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I When can one clear the operator

  1. Apr 6, 2016 #1
    Hi all!
    I'm having problems understanding the operator algebra. Particularly in this case:
    Suppose I have this projection ## \langle \Phi_{1} | \hat{A} | \Phi_{2} \rangle ## where the ##\phi's ## have an orthonormal countable basis.
    If I do a state expansion on both sides then I suppose I'd get this: [tex]\sum_{n,m} \langle n | b_{n}^* \, \hat{A} \, c_{m} | m \rangle [/tex]
    And what's to stop me from moving the operator to the left and getting a kronecker-delta?
     
  2. jcsd
  3. Apr 6, 2016 #2

    strangerep

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    I presume your expansion states are eigenstates of ##\hat A##? If so, then I'd say "yes..."
     
  4. Apr 6, 2016 #3
    Ah no... Sorry for not pointing that out. They are just of the same same arbitrary orthonormal basis.
     
  5. Apr 6, 2016 #4

    strangerep

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    Oh, well, if your expansion states are not eigenstates of ##\hat A##, then... what are they? E.g., what is $$\hat A |m\rangle ~=~ ?$$
     
  6. Apr 6, 2016 #5
    I don't know, that's the problem. I've seen bras and kets being moved around without being eigenstates of an operator. Like in this case $$ \langle a | b \rangle \langle c1 | A | c2 \rangle \langle d | e \rangle = \langle a | b \rangle \langle d | e \rangle \langle c1 | A | c2 \rangle $$
     
  7. Apr 6, 2016 #6
    Is the operator like glued to the left ket? Can I move the bra around?
     
  8. Apr 6, 2016 #7

    strangerep

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    Think of the bra ##\langle a|## as a row vector "##a^T##'', the ket ##|b\rangle## as a column vector "##b##", and the operator ##\hat A## as a matrix ##A##. I'll write it the overall expression as ##a^T A b##. Does that much make sense?
    (I'm not sure how much ordinary linear algebra you already know.)
     
  9. Apr 6, 2016 #8
    Ok it hadn't occurred to me to think of them in that way. It wouldn't make sense to do a commutation there if the rows didn't equal the columns. Nevertheless, moving scalars around wouldn't be a problem.

    Thanks, it was actually really easy...
     
  10. Apr 6, 2016 #9

    strangerep

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    More generally, see the second line of my signature below... :oldbiggrin:

    Ballentine, ch1, can help a lot with these sort of questions.
     
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