The general Baker-Hausdorff formula is :
\begin{equation} e^{(\hat{A} +\hat{B})t} = e^{t\hat{A}} e^{t\hat{B}} e^{-\frac{t^2}{2}[\hat{A},\hat{B}]} e^{t\hat{B}} e^{\frac{t^3}{6}(2[\hat{B},[\hat{A},\hat{B}]] + [\hat{A},[\hat{A},\hat{B}]])} ... \end{equation}
It should be easy to find the Trotter decomposition:
\begin{equation} e^{(\hat{A} +\hat{B})t} = lim_{n \rightarrow \infty} \left( e^{\frac{\hat{A}t}{n}} e^{\frac{\hat{B}t}{n}} \right)^n \end{equation}
I tried to express the exponential as it's Taylor expansion but I get nowhere. Do you have any idea how to start the derivation. Thanks !
You don't have to use the Baker-Hausdorff formula, just use the definition of an exponentiated operator $e^H = lim_{n \rightarrow \infty} \left( 1+\frac{H}{n} \right)^n $
\begin{equation} e^{(\hat{A} +\hat{B})t} = lim_{n \rightarrow \infty} \left( 1+\frac{(\hat{A}+\hat{B})t}{n} \right)^n \end{equation}
\begin{equation} = lim_{n \rightarrow \infty} \left( (1+\frac{\hat{A}t}{n}) (1+\frac{\hat{B}t}{n})\right)^n \end{equation}
\begin{equation} = lim_{n \rightarrow \infty} \left( e^{\frac{\hat{A}t}{n}} e^{\frac{\hat{B}t}{n}} \right)^n \end{equation}