Product of exponential of operators

[cpsinkule]

How does one show that
e^Ae^B=e^A+Be^[A,B]/2
where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products which lead nowhere.

 

[micromass]

It's false. But it's a good approximation. Here is the general formula: https://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula#The_Zassenhaus_formula