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Two-state perturbation problem (QM)

  1. Apr 3, 2008 #1
    1. The problem statement, all variables and given/known data

    Given is a hamiltionan H0:

    E1 0
    0 E2

    with E1 and E2 being eigenvalues of two eigenstates phi1 and phi2

    A distortion W, with W12 = W21* (complex conjugate):

    0 W12
    W21 0

    Calculate the eigenvalues and eigenstates of H = H0 + W


    2. Relevant equations

    See for most info:
    http://farside.ph.utexas.edu/teaching/qm/lectures/node50.html

    it's a two-state perturbation problem


    3. The attempt at a solution

    I found out most things like how to calculate the eigenvalues. A lot of it is explained at the mentioned site (http://farside.ph.utexas.edu/teaching/qm/lectures/node50.html) The thing I don't get is how they got from the eigenvalues to the eigenstates.

    I have tried the following:

    ( E1 - E1' W12 )(a) (0)
    ( W21 E2 - E1' )(b) = (0)

    with E1' being the first new eigenvalue: E1'=E1 + |W12|^2/(E1-E2)

    But I do not get to the answer provided on the site.


    I hope I have been clear about my question, so someone can explain it to me. Thanks in advance!

    Tim
     
  2. jcsd
  3. Apr 3, 2008 #2
    Note that since eigenvectors are only defined up to a scaling, your matrix only represents one equation. So define a=kb, and solve for k. (This is the canonical way to proceed. In practice people just stare at it and then write down the solution.)
     
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