Can a Unitary Operator Be Expressed in Exponential Form?

[Juan Carlos]

Homework Statement

I'm working on this problem:

Let \hat{U} an unitary operator defined by:
\hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}} with \hat{G} hermitian. Show that \hat{U} can be written as: \hat{U}=Exp[i\hat{K}] where \hat{K} is hermitian.

Homework Equations

\hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}} , \hat{U}=Exp[i\hat{K}]

The Attempt at a Solution

My attempt at a solution: I have to show who is \hat{K}=\hat{K}(\hat{G}) (as a function) so after several algebra manipulation,equating the two relevant equations I arrive to:

\hat{G}=tan(\frac{\hat{K}}{2})

I would like to simply apply the inverse of tan, in that way:

\hat{K}=2 arctan(\hat{G}). I do not know if it is arctan defined for an operator, if it is, I think that is via taylor Series.

Some help please

 

[ShayanJ]

Try taylor-expanding \hat U w.r.t. \hat G!

 

[Juan Carlos]

I've tried, but i think this is not the way because when you expand both sides there is no clear relation between G and K.

 

[DEvens]

Don't forget those important words: Hermition and unitary. You probably need to use those.

You can get there in a couple ways. Expansion by a Taylor series is one way. Be careful that you don't depend things commuting if they don't.

 

[Juan Carlos]

I agree. And I used those properties. Thanks

 

[Quantum Braket]

You may be over-thinking this problem.

Simply equate Exp[i K] = (I +iG)/(I-iG) and solve by taking the log of both sides. (Note issues of phase in your equality after taking the log etc.)

Now you have a form of K in terms of G, which you must show is Hermitian. This is where a series expansion comes into play. Expand your K as a series expansion, and show that by daggering (taking the Hermitian conjugate of) each term in the expansion you leave the expansion unchanged, thus showing that K is Hermitian, and showing the solution.