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Relation between commutator, unitary matrix, and hermitian exponential operator

  1. Jan 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!....


    2. Relevant equations
    U=exp(iC)
    C=C*
    U*U=I
    U=A+iB
    exp(M) = sum over n: ((M)^n)/n!


    3. The attempt at a solution
    I am really stumped. I tried (A+iB)(A*-iB*)=I, and I can get the commutator to come out of that, but I have these A*A and B*B terms which I am unsure how to use. I am also not using the exponential term in any way. I know it has something to do with the taylor expansion, just not sure how to get A+iB into that expansion.
     
  2. jcsd
  3. Jul 13, 2012 #2
    Remember that A and B are real matrices. For the first question, try diagonalizing U first.
     
    Last edited: Jul 13, 2012
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