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Perturbation theory problem

  1. Sep 1, 2016 #1
    1. The problem statement, all variables and given/known data
    I have the particle in the infinite square well and need to calculate the first order correction energy and the wave function. L is the width and the potential is:
    1/2 mw2x2 in the -L/2 < x < L/2
    and infinity in x <= -L/2 and x>=L/2

    2. Relevant equations
    H'=H-H0


    3. The attempt at a solution
    I have stated that the perturbed Hamiltonian is equal to 1/2 mw2x2.
    I am confused by the integral itself; I am not sure which boundaries to use. I am assuming that since the potential is infinite in L/2 and -L/2 I am not supposed to use those boundary conditions?
     
  2. jcsd
  3. Sep 1, 2016 #2

    Dr Transport

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    if the potential is infinite, then the wave function is zero
     
  4. Sep 1, 2016 #3
    Yes, but which boundaries to use then when integrating? Is the perturbed Hamiltonian correct?
     
  5. Sep 1, 2016 #4

    Dr Transport

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    only integrate where the wave function is non-zero
     
  6. Sep 1, 2016 #5
    That is what I'm saying. I don't know how to put boundaries since I can't use -L/2 and L/2. In all the other examples I've had, the <= and >= signs were where the potential isn't infinite, as it is in the general infinite square well problem.
     
  7. Sep 1, 2016 #6

    Dr Transport

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    the original wave function is valid between [itex] \pm L/2[/itex] so those are the limits of integration.
     
  8. Sep 1, 2016 #7
    So, I approach this problem the same, regardless this < sign? It confuses me because it says than when x is + or -L/2, the potential is infinite, hence wave function is zero. How is it right to put those boundaries once the integral is solved?
     
  9. Sep 1, 2016 #8

    Dr Transport

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    it is a perturbation, it is supposed to be small, if the width of the well is not too large the perturbation at the bottom will only slightly affect the energies and wave functions.
     
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