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Degenerate Perturbation Theory

  1. Oct 8, 2014 #1
    1. The problem statement, all variables and given/known data
    We have spin-1 particle in zero magnetic field.
    Eigenstates and eigenvalue of operator [itex] \hat S_z [/itex] is [itex] - \hbar |-1> [/itex], [itex] 0 |0> [/itex]
    and [itex] \hbar |+1> [/itex].

    Calculate the first order of splitting which results from the application of a weak magnetic field in the x direction.

    2. Relevant equations
    Hamiltonian is perturbed by [itex] H' = \gamma B \hat S_x = - ( \gamma B/2) (\hat S_+ + \hat S_-)[/itex]

    3. The attempt at a solution

    We have to solve it using degenerate perturbation theory in the basis mentioned, and check it which the basis of eigenvectors of [itex] \hat S_x [/itex].

    I am really confused with quantum mechanics, thus would like to know how do we start the question.
    Is there anyone that can help us by going through step by step how should we go about it and explain it as well?
     
  2. jcsd
  3. Oct 8, 2014 #2

    vela

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    No, we don't supply solutions here. You need to show some effort in trying to figure out the problem yourself.

    You already know how to start the question. As you said, you need to apply degenerate perturbation theory. This topic is surely covered in your textbook.
     
  4. Oct 8, 2014 #3
    How do we apply the theory?
    Do I need to assume anything?

    For unperturbed,
    [itex] |-1> => E^0_{-1} =-\hbar [/itex]
    [itex] |0> => E^0_{0} = 0
    [itex] |+1> => E^0_{+1} =+\hbar [/itex]

    Correct?
    How do we create the new basis for the H' then?
     
  5. Oct 8, 2014 #4

    vela

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    No, that's not correct. If the unperturbed energies are different, you don't need degenerate perturbation theory, do you?

    Can you describe the general idea behind degenerate perturbation theory?
     
  6. Oct 8, 2014 #5
    Is it that when we affect the system by a little, then we are to find out the how much the system changes?
    And by degenerate, it means that energy eigenvalues are the same for all states it act on?
     
  7. Oct 8, 2014 #6

    vela

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    Yes, that's what degenerate perturbation theory is, but I want you to describe the basic idea of what you're doing when you apply the theory to a situation. In other words, what's the math problem you are solving? Are you diagonalizing a matrix? If so, what matrix? What does it represent and in what basis?
     
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