# Eigenvalues of unitary operators

1. Apr 7, 2013

### black_hole

1. The problem statement, all variables and given/known data

We only briefly mentioned this in class and now its on our problem set...

Show that all eigenvalues i of a Unitary operator are pure phases.
Suppose M is a Hermitian operator. Show that e^iM is a Unitary operator.

2. Relevant equations

3. The attempt at a solution

Uf = λf where is is an eigenfunction, U dagger = U inverse
multiply by either maybe...

2. Apr 7, 2013

### qbert

Uf = λf

a.
$(f, U^\dagger U f) = ?$
calculate this two ways (in terms of λ and λ*)
$(f, U^\dagger U f ) = (Uf, Uf) = ?$
$(f, U^\dagger U f ) = (f, f)$ [since $U^\dagger = U^{-1}$ ]
so what does this say about λ and λ*?

b.
The second part follows from

1. $$\left( e^A \right)^\dagger = \left( 1 + A + \frac{1}{2!}A^2 + \cdots \right)^\dagger = \left( 1 + A^\dagger + \frac{1}{2!}(A^\dagger)^2 + \cdots \right) = e^{(A^\dagger)}.$$

2. For commuting matrices (operators)
$$e^A e^B = e^{(A+B)} .$$

now you need to show show U = e^(iM) satisfies UU^(dagger) = 1.

can you fill in the rest?