Consider the attached exercise. I am having some trouble understanding exactly what time dependent hamiltonian it refers to. Because from the equation it refers to it seems that the hamiltonian is by definition time independent. Am I to assume that the H diagonal is a time independent hamiltonian which is perturbed by a time dependent potential V(t) or am I to assume that it wants the time evolution in the Heissenberg picture where for an operator: A(t) = exp(iHt)Aexp(-iHt) (1) Because then I can just expand the operators a_dagger, a in time and recover (1). But that would hold also for a non diagonalized hamiltonian. Can anyone explain to me the difference between H(t) and H in this case and why it is not always just given by the formula I am to prove for a diagonalized H - it seems intuitive for me that the time evolution of H should follow the time evolution of a_dagger and a.