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Eigenfunctions and hermitian operators

  1. Oct 13, 2011 #1
    Hi. I'm just a bit stuck on this question:

    Write down two equations to represent the fact that a given wavefunction is simultaneously an eiigenfunction of two different hermitian operators. what conclusion can be drawn about these operators?

    Im not quite sure how to start it.

    Thanks!
     
  2. jcsd
  3. Oct 13, 2011 #2

    Hootenanny

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    If [itex]\psi[/itex] is an eigenfunction of the hermitian operator [itex]A_1[/itex], what does this mean? Can you write the eigenvalue problem?
     
  4. Oct 13, 2011 #3
    is it.

    phi|A>= a|A> ?
     
  5. Oct 13, 2011 #4

    Hootenanny

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    Yes. So, we have (preserving the index):

    [tex]A_1 \psi = a_1\psi.[/tex]

    Suppose we now have a second hermitian operator, [itex]A_2[/itex]. Can you write a similar equation?
     
  6. Oct 13, 2011 #5
    [tex]A_2 \psi = a_2\psi.[/tex]
     
  7. Oct 13, 2011 #6

    Hootenanny

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    Excellent!

    So what happens if you first operate on [itex]\psi[/itex] with [itex]A_1[/itex], followed by [itex]A_2[/itex]? Compare this with what happens when you do it the other way round.
     
  8. Oct 13, 2011 #7
    [itex]A_2[/itex][itex]A_1[/itex][itex]\psi[/itex] = [itex]A_2[/itex]([itex]A_1[/itex][itex]\psi[/itex]) =[itex]A_2[/itex]([itex]a1[/itex][itex]\psi[/itex])=[itex]a_1[/itex][itex]A_2[/itex][itex]\psi[/itex]= [itex]a_1[/itex][itex]a_2[/itex][itex]\psi[/itex]

    [itex]A_1[/itex][itex]A_2[/itex][itex]\psi[/itex]=[itex]a_2[/itex][itex]a_1[/itex][itex]\psi[/itex]

    Therefore they commute!! XD
     
  9. Oct 13, 2011 #8

    Hootenanny

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    Indeed they do! :approve:
     
  10. Oct 13, 2011 #9
    Thanks! :D


    another similar question to that one I have is :

    write down two equations to represent the fact that two different wavefunctions are simultaneously eigenfunctions of the same hermation operator, with different eigenvalues. what conclusion can be drawn about these wavefunctions. So far I have

    [itex]A_1[/itex][itex]\psi[/itex]=[itex]a_1[/itex][itex]\psi[/itex]

    [itex]A_1[/itex]θ=[itex]a_1[/itex]θ
     
  11. Oct 13, 2011 #10

    dextercioby

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    Shouldn't there be a2 for the second line ?
     
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