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Kubo formula derivation

  1. Dec 5, 2013 #1
    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.
     

    Attached Files:

  2. jcsd
  3. Dec 5, 2013 #2

    TSny

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    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.
     
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