- #1
Juan Carlos
- 22
- 0
Homework Statement
I'm working on this problem:
Let [itex]\hat{U}[/itex] an unitary operator defined by:
[itex]\hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}}[/itex] with [itex]\hat{G}[/itex] hermitian. Show that [itex]\hat{U}[/itex] can be written as: [itex]\hat{U}=Exp[i\hat{K}][/itex] where [itex]\hat{K}[/itex] is hermitian.
Homework Equations
[itex]\hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}}[/itex] , [itex]\hat{U}=Exp[i\hat{K}][/itex]
The Attempt at a Solution
My attempt at a solution: I have to show who is [itex]\hat{K}=\hat{K}(\hat{G})[/itex] (as a function) so after several algebra manipulation,equating the two relevant equations I arrive to:
[itex]\hat{G}=tan(\frac{\hat{K}}{2})[/itex]
I would like to simply apply the inverse of tan, in that way:
[itex]\hat{K}=2 arctan(\hat{G})[/itex]. 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