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Current density free electron gas

  1. Jun 26, 2015 #1
    Hello,
    I am studying transport in the free electron gas model and I don't understand how to compute the average of the electron density current.
    We are given the hamiltonian
    ## H=\int \psi^\dagger(r,t)(-\frac{\hbar^2\nabla^2}{2m}+e\vec{E}\cdot\vec{r})\psi(r,t)##
    where the ##psi## operator is the solution of the Heisenberg equation for field operators
    ## [H,\psi]=-i\hbar\frac{d\psi(r,t)}{dt} ##,
    namely
    ## \psi(r,t) = \frac{1}{\sqrt{V}}\sum_k e^{i(\vec{k}-e\vec{E}t/\hbar)\cdot\vec{r}} e^{\int_0^t \epsilon(k-iEt'/\hbar)dt'}c_k##
    where
    ## \epsilon(k)=\frac{\hbar^2 k^2}{2m}##
    and I need to compute the average of
    ## J(r,t)=\frac{ie\hbar}{2m}[\psi^\dagger(r,t) \nabla \psi(r,t) - \nabla\psi^\dagger(r,t) \psi(r,t)]##
    Now, the result is Drude's one:
    ##n e^2 \tau/m ##
    Could someone please give me some hint on how to sketch out this calculation?
    I know that I have to take the average over k weighting via the Fermi distribution and I also have to average over collision times, namely
    ## \int e^{-t/\tau}/\tau (...) dt ##
    but why does the dependence on position disappear?
    Thank you in advance
     
    Last edited: Jun 26, 2015
  2. jcsd
  3. Jul 1, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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