- #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: