a different Larmor frequency, ${\omega }_2$.

1. Construct the original Hamiltonian, $H_0$, and the perturbation Hamiltonian, $H^{\prime }$. Use the matrix representation.
2. What are the original energy eigenvalues?
3. Use Perturbation Theory to determine the first-order energy corrections. What do you notice?
4. Use Perturbation Theory to determine the second-order energy corrections.
5. Draw a sketch of the energy levels and how they are affected by increasing $B_2$. What do you notice about the shape of the curves?
6. Compare and contrast your results here with the results from our previous in-class activity that used Perturbation Theory.