Something that always confused me when first hearing about second quantization were the dependencies of the creation an annihilation operators.

On the one hand I have seen expressions such as $$ \hat{H} = \sum_k \epsilon_k\hat{a}^\dagger_k\hat{a}_k $$ which just represent the static energy of a system and on the other hand one can describe electron-electron scattering using the hamiltonian $$ \hat{H} = \sum_{k,k',q}V_q \hat{a}^\dagger_{k+q}\hat{a}^\dagger_{k'-q} \hat{a}_{k}\hat{a}_{k'}$$ which seems to be a time dependent process. Furthermore, I am confused about the picture these operators are used in since one can "always" calculate there equation of motion using the Heisenberg equation of motion (for example for operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$) $$ \frac{\partial}{\partial t} \hat{a}_{k}^\dagger\hat{a}_{k'} = \frac{i}{\hbar} \left[\hat{H}, \hat{a}_{k}^\dagger\hat{a}_{k'}\right]$$ I cannot wrap my head around, why this time dependence seems to arise from a seemingly "static" hamiltonian.

More confusion arises from the fact that operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$ seem to be "instantaneous", but can lead to complicated behaviour of the system (e.g. oszillations in driven two-level-systems).

Something that always confused me when first hearing about second quantization were the dependencies of the creation and annihilation operators.

On the one hand I have seen expressions such as $$ \hat{H} = \sum_k \epsilon_k\hat{a}^\dagger_k\hat{a}_k $$ which just represent the static energy of a system and on the other hand one can describe electron-electron scattering using the hamiltonian $$ \hat{H} = \sum_{k,k',q}V_q \hat{a}^\dagger_{k+q}\hat{a}^\dagger_{k'-q} \hat{a}_{k}\hat{a}_{k'}$$ which seems to be a time dependent process. Furthermore, I am confused about the picture these operators are used in since one can "always" calculate there equation of motion using the Heisenberg equation of motion (for example for operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$) $$ \frac{\partial}{\partial t} \hat{a}_{k}^\dagger\hat{a}_{k'} = \frac{i}{\hbar} \left[\hat{H}, \hat{a}_{k}^\dagger\hat{a}_{k'}\right]$$ I cannot wrap my head around, why this time dependence seems to arise from a seemingly "static" hamiltonian.

More confusion arises from the fact that operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$ seem to be "instantaneous", but can lead to complicated behaviour of the system (e.g. oszillations in driven two-level-systems).

Something that always confused me when first hearing about second quantization were the dependencies of the creation an annihilation operators.

On the one hand I have seen expressions such as $$ \hat{H} = \sum_k \epsilon_k\hat{a}^\dagger_k\hat{a}_k $$ which just represent the static energy of a system and on the other hand one can describe electron-electron scattering using the hamiltonian $$ \hat{H} = \sum_{k,k',q}V_q \hat{a}^\dagger_{k+q}\hat{a}^\dagger_{k'-q} \hat{a}_{k}\hat{a}_{k'}$$ which seems to be a time dependent process. Furthermore, I am confused about the picture these operators are used in since one can "always" calculate there equation of motion using the Heisenberg equation of motion (for example for operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$) $$ \frac{\partial}{\partial t} \hat{a}_{k}^\dagger\hat{a}_{k'} = \frac{i}{\hbar} \left[\hat{H}, \hat{a}_{k}^\dagger\hat{a}_{k'}\right]$$ I cannot wrap my head around, why this time dependence seems to arise from a seemingly "static" hamiltonian.

Something that always confused me when first hearing about second quantization were the dependencies of the creation an annihilation operators.

On the one hand I have seen expressions such as $$ \hat{H} = \sum_k \epsilon_k\hat{a}^\dagger_k\hat{a}_k $$ which just represent the static energy of a system and on the other hand one can describe electron-electron scattering using the hamiltonian $$ \hat{H} = \sum_{k,k',q}V_q \hat{a}^\dagger_{k+q}\hat{a}^\dagger_{k'-q} \hat{a}_{k}\hat{a}_{k'}$$ which seems to be a time dependent process. Furthermore, I am confused about the picture these operators are used in since one can "always" calculate there equation of motion using the Heisenberg equation of motion (for example for operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$) $$ \frac{\partial}{\partial t} \hat{a}_{k}^\dagger\hat{a}_{k'} = \frac{i}{\hbar} \left[\hat{H}, \hat{a}_{k}^\dagger\hat{a}_{k'}\right]$$ I cannot wrap my head around, why this time dependence seems to arise from a seemingly "static" hamiltonian.

More confusion arises from the fact that operator combinations such as $\hat{a}_{k}^\dagger\hat{a}_{k'}$ seem to be "instantaneous", but can lead to complicated behaviour of the system (e.g. oszillations in driven two-level-systems).