This answer has been edited to fix the grammar and punctuation errors.

[aaaa202]

Homework Statement

I am supposed to show equation 6.6 on page 97 of http://www.phys.lsu.edu/~jarrell/CO...hysics Henrik Bruus and Karsten Flensberg.pdf
I have tried to plug A and the expression for ln(t)> into 6.3 a) on page 96 but end up with a double integral after doing the linear approximation of the time evolution operator. Is my method correct and if so, how do I conjugate transpose the expression for U when given by the first order integral? Second of all I am quite confused by the notation <A>_0. Does that simply mean <A(t0)> or am I missing something?

Homework Equations

The Attempt at a Solution

I have attached my attempt.

 

[TSny]

I have tried to plug A and the expression for ln(t)> into 6.3 a) on page 96 but end up with a double integral after doing the linear approximation of the time evolution operator. Is my method correct

Yes, you have the correct approach. But I think you should be careful with the notation and use a carat ^ to denote any operator or state in the interaction picture (as introduced in section 5.3 of the notes.) Also, there is no operator $\hat{H}''$ with a double prime.

You only want to keep terms up to first order in $\hat{H}'$. So, you should not get a double integral. You will have two single integrals that you can combine into one single integral.

how do I conjugate transpose the expression for U when given by the first order integral?

It should be easy to take the Hermitian conjugate of $\hat{U}$ since $\hat{H}'(t)$ is a Hermitian operator.

Second of all I am quite confused by the notation <A>_0. Does that simply mean <A(t0)> or am I missing something?

See just below equation (6.6). The notation $\langle \rangle_0$ means an equilibrium average with respect to the Hamiltonian $H_0$. Thus $\langle A \rangle_0$ means exactly the expression defined by equation (6.1a). Note that $\langle A \rangle_0$ is time independent, so you don't need to worry about the time for this quantity.