Discussion
A system described by the Hamiltonian
\[ H = \begin{pmatrix} 5 & 2 & 0 \\ 2 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix} \]
is perturbed by
\[ H' = \varepsilon \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & 1 \end{pmatrix} \]
where (\(\varepsilon \ll 1\)). A pair of eigenvalues of the perturbed system is :
Answer: C
A pair of eigenvalues is 3, 7 + 2\(\varepsilon\).
The unperturbed H has a 2 by 2 block with eigenvalues 7 and 3 and eigenvectors \((1, 1, 0)/\sqrt{2}\) and \((1, -1, 0)/\sqrt{2}\), plus the eigenvalue 2 with \((0, 0, 1)\). First-order shifts are the diagonal elements of H' in this basis:
\[ \begin{aligned} E_{7} &: \tfrac{1}{2}(1 + 1 + 1 + 1)\varepsilon = 2\varepsilon \\ E_{3} &: \tfrac{1}{2}(1 - 1 - 1 + 1)\varepsilon = 0 \\ E_{2} &: H'_{33} = \varepsilon \end{aligned} \]
So the levels become 7 + 2\(\varepsilon\), 3 and 2 + \(\varepsilon\). Only the third option lists two of them; 3 + 2\(\varepsilon\) gives the shift of the 7 level to the 3 level, and 2 + 2\(\varepsilon\) doubles the shift of the lowest level.
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