- #1
peterprp
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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
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: