The evolution operator in the interaction picture is defined as


Where $H_S=H_0+V$

I am trying to find the evolution of the operator $U_I$. In literature it is given by:


However when I take the derivative of the operator:

$\begin{align}i\dfrac{dU_I}{dt}&=i^2H_0e^{iH_0t}e^{-iH_St}e^{-iH_0t}+ie^{iH_0t}(-iH_s)e^{-iH_St}e^{-iH_0t}+ie^{iH_0t}e^{-iH_St}(-iH_0)e^{-iH_0t}\\ &=-H_0e^{iH_0t}e^{-iH_St}e^{-iH_0t}+e^{iH_0t}(H_0+V)e^{-iH_St}e^{-iH_0t}+e^{iH_0t}e^{-iH_st}H_0e^{-iH_0t}\end{align}$

Since $H_0$ and $e^{iH_0t}$ commute, the first two terms cancel and we are left with

$\begin{align} i\frac{dU_I}{dt}&=e^{iH_0t}Ve^{-iH_St}e^{-iH_0t}+e^{iH_0t}e^{-iH_st}H_0e^{-iH_0t}\\&=e^{iH_0t}Ve^{-iH_0t}e^{iH_0t}e^{-iH_St}e^{-iH_0t}+e^{iH_0t}e^{-iH_st}H_0e^{-iH_0t}\\&=V_IU_I+e^{iH_0t}e^{-iH_st}H_0e^{-iH_0t} \end{align}$

In the second line I inserted $1=e^{-iH_0t}e^{iH_0t}$, so I arrive at the correct term, $V_IU_I$ but + some extra term. What is going wrong here?


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