# Time dependent hamiltonian

1. Dec 8, 2013

### aaaa202

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.

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• ###### Hamiltonian.png
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2. Dec 11, 2013

### BruceW

I think this problem is just meant to show how you can go through the calculation to explicitly show that H is not time dependent in this case. And you are not meant to assume $a_\nu(t) = \exp(-i\varepsilon_\nu t) a_\nu$ from the start. (By the way, I'm guessing the stuff under the red line is the solution to the problem, right?)