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Perturbation Hamiltonian

  1. Jun 15, 2015 #1
    1. The problem statement, all variables and given/known data

    The e-states of H^0 are

    phi_1 = (1, 0, 0) , phi_2 = (0,1,0), phi_3 = (0,0,1) *all columns
    with e-values E_1, E_2 and E_3 respectively.

    Each are subject to the perturbation

    H' = beta (0 1 0
    1 0 1
    0 1 0)

    where beta is a positive constant

    a) If E_1 =/ E_2 =/ E_3

    What are the new energy levels according to first and second-order perturbation theory

    b) If E_1 = E_2 = E_3

    What are the new energy levels according to first degenerate perturbation theory

    c) If E_1 =/ E_2 = E_3

    What are the new energy levels according to first perturbation theory

    2. Relevant equations

    For first order non degenerate perturbation:

    E_n ^1 = <phi_n ^ 0 | H' | phi_n ^ 0>

    For second order perturbation

    E_n ^2 = Σ (m=/n) of (|phi_m ^0 | H' | phi_n ^ 0>|^2)/(E_n ^ 0 - E_m ^0)



    3. The attempt at a solution

    a)

    E_1 ^1 = < (1 | H' | (1 >
    0 0
    0) 0)

    I am not sure how to deal with this as I just get zero


    Any help pushing me in the right direction would be appreciated
     
    Last edited: Jun 15, 2015
  2. jcsd
  3. Jun 15, 2015 #2

    Orodruin

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    Getting a zero only means that the first order correction to the energy vanishes. You should comtinue with the eigenstate perturbation and the second order energy correction.
     
  4. Jun 16, 2015 #3
    Ok, thanks I have figured out a) but am still have trouble with the degenerate case
     
  5. Jun 17, 2015 #4

    Orodruin

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    Can you show what you have attempted for the degenerate case?
     
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