Exponential of Operators

[yurell]

I didn't know whether to post this in 'School' or here — I settled for here since it's a science question and I'm just studying for the exam, and this has the best chance of someone spotting it.
Essentially, I'm trying to prove that:

Exp(A+B)=Exp(A)Exp(B)Exp(-(1/2)[A,B])

Where A and B are operators, and [A,B] is the commutator AB-BA.
I obviously need to expand is as the infinite sum Exp(x)=Sum[x^n/n!,{n,0,inf}], but I haven't been able to do anything with that, so ... halp please!

[SU3SU2U1]

Does the commutator commute with A and B?

Anyway, I'd suggest working with the log of the equation. Also, in case you are unaware, you are essentially looking at a Campbell-Baker-Hausdorff formula.

[yurell]

Oh, right, forgot to add A and B don't commute, but [A,[A,B]] = [B,[A,B]] = 0

[Charlie!]

If A and B could commute, sum of (A+B)^n / n! would look exactly like e^A e^B. If A and B can't commute, where do the commutators come in? Can you turn that into the exponential?

[Yakk]

I'd be tempted by Charlie!'s approach. In fact, I'd be tempted to find a proof that e^(x+y) = e^x e^y proven via the power series that assumes x and y commute, then see if I could modify the proof to extract out (xy-yx) terms.

I'm relatively sure if x and y commute, then e^x and e^y commute, but not absolutely so.