Baker-Campbell-Hausdorff formula

For two possibly noncommuting operators $A,B$ , BCH states

\begin{align*} e^{A + B} = e^A e^B e^{-[A,B]/2} e^{-([A,[A,B]] + [B,[B,A]])/12} \cdots. \end{align*}

In most cases we consider, $[A,B]$ is a constant so we stop after the third term. Here the operator exponential is defined in terms of its power series.