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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

λ = e

^{†}U =**I**U

^{-1}=U^{†}λ = e

^{iΦ{/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. 1 = |λ|2 where <a|a> = 1 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?}