# Expectation value in Heisenberg picture: creation and annihilation

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Bruno Cardin
TL;DR Summary
Hi. I posted this in homework, but it isn't really homework. I'm just someone who has spent 2 years in classical general relativity and find myself lost trying to re-do my final exam.
So, I have a hamiltonian for screening effect, written like:

$$H=\sum_{k}^{}\epsilon_{k}c_{k}^{\dagger}c_{k}+ \frac{1}{\Omega}\sum_{k,q}^{}V(q,t)c_{k+q}^{\dagger}c_{k}$$

And I have to find an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}##.

I wrote this, initially

$$i\hbar\frac{d}{dt}c_{k-Q}^{\dagger}(t)c_{k}(t)= [c_{k-Q}^{\dagger}c_{k} , H]$$

as the time evolution equation for the operator in the Heisenberg picture. What I procceed to do is to plug a bra in the left <phi| and a ket in the right |phi> , with phi being an energy eigenstate, and then start raising and lowering energy levels since the operators ##c_{k}## are the anihilation operators (and with the dagger they switch to creation operators). But the result I have to get to, according to the exam's solution is:

$$i\hbar\frac{d}{dt} < c_{k-Q}^{\dagger}(t)c_{k}(t) > = (\epsilon_{k}-\epsilon_{Q-k})< c_{k-Q}^{\dagger}c_{k}>+\frac{1}{\Omega}\sum_{k}^{}V(q,t)[<c_{k-Q}^{\dagger}c_{k-q} >- <c_{k+q-Q}^{\dagger}c_{k}>]$$

which has the expression of V in it.. this means I have to "open" the hamiltonian. I'm so rusty that that didn't even cross my mind. I don't get it. Could anyone help? Thank you in advance.

• Delta2

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