relation between commutator, unitary matrix, and hermitian exponential operatorby silverdiesel Tags: commutator, exponential, hermitian, matrix, operator, relation, unitary 

#1
Jan2612, 05:04 PM

P: 65

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
Jul1312, 09:11 PM

P: 9

Remember that A and B are real matrices. For the first question, try diagonalizing U first.



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