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Quantum Mechanics algebra

  1. Dec 30, 2009 #1
    Quantum Mechanics algebra - time independant peturbation theory

    Hi

    1. The problem statement, all variables and given/known data
    The potential shown is operating on the eigenstate as shown in the pic. I am having trouble getting the second term using orthogonality (got the first term :-) ). Please Help!


    2. Relevant equations
    see pic


    3. The attempt at a solution
    put the i's equal to j (i.e. orthogonality) but don't know what to do next.

    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

    Last edited: Dec 30, 2009
  2. jcsd
  3. Dec 30, 2009 #2
    Just note that, in the sum, the only nonzero term in the second part will be the one where
    i + 1 = j, so i will be equal to...
     
  4. Dec 30, 2009 #3
    This was part of a hint for a question on time independant pertubation theory (see pic). i was reluctant to post the whole question because i wanted a good crack at it without help but i am now stuck. i can't seem to get the 2nd order term and can't work out what happens with the first order term.

    Thank you
     

    Attached Files:

    Last edited: Dec 30, 2009
  5. Dec 30, 2009 #4

    diazona

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    Homework Helper

    Well, you're given a formula for the energy [itex]E_0[/itex] in terms of [itex]\hat{V}[/itex], and you're given a formula for [itex]\hat{V}[/itex]. What happens when you plug [itex]\hat{V}[/itex] in to the energy formula? If you get stuck, show your work.
     
  6. Dec 31, 2009 #5
    Thanks for the replies. This is my working so far (see attachment). I have plugged V in but am stuck on the last 2 lines of my working. Using orthogonality i think you get lambda and lambda* for the 1st terms of the second-last and last-lines respectively but what about the second terms in those lines?
    Thanks
     

    Attached Files:

  7. Dec 31, 2009 #6
    Ah, do they equal zero (the last terms) because i is from 0 to infinity and those two terms require i=-1?
     
  8. Dec 31, 2009 #7

    diazona

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    Homework Helper

    Yeah, you can use that argument for the second term on each of the last two lines.
     
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