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Equation of motion and operators in the interaction picture.

  1. May 20, 2010 #1
    1. The problem statement, all variables and given/known data

    I have a question that says:
    What is the equation of motion for a general operator in the interaction picture. I.e. how does the time derivative of the operators behaves ? Show this.

    And then I have to find the time development for the annihilation and creation operator ([itex]\hat{a}[/itex] and [itex]\[{{\hat{a}}^{\dagger }}\][/itex]) in the interaction picture.


    2. Relevant equations


    3. The attempt at a solution

    The first question I THINK this is how it is supposed to be done, but I'm not sure.
    I have that:

    [tex]\[\frac{d{{A}_{I}}}{dt}=\frac{1}{i\hbar }\left[ {{A}_{I}},{{H}_{0}} \right]\][/tex]

    where:

    [tex]{{A}_{I}}={{e}^{i{{H}_{0}}t/\hbar }}{{A}_{s}}{{e}^{-i{{H}_{0}}t/\hbar }}[/tex]

    So my thought is, that I just take the derivative of [itex]{{A}_{I}}[/itex], and then I think, if my math is correct, I end up with something where I'm able to write the commutator as in the first equation. And then I get what is says. And if I'm now mistaken that is the equation of motion I need to find ?

    The second question I'm not entirely sure about how to do.
    For the Harmonic Oscillator, which is what I'm working with here, the commutator of the two operators is [itex]\left[ {{a}_{-}},{{a}_{+}} \right]=1[/itex].
    But then I don't know what my next step is.

    So if someone could help I would be grateful :)


    Regards
     
  2. jcsd
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