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Derivative of operator

  1. Nov 24, 2013 #1
    For the time evolution operator:

    exp(-iHt)

    How do I take the derivative of an operator like this keeping the order correct? I mean I of course know how to differentiate an exponential function, but this is the exponential of an operator.
     
  2. jcsd
  3. Nov 24, 2013 #2

    hilbert2

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    You mean differentiating that with respect to ##t##? It's just ##\frac{d}{dt}e^{-iHt}=-iHe^{-iHt}=-ie^{-iHt}H##, because ##H## commutes with ##e^{-iHt}##. (you can see this by expanding the exponential to power series and noting that ##H## commutes with any power of itself)

    If the problem were to differentiate ##e^{At}e^{Bt}##, where ##A## and ##B## are non-commuting operators, you would have to be more careful:

    ##\frac{d}{dt}\left(e^{At}e^{Bt}\right)=\left(\frac{d}{dt}e^{At}\right)e^{Bt}+e^{At}\left(\frac{d}{dt}e^{Bt}\right)=Ae^{At}e^{Bt}+e^{At}Be^{Bt}##.
     
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