Product of two exponentials of different 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.

 

[DrDu]

Search for ."Baker-Campbell-Hausdorf" formula.

 

[andresB]

Do note that the formula only works if A and B commute with [A,B]. I think ther eis a proof in the Cohen-Tannoudji book

 

[moss]

Its in Sakurai's Modern Quantum Mechanics and indeed its Baker-Campbell-Hausdorf formula.
If i remember good then there is a proof or at least a partial proof in Sakurai.