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Multi-electron eigenfunction problem

  1. Apr 16, 2012 #1
    1. The problem statement, all variables and given/known data
    "Prove that any two different nondegenerate bound eigenfunctions [itex]\psi[/itex]j(x) and [itex]\psi[/itex]i(x) that are solutions to the time-independent Schroedinger equation for the same potential V(x) obey the orthogonality relation

    [itex]\int[/itex]-∞ [itex]\psi [/itex]j*[itex]\psi[/itex]i(x)dx=0

    "


    2. Relevant equations
    I believe I have to find equations for which both eigenfunctions are solutions?


    3. The attempt at a solution
    I'm lost on how to get the problem started. I cannot think of any eigenfunctions to use. I might be putting more thought to it than I need to though.
     
  2. jcsd
  3. Apr 17, 2012 #2

    collinsmark

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    Yes, the key is that they are eigenfunctions. But I think you're supposed to prove the orthogonality in the general sense, rather than use specific examples of eigenfunctions.

    Suppose that [itex] \psi_a(x) [/itex] and [itex] \psi_b(x) [/itex] are eigenstates of the operator [itex] O [/itex], with corresponding eigenvalues [itex] a [/itex] and [itex] b [/itex] respectively.

    [tex] O \psi_a(x) = a \psi_a(x) [/tex]
    [tex] O \psi_b(x) = b \psi_b(x) [/tex]

    Now consider each of these cases:

    [tex] \int_{-\infty}^{\infty} [O \psi_a(x)]^* \psi_b(x)dx = \ \ ???[/tex]

    [tex] \int_{-\infty}^{\infty} \psi^*_a(x) [O \psi_b(x)]dx = \ \ ???[/tex]

    I'll let you take it from there. :wink:
     
    Last edited: Apr 17, 2012
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