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QM: orthonormality

  1. Feb 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the eigenstates of the simple harmonic oscillator using Dirac notations are orthonormal.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 22, 2009 #2

    malawi_glenn

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    attempt? ...
     
  4. Feb 22, 2009 #3
    Sorry malawi_glenn. I didn't post the attempt because I don't know how to write equations like everybody else does!
    But I'll try to explain.
    The first thing that came up to my mind is, to prove orthonormality I have to show,
    <psi_m|psi_n>=0
    and that is,

    <psi_0|[(a^dagger)^m]/sqrt(m!)*[(a^dagger)^n]/sqrt(n!)|psi_0>=0

    and then, I don't know how to evaluate further.

    Once again, I'm sorry. I hope you don't get annoyed by my poor explanation.
    Thanks
     
  5. Feb 22, 2009 #4

    malawi_glenn

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    we can read it, do your best effort

    now the dual correspondance to (a^dagger |psi>) is (<psi| a)

    So, start over again.
     
  6. Feb 22, 2009 #5

    malawi_glenn

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    also "show that"... is quite unprecise, since you construct states |n> according to (a^dagger)^n]/sqrt(n!)|psi_0> in order to make orthonormality.

    What may be used? The canonical commutator relation only? Or what? Boring excerice I would say ;-)
     
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