Horrifying electron gas model
I am given the Hamiltonian, in an exercise called plasmons, and where $\langle, \rangle $ denotes the expectation value.
$$ H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_{k_1,k_2,q} V_q a_{k_1+q}^{\dagger} a_{k_2+q}^\dagger a_{k_2} a_{k_1}$$
and I am supposed to write down the time-dependence equation for $\langle a_{k-Q}^{\dagger} a_k \rangle$.
Now, I know that this is the Heisenberg equation of motion which is
$$-i \hbar \partial_t \langle a_{k-Q}^{\dagger} a_k \rangle = \langle [H, a_{k-Q}^{\dagger} a_k ] \rangle $$ in that case.
I am supposed to end up with
$$- i \hbar \partial_t \langle a_{k-Q}^{\dagger} a_k \rangle = (\varepsilon_{k-Q} - \varepsilon_{k} ) \langle a_{k-Q}^{\dagger} a_k \rangle +V_Q(\langle a_{k}^{\dagger} a_k \rangle - \langle a_{k-Q}^{\dagger} a_{k-Q} \rangle ) \sum_{k_2} \langle a_{k_2-Q}^{\dagger} a_{k_2} \rangle + \sum_{q} V_q (\langle a_{k-q}^{\dagger} a_{k-q} \rangle - \langle a_{k-Q+q}^{\dagger} a_{k-Q+q} \rangle)\langle a_{k-Q}^{\dagger} a_k \rangle$$
Now I actually managed to get the first two terms, but I don't see how to the potential:
$$+V_Q(\langle a_{k}^{\dagger} a_k \rangle - \langle a_{k-Q}^{\dagger} a_{k-Q} \rangle ) \sum_{k_2} \langle a_{k_2-Q}^{\dagger} a_{k_2} \rangle + \sum_{q} V_q (\langle a_{k-q}^{\dagger} a_{k-q} \rangle - \langle a_{k-Q+q}^{\dagger} a_{k-Q+q} \rangle)\langle a_{k-Q}^{\dagger} a_k \rangle$$
We are also allowed to use Hartree Fock factorizations in the potential in order to avoid a coupling of the "calculated expectation values to higher expectation values", but I am not sure what this actually means.
From the lecture I would guess that this means something like $$\langle a_1^{\dagger} a_2^{\dagger} a_3 a_4 \rangle \sim \langle a_1^{\dagger}a_4 \rangle \langle a_2^{\dagger} a_3 \rangle - \langle a_1^{\dagger} a_3 \rangle \langle a_2^{\dagger} a_4 \rangle.$$
By the way: I have one rather simples questions about this: Does anybody know if this model assumes that $k$ is discrete or continuous?
-(Maybe you could post this in the comments).
If anything is unclear please let me know. 669