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Derivative of an operator

  1. May 31, 2012 #1
    1. The problem statement, all variables and given/known data
    calculate [itex]\frac{d}{dt}e^{\hat{A}t}[/itex] where [itex]\hat{A} \neq \hat{A}(t)[/itex] in other words operator A doesn't depend explicitly on t.

    2. Relevant equations

    3. The attempt at a solution

    [itex]\frac{d}{dt}e^{\hat{A}t} = (\frac{d}{dt}(\hat{A})t + \hat{A})e^{\hat{A}t} = (\sum^{n}_{i=0}\frac{d\hat{A}}{dx_{i}}\frac{dx_{i}}{dt}t + \hat{A})e^{\hat{A}t} [/itex]

    if the xi ≠ xi(t) we get [itex]\hat{A}e^{\hat{A}t} [/itex]

    but is this correct I know how to define the derivative of an operator if it is explicitly dependent on the variable of differentiation but not in this case.
     
    Last edited: May 31, 2012
  2. jcsd
  3. May 31, 2012 #2
    First off are you sure this isn't just a partial differntiation in which case there is no problem. Otherwise this looks quite allright.
     
  4. Jun 1, 2012 #3
    Yup, there's nothing wrong with your solution.
     
  5. Jun 3, 2012 #4

    dextercioby

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    Science Advisor
    Homework Helper

    It makes a world of difference if the operator in the exponent is bounded or not. Either way, there's a strict definition of such a derivative in terms of limits which can be found in almost all books on functional analysis.
     
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