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Proof for the Hermitian operator

  1. Aug 21, 2016 #1
    1. The problem statement, all variables and given/known dataprove the following statement:
    Hello, can someone help me prove this statement

    PzXRBBp.jpg

    A is hermitian and {|Ψi>} is a full set of functions


    2. Relevant equations
    Σ<r|A|s> <s|B|c>



    3. The attempt at a solution
    Since the right term of the equation reminds of the standard deviation, I tried using its definition but it didn't yield any results. Also, I tried to use the hermicity of the operator A to get the complete set but after that I got stuck.
     
  2. jcsd
  3. Aug 21, 2016 #2
    This is prety simple if you assume ##\psi_i## form a basis for the Hilbert space of states. Just use the complex relation ##|z|^2=z z^*## for the left hand side, and use the completeness relation for ##\psi_i##. Is this what you meant by a full set of functions?
     
  4. Aug 21, 2016 #3
    Yes, I did, I apologize, English is not my mother language. I tried but I can't get those two terms on the right side
     
  5. Aug 21, 2016 #4
    No problem. The two terms come as a property of the summation. The completeness relation assumes you sum over all ##i##. The first term comes from the completeness relation. The second term comes from the fact that you are missing the ##i=j## in the summation on the left hand side. The key equation you need to use is
    $$
    \langle \psi_j | A^2 | \psi_j \rangle=\sum_i \langle \psi_j | A | \psi_i \rangle \langle \psi_i | A | \psi_j \rangle
    $$
     
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