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Homework Help: Hermitian Operators

  1. Dec 31, 2005 #1
    Hi there,

    Was wondering if anyone could point me in the right direction for this one?

    Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal?

    Thanks
     
  2. jcsd
  3. Dec 31, 2005 #2

    LeonhardEuler

    User Avatar
    Gold Member

    Alright, say you have an hermitian operator, O with eigenfunctions |a1> and |a2>, with eigenvalues of a1 and a2 respectively. Then:
    O|a1>=a1|a1> (1)
    <a1|O=a1*<a1| (2)
    O|a2>=a2|a2> (3)
    and
    <a2|O=a2*<a2| (4)
    Now right multiply |a1> in equation (4) and left multiply by <a2| in equation (1) to get two expressions for <a2|O|a1>. Subtract the two equations and observe.
    -edit: keep in mind that the eigenvalues of hermitian operators are real. You can prove this by letting a1=a2
     
    Last edited: Dec 31, 2005
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