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Degenerate perturbation theory - PLEASE HELP

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Ok, so i have this online test to be completed by tomorrow and i have NO IDEA how to go about it, my notes are useless, they dont explain anything. On the up side all the questions seem to be on a very similar topic so if i could understand some key ideas then i should be able to have a go at most of them. Does anyone have any pointers? Where can i find information on the net to understand and digest so i can answer these? I have tried reading about DPT but had no luck there.


    The main aim of this test is to develop and test your understanding of the matrices arising in degenerate perturbation theory (DPT). In degenerate pertrubation theory (DPT) one constructs a matrix with elements < i | H' | j >, where i and j represent unperturbed orthonormal (i.e. < i | j > = 0 for i not equal to j, or 1 for i = j) eigenstates and H' is the perturbation. For simplicity, we will assume that the eigenstates are real.

    TRUE OR FALSE

    Question 1

    The units of < i | H' | j > are energy.




    Question 2


    The eigenvalues of a matrix are unchanged if a constant is added to each diagonal element.






    Question 3


    If | i > and | j > are eigenstates of H', then all the off-diagonal elements of the matrix are zero.





    Question 4


    One can add a constant times the identity matrix to the DPT matrix without changing the physics, because the zero of the energy scale is arbitrary.





    Question 5


    One can add a constant to each element of the DPT matrix because the zero of the energy scale is arbitrary.





    Question 6


    For a 2 - state system, the eigenenergies are proportional to < 0 | H' | 1> squared if the diagonal elements of the DPT matrix are zero.





    Question 7


    If | 0 > and | 1 > have opposite parity and H' is an odd function then < 0 | H' | 1 > is zero.





    Question 8


    For a 2 - state system, the eigenvalues are propotional to < 0 | H' | 1 > if the off-diagonal terms are much smaller than the difference between the diagonal terms.





    Question 9


    For a 2 - state system, if the diagonal terms are zero then the eigenvalues are plus and minus < 0 | H' | 0 >.





    Question 10


    For a 2 - state system with real unperturbed states, < 0 | H' | 1 > = < 1 | H' | 0 >.





    Question 11
    1.

    For a 2 - state system with degenerate unperturbed states | 0 > and | 1 >,

    < 0 | H' | 0 > = < 1 | H' | 1>





    Question 12



    For a 2 - state system, a perturbation expansion of the eigenvalues is valid when the off-diagonal terms are large compared to the difference between the diagonal terms.





    Question 13
    1.

    The 'first order' shift in the energy of state i due to the perturbation H' is given by < i | H' | i >.





    Question 14


    For a 2 - state system, the 'second order' shift in the energy of state | 0 > is inversely proportional to < 1 | H' | 1 > - < 0 | H' | 0 >.





    Question 15


    One can add a constant to each element on the diagonal of the matrix without changing the physics because the zero of the energy scale is arbitrary.
     
    Last edited: Nov 11, 2009
  2. jcsd
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