Proof of exp(a)exp(b) = exp(a+b)exp(1/2 [a,b])

[asantas93]

Homework Statement

a, b are operators that commute with their commutator

(1) Show that f(x) = e^xae^xb satisfies df/dx = (a + b + x[a,b])f

(2) use (1) to show that e^ae^b = e^a+be^(1/2)[a,b]

Homework Equations

[a,[b,a]] = [b,[b,a]] = 0

The Attempt at a Solution

I tried a Taylor expansion but just got a mess.

 

[Oxvillian]

Did you get part (1)?

If so, the strategy is to slip an x into the exponent of the object in (2) and then differentiate it with respect to x. You should find that the object in (2) satisfies the same differential equation, with the same boundary condition, as the object in (1). It's therefore the same object. Then at the end you just set x=1 to get the answer.

Btw this is an example of a "BCH" relation (Baker-Campbell-Hausdorff) and if you search around for that you should find loads of stuff.

 

[dextercioby]

This is not an exercise about Taylor series, it's simply a neat trick to get a useful relation between exponentials of particular bounded operators.

 

[asantas93]

Thanks, knowing the name of this sort of equation helped me find a proof.