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[QM] Help understanding this bra-ket solution

  1. Jun 27, 2013 #1
    Hello,

    I am working my way though Sakurai's book on Quantum MEchanics and am having some problems understanding the bra-ket notation. I keep believing I understand everything there is to it but then he will do something in a single line that I cannot understand. This is one of them. If someone could help me out it would be great.

    1. The problem statement, all variables and given/known data

    Show why the following in correct: [itex]<a''|A|a'>=<a'|A|a'>\delta_{a'a''}= a'\delta_{a'a''}[/itex]

    A is an hermitian operator. a' and a'' are the eigenkets and eigenvalues of A.

    2. Relevant equations



    3. The attempt at a solution

    The only method I can think of to coming up with the final solution is the following, which may not even be correct.

    Use [itex]A|a'> = a'|a'>[/itex] since A is hermitian and rewrite as [itex]<a''|a'|a'>[/itex]
    Since a' is real [itex]a'=a'^* [/itex] so we can rewrite as [itex]a'<a''|a'>[/itex]
    From here we can use the fact that a'' and a' are orthonormal eigenkets from the same operator so [itex]<a''|a'> = \delta_{a'a''} [/itex] and we finally have [itex]a'\delta_{a'a''}[/itex]

    However this method does no provide the middle expression which has me really thrown off. I am not sure if I am doing everything correct and I do not know how Sakurai came to that. I am also not 100% on what all this means either.
     
  2. jcsd
  3. Jun 27, 2013 #2
    Ugh I don't know how I just saw what they did. Multiply by the identity operator (a') between <a''| and A. It all falls together then. Still not 100% sure what all it means but I'll work on it.
     
  4. Jun 27, 2013 #3

    rude man

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    I think this post belongs under Advanced Physics.
     
  5. Jun 27, 2013 #4
    I checked the rules for the advanced physics and it said that just because it's QM doesn't mean it belongs there. So I figured this would be a good place to put it. Either way this question can be closed now as I figured it out. Just cant find out how to mark it for closure.
     
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