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Eigenvalues of unitary operators

  1. Apr 7, 2013 #1
    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. jcsd
  3. Apr 7, 2013 #2
    Uf = λf

    denote your inner product (a,b)

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

    b.
    The second part follows from

    1. [tex] \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)}.[/tex]

    2. For commuting matrices (operators)
    [tex] e^A e^B = e^{(A+B)} . [/tex]

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

    can you fill in the rest?
     
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