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

 

[paranormal]

Hi there,

Thank you for your question. It is important to note that operators and matrices are two different mathematical objects, although they can be related to each other. Operators are abstract mathematical objects that represent transformations or operations on a vector space, while matrices are concrete representations of these operators in a specific basis.

In this case, we are dealing with a unitary operator U, which is a type of linear operator that preserves the inner product of a vector space. This means that for any two vectors x and y, the inner product <Ux, Uy> is equal to <x,y>.

Now, to show that U can be written as a fraction of Hermitian operators, we can use the fact that any unitary operator can be written as e^{iH}, where H is a Hermitian operator. This is known as the polar decomposition of a unitary operator.

Using this, we can write U as:

U=e^{iH}=\frac{e^{iH}+e^{-iH}}{2}+\frac{e^{iH}-e^{-iH}}{2}=\frac{\mathbf{1}+iK}{\mathbf{1}-iK}

where K is the Hermitian operator H divided by i. This shows that U can indeed be written as a fraction of Hermitian operators.

I hope this helps clarify things for you. Let me know if you have any other questions.