Parity violating interaction

Consider a hypothetical parity violating interaction \[\hat{V}'=\lambda\hbar\frac{\mathbf{S}\cdot\mathbf{r}}{r}\] added to the hydrogen atom Hamiltonian. The symbol \(\lambda\) is the strength of the interaction. Consider how it affects the \(n=2\) energy levels, but neglect the fine structure. So \(\hat{H}=\hat{H}_0+\hat{V}'\) where \[\hat{H}_0=\frac{\hat{p}^2}{2\mu}-\frac{Ze^2}{\hat{r}}\] where \(Z\) is the atomic number and \(\mu\) is the reduced mass.
 
To first order in \(\hat{V}'\) compute the energy levels and degeneracies of \(\hat{H}\) for the case \(n=2\). (Note that \(Z\) is not necessarily 1 here).
 
HINT: Think of which operators commute with \(\hat{H}\).

Two spin-half particles

A system of two “nailed-down” spin-\(\frac{1}{2}\) particles are described by \[\hat{H}=A\mathbf{S}_1\cdot\mathbf{S}_2+BS_1^z.\] a.) Compute the energy levels exactly.
b.) Treat \(BS_1^z\) as a perturbation. Find the unperturbed energy levels \(E_n^0\). Find the perturbed levels through second order in \(S\). Compare the exact to the perturbed results by using a Taylor expansion.

Sequel to problem 2 from HW#7

Suppose the matrix \(V^k\) also vanishes at second order. Show that \(\left|k,n_k\right\rangle_\parallel^{(3)}\) can be found in third order and that the energy shifts are found from \[\det\left(\hat{P}_k\hat{V}\frac{\hat{Q}}{E_{k}^0-\hat{H}_0}\hat{V}\frac{\hat{Q}}{E_{k}^0-\hat{H}_0}\hat{V}\hat{P}_k-E_k^{(3)}\hat{P}_k\right)=0.\]