Derivative of Time Evolution Operator: Exp(-iHt)

[aaaa202]

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.

 

[hilbert2]

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:

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