Summary
Chinese version here Baker-Campbell-Hausdorff Formula can be used to compute operator evolution in the Heisenberg picture: $e^X Y e^{-X}=Y+[X,Y]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots$ This formula is actually just a younger sibling of the BCH formula. Because the evolution rule of operators in the Heisenberg picture is $A\rightarrow UAU^{\dag}$, where $U$ is a unitary evolution operator. If $U$ is generated by $H$, then it becomes $A\rightarrow e^{\frac{t}{i\hbar}H}Ae^{-\frac{t}...