Register to reply

Relation between commutator, unitary matrix, and hermitian exponential operator

Share this thread:
silverdiesel
#1
Jan26-12, 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.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Latrace
#2
Jul13-12, 09:11 PM
P: 9
Remember that A and B are real matrices. For the first question, try diagonalizing U first.


Register to reply

Related Discussions
A Unitary Matrix and Hermitian Matrix Linear & Abstract Algebra 2
Unitary matrix of a hermitian form matrix Calculus & Beyond Homework 3
Unitary operator/matrix Quantum Physics 6
Commutator and hermitian operator problem Advanced Physics Homework 10
Prove that Hermitian/Skew Herm/Unitary Matrix is a Normal Matrix Calculus & Beyond Homework 2