Solving a 2x2 Perturbed Hamiltonian Exactly

Problem

Consider Hamiltonian $$H = H_0 + \lambda H'$$ with

$$H_0 = \Bigg(\begin{matrix}E_+ & 0 \\ 0 & E_-\end{matrix}\Bigg)$$ $$H' = \vec{n}\cdot\vec{\sigma}$$ for 3D Cartesian vector $$\vec{n}$$ and $$\sigma_i$$ the Pauli matrices.

Solve exactly for $$E_+ = E_i$$ and $$E_+ \ne E_-$$. (NOTE: As an earlier part of this problem, both cases were solved to second-order energy and state corrections, and those results are suppose to be compared to these results.)

My Question

As part of my pertubation solutions, I found the eigenenergies and eigenstates for both $$H$$ and $$H'$$. Before I do a brute force approach and actually diagonalize $$H$$, I want to make sure there isn't a more elegant approach. It seems like I should be able to use the information about the components---namely the eigenstates and eigenvalues of $$H_0$$ and $$H'%---to find information about the sum$$H$. •$H_0$and$H'$don't commute so diagonalizing each individually does not give you the diagonalization of the sum. – AHusain Nov 13 '18 at 19:30 • You want to solve the total Hamiltonian exactly. This means that you should find the eigenvalues of$H=H_0 + \lambda H^\prime\$, I think that is needed. – Dani Nov 13 '18 at 19:31

If you want to make your life a little bit easier, first prove that the spectrum of $$H$$ can only depend on $$n_x^2 + n_y^2$$, and not on $$n_x$$ and $$n_y$$ individually. Then you can set $$n_y=0$$ before you compute the eigenvalues.
Solve exactly for $$E_+=E_i$$ and $$E_+\ne E_-$$.
What is $$E_i\,$$? $$\let\l=\lambda \let\s=\sigma \def\half{{\textstyle{1 \over 2}}} \def\vs{\vec\s} \def\vN{\vec N}$$
I'll give a hint to solution. Write $$H_0 = \half(E_+ + E_-)\,I + \half(E_+ - E_-)\,\s_3.$$ Then $$H + \l\,H' = \half(E_+ + E_-)\,I+ \vN \cdot \vs$$ where $$\vN = \left(\l\,n_1,\ \l\,n_2,\ \half(E_+ - E_-) + \l\,n_3\right)\!.$$
Can you find eigenvalues of $$\vN \cdot\s\,$$? Try evaluating $$(\vN\cdot\vs)^2$$.