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Perturbed Hamiltonian and its affect on the eigenvalues

  1. Dec 6, 2015 #1
    1. The problem statement, all variables and given/known data

    ps4 QM.PNG

    2. Relevant equations

    $$E_n^{(2)}=\sum_{k\neq n}\frac{|H_{kn}'|^2}{E_n^{(0)}-e_k^{(0)}}$$

    3. The attempt at a solution

    Not sure where to start here. The question doesn't give any information about the unperturbed Hamiltonian. Some guidance on the direction would be great! Cheers.
     
  2. jcsd
  3. Dec 6, 2015 #2

    blue_leaf77

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    Start by writing out ##H_T = H_0 + H'##, where ##H_T## is the total Hamiltonian, the matrix of which is given in the question, ##H_0## the unperturbed Hamiltonian whose elements do not contain ##\lambda##, and ##H'## the perturbation term which is proportional to ##\lambda##.
     
  4. Dec 6, 2015 #3
    Hi blue_leaf! Thanks for that, I just realised that I interpreted the question wrong. I thought the matrix given was only the unperturbed Hamiltonian.

    So I have separated it into its un/pertrubed matrices. I just don't think I understand what the question is asking, when it says the change in eigenvalues.

    ##E_1\approx E_1^{(0)}+E_1^{(1)}+E_1^{(2)}##
    ##~~~~~=E_0+0-\frac{4}{7}E_0\lambda^2##

    ##E_2\approx E_2^{(0)}+E_2^{(1)}+E_2^{(2)}##
    ##~~~~~=8E_0+0+\frac{4}{7}E_0\lambda^2##

    ##E_3\approx E_3^{(0)}+E_3^{(1)}+E_3^{(2)}##
    ##~~~~~=3E_0+E_0\lambda+0##

    ##E_4\approx E_4^{(0)}+E_4^{(1)}+E_4^{(2)}##
    ##~~~~~=7E_0+0+0##

    So are the changes in the eigenvalues ##-\frac{4}{7}E_0\lambda^2,~\frac{4}{7}E_0\lambda^2,~E_0\lambda \text{ and } 0## respectively ?
     
  5. Dec 6, 2015 #4

    blue_leaf77

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    I think the full expressions of the new energies are what the questions asks.
     
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