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Dealing with normalized quantum functions

  1. Oct 20, 2011 #1
    1. The problem statement, all variables and given/known data

    If <∅| is normalized, show that:

    <∅|∅>=1=<∅|n><n|∅>

    (where ∅ is a non-eigenfunction wave function composed of Ʃc(n)ψ(n).

    2. Relevant equations



    3. The attempt at a solution

    I can show that <∅|∅>=Ʃc*(n)c(n) (=1). But the next part of the question asks to use your proof to show Ʃc*(n)c(n)=1 so that's not the way it is supposed to be done. I feel like I'm missing something very simple.
     
  2. jcsd
  3. Oct 20, 2011 #2
    It seems to me that the problem statement must be


    <∅|∅>=1=Ʃn<∅|n><n|∅>.

    This is just the usual expansion over basis states. What you wrote can't be true for arbitrary n; it only works when you sum over all the basis states.

    In matrix language, Ʃn|n><n| is just the identity matrix.

    BBB
     
  4. Oct 20, 2011 #3
    Aha that was what I suspected.

    With the correction, is there a way to prove that it is equal to one without having to use that Ʃc*(n)c(n) (=1)?
     
  5. Oct 20, 2011 #4
    I would just say Ʃn|n><n| = I, the identity matrix, because that is the definition of a complete set of states {|n>}. Then

    <θ|I|θ> = <θ|θ> =1

    because |θ> is normalized. This is not the complicated part of quantum mechanics.

    BBB
     
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