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Unitary operator U ?

  1. Jan 30, 2012 #1
    Hi,

    I have to show that a unitary operator [itex] U [/itex] can be written as

    [tex]
    U=\frac{\mathbf{1}+iK}{\mathbf{1}-iK}
    [/tex]

    where [itex] K [/itex] is a Hermitian operator.

    Now how could you possibly have a fraction of operators if those can be represented by matrices? Not sure what to do here.
     
  2. jcsd
  3. Jan 30, 2012 #2
    Also doesn't the equality fail for when U=-1 (the negative of a 1 matrix)? No matter what K is I can't see how it would hold since we'd basically end up with -1 + iK = 1 + iK
     
  4. Jan 30, 2012 #3
    Hi erogard,

    First think about what an Unitary operator is by definition.

    with just a quick look at wikipedia you'll be able to see that
    \begin{equation}
    U^{*}U = UU^* = I
    \end{equation}
    so given that K is hermitian
    \begin{equation}
    K^*=K
    \end{equation}
    and that
    \begin{equation}
    I^*I = II^*=I
    \end{equation}
    just see if the above identity holds,

    Nik
     
  5. Jan 30, 2012 #4
    Stuff written like that generally just means inverse eg
    [itex]A=\frac{1}{\mathbf{(1-B)}}=\mathbf{(1-B)}^{-1}[/itex]

    So you just want to show that U us unitary eg [itex]UU^{\dagger}=U^{\dagger}U=I[/itex] (where[itex]\dagger[/itex] is the hermitian conjugate operation)
    Which is a pretty simple operation
     
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