Prove formula for the product of two exponential operators

[astrocytosis]

Homework Statement

Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that

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

Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of T(s). Then use the Baker-Hausdorff lemma, and finally simply integrate your expression.

Homework Equations

[/B]
Baker-Hausdorff Lemma

e^-B A e^B = A + [B,A] + 1/2! [B,[B,A]] + 1/3! [B,[B,[B,A]]] +...

The Attempt at a Solution

I did what the hint said and took the derivative of T(s)

T'(s) = Exp(A s)Exp(B s) B + Exp(B s)Exp(A s) A

T'(s) = T(s) * (A + B)

but I am very lost as to how to proceed for here. I don't see how the Baker-Hausdorff lemma can be applied to this. I looked online for a derivation of this formula, but they all seemed more complicated than what the problem is asking for, and none of them defined a function like T(s) (that I saw). I think I must be fundamentally misunderstanding something here but I can't figure out what it is. I tried computing T(s) for the case where A, B depend on s, but that just made things more confusing.

 

[nrqed]

Homework Statement

Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that

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

Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of T(s). Then use the Baker-Hausdorff lemma, and finally simply integrate your expression.

Homework Equations

[/B]
Baker-Hausdorff Lemma

e^-B A e^B = A + [B,A] + 1/2! [B,[B,A]] + 1/3! [B,[B,[B,A]]] +...

The Attempt at a Solution

I did what the hint said and took the derivative of T(s)

T'(s) = Exp(A s)Exp(B s) B + Exp(B s)Exp(A s) A

You moved the Exp(As) A to the right of Exp(Bs) which is not allowed if A and B don't commute. Be careful to not switch the order of the A and B operators,

 

[astrocytosis]

I thought an analytic function of an operator returned a function of its eigenvalue so it wouldn't matter... but then how can I write it in terms of T(S)?