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Time-Dependent Degenerate Pertubation Theory for 3x3 matrix

  1. Apr 9, 2013 #1
    1. The problem statement, all variables and given/known data
    H0 = [2,0,0;0,2,0;0,0,4]
    H1 = [0,1,0;1,0,1;0,1,0]

    Find energy eigenvalues to 2nd order.

    2. Relevant equations



    3. The attempt at a solution
    I know that I need to diagonalize the perturbation in the 2x2 subspace (for my 2 degenerate eignevalues of 2 but I'm not sure how to diagonalize my perturbation in this subspace. As far as I can tell H1 is not block diagonal so I can't separate it. What am I missing? From there I think I can do the rest of the problem. Just stuck here.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 9, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    Let ##|u_1\rangle## and ##|u_2\rangle## be the eigenvectors of H0 that correspond to the degenerate eigenvalue.

    Form the 2x2 matrix ##V## with elements ##V_{11} = \langle u_1|H_1|u_1\rangle##, ##V_{12} = \langle u_1|H_1|u_2\rangle##, etc.

    That's the 2x2 matrix that you need to diagonalize.
     
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