Derive approximate expressions for the eigenvalues of a Hamiltonian $H+\lambda V$, working to second order in the parameter $\lambda$ and assuming the eigenstates and eigenvalues of $H$ are known and non-degenerate.

Let $\mathbf{J}=\left(J_{1}, J_{2}, J_{3}\right)$ be angular momentum operators with $|j m\rangle$ joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$. What are the possible values of the labels $j$ and $m$ and what are the corresponding eigenvalues of the operators?

A particle with spin $j$ is trapped in space (its position and momentum can be ignored) but is subject to a magnetic field of the form $\mathbf{B}=\left(B_{1}, 0, B_{3}\right)$, resulting in a Hamiltonian $-\gamma\left(B_{1} J_{1}+B_{3} J_{3}\right)$. Starting from the eigenstates and eigenvalues of this Hamiltonian when $B_{1}=0$, use perturbation theory to compute the leading order corrections to the energies when $B_{1}$ is non-zero but much smaller than $B_{3}$. Compare with the exact result.

[You may set $\hbar=1$ and use $J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle .$ ]