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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 $H$.

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  • $\begingroup$ $H_0$ and $H'$ don't commute so diagonalizing each individually does not give you the diagonalization of the sum. $\endgroup$ – AHusain Nov 13 '18 at 19:30
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    $\begingroup$ 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. $\endgroup$ – Dani Nov 13 '18 at 19:31
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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.

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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 $(\vN\cdot\vs)^2$.

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