Expectation value in Heisenberg picture: creation and annihilation

In summary, the conversation discusses finding an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}## in the Heisenberg picture. The desired equation involves the Hamiltonian and includes the expression for V. More information is needed about the (anti)commutation relations satisfied by the ##c_k##'s and the definition of the vacuum in order to fully understand the solution.
  • #1
Bruno Cardin
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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.
 
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  • #2
This needs a bit more information. What (anti)commutation relations are satisfied by the ##c_k##'s ? Is your vacuum annihilated by ##c_k##? If not, then how is your vacuum defined?
 
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Related to Expectation value in Heisenberg picture: creation and annihilation

1. What is the Heisenberg picture in quantum mechanics?

The Heisenberg picture is one of the two commonly used formulations of quantum mechanics, along with the Schrödinger picture. In this picture, the operators representing physical observables, such as position and momentum, are time-independent, while the state of the system evolves with time.

2. What is the expectation value in the Heisenberg picture?

The expectation value in the Heisenberg picture is the average value of an observable in a given quantum state. It is calculated by taking the inner product of the state vector with the operator representing the observable.

3. What is the role of creation and annihilation operators in the Heisenberg picture?

In the Heisenberg picture, creation and annihilation operators are used to represent the creation and annihilation of particles in a given quantum state. These operators act on the state vector and produce a new state with a different number of particles.

4. How is the expectation value of a creation or annihilation operator calculated in the Heisenberg picture?

The expectation value of a creation or annihilation operator in the Heisenberg picture is calculated by taking the inner product of the state vector with the operator and its Hermitian conjugate. This gives the average number of particles created or annihilated in the given state.

5. How does the Heisenberg uncertainty principle relate to the expectation value in the Heisenberg picture?

The Heisenberg uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. In the Heisenberg picture, the expectation value of the position and momentum operators can be used to calculate the uncertainties in these quantities, thus demonstrating the uncertainty principle.

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