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Lawrencel2
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Homework Statement
I know that Unitary operators act similar to hermitean operators.
I want to prove that the eigenvalues of unitary operators are complex numbers of modulus 1, and that Unitary operators produce orthogonal eigenvectors.
Homework Equations
U†U = I
U-1=U†
λ = eiΦ{/SUP] (eigenvalue form)
U-1=U†
λ = eiΦ{/SUP] (eigenvalue form)
The Attempt at a Solution
For Eigenvalues being modulus 1, I wasn't sure if i started far enough back, but i have:
- I read in Sakurai that Eigenvalues of unitary operators have form λ = eiΦ{/SUP]
U |a> = λ |a>
<a| U† = λ*<a|
<a| U† U |a> = λ*λ<a|a>
<a|I|a> = |λ|2<a|a>
- But then I get to the point where I am already trying to assume Orthonormality and not showing it is complex.
- The only other way i can assume to to show this, is by asserting that the form IS: λ = eiΦ{/SUP]. Which then i can expand with Euler's Formula and use the basic Mod sq formatting. This gives me 1. At this point I don't know where to start when trying to show orthogonality...
[*]We used this argument in lecture to show orthogonality of a hermitian matrix
<a'|A†|a>- <a'|A|a> = (λ*-λ)<a'|a>
And since A is self adjoint The LHS equals zero..
Any tips on how i should be viewing this?
And since A is self adjoint The LHS equals zero..
Any tips on how i should be viewing this?