# Degenerate Perturbation: Calculating Eigenvalues

• ergospherical
In summary, the conversation discusses a model hamiltonian with unperturbed eigenvalues E1 and E2 = E3 that is subjected to a perturbation V with specific matrix elements. The problem at hand is to calculate the corrected eigenvalues, but in the degenerate subspace, diagonalizing V seems to be impossible due to all matrix elements being zero. It is suggested that a higher order correction may be needed to lift the degeneracy. The conversation ends with a suggestion to use the usual method of diagonalization by solving the characteristic equation, which results in three non-degenerate eigenvalues.
ergospherical
Homework Statement
See below
Relevant Equations
N/A
Say a model hamiltonian with unperturbed eigenvalues E1 and E2 = E3 is subjected to a perturbation V such that V12 = V21 = x and V13 = V31 = x2, with all other elements zero. I'm having trouble calculating the corrected eigenvalues. In the degenerate subspace spanned by |2> and |3> I need to diagonalise V, but all of these matrix elements are zero?

It sounds like the first-order correction is zero, and you'll need to go to higher orders to lift the degeneracy.

DrClaude
If I understand you correctly, you have a perturbed matrix of the form $$H=\begin{pmatrix} E_1 & V_{12} & V_{13} \\ V_{12}& E_2 & 0 \\ V_{13} & 0 & E_2 \end{pmatrix}.$$Why can you not diagonalize the usual way? Just say$$\det\begin{bmatrix} E_1-\lambda & V_{12} & V_{13} \\ V_{12}& E_2-\lambda & 0 \\ V_{13} & 0 & E_2-\lambda \end{bmatrix}=0$$ and solve the characteristic equation. That is easy to do because it factors into ##(E_2-\lambda)## times a quadratic in ##\lambda.## You get three non-degenerate eigenvalues.

First of all, there is no reason not to make a rotation in the degenerate subspace such that ##V_{13} = 0##. After that rotation it should be clear that ##E_2## is still an eigenvalue for one state. You can then apply non-degenerate perturbation theory to the remaining 2-dimensional subspace.

Replies
0
Views
252
• Quantum Physics
Replies
12
Views
2K
Replies
28
Views
6K
• Quantum Physics
Replies
18
Views
2K
Replies
6
Views
3K
• Quantum Physics
Replies
2
Views
1K
Replies
2
Views
2K
Replies
5
Views
1K