# Kubo formula derivation

1. Dec 5, 2013

### aaaa202

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. The attempt at a solution
I have attached my attempt.

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2. Dec 5, 2013

### TSny

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.

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

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.