Showing Unitary Operator U is a Fraction of Hermitian Operators

[erogard]

Hi,

I have to show that a unitary operator $U$ can be written as

$$U=\frac{\mathbf{1}+iK}{\mathbf{1}-iK}$$

where $K$ 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.

 

[erogard]

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

 

[nik86]

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

 

[genericusrnme]

Hi,

I have to show that a unitary operator $U$ can be written as

$$U=\frac{\mathbf{1}+iK}{\mathbf{1}-iK}$$

where $K$ 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.

Stuff written like that generally just means inverse eg
$A=\frac{1}{\mathbf{(1-B)}}=\mathbf{(1-B)}^{-1}$

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