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Time dependent hamiltonian

  1. Dec 8, 2013 #1
    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.
     

    Attached Files:

  2. jcsd
  3. Dec 11, 2013 #2

    BruceW

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    Homework Helper

    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?)
     
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