Operators problem

[y35dp]

1. The problem statement, all variables and given/known data

Use the operator expansion theorem to show that

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

when [A,B] = \lambda and \lambda is complex. Relationship (1) is a special case of the Baker-Hausdorff theorem.

2. Relevant equations

Operator expansion theorem

Exp(A)\astB\astExp(-A) = B + [A,B] (2)

3. The attempt at a solution

Take Exp(A+B) and write in terms of a complex number parameter

Exp(xA)\astExp(xB) = C(x)

differentiate wrt parameter x

C'(x) = A\astExp(xA)\astExp(xB) + Exp(xA)\astB\astExp(xB)

Now here is where I'm stuck I think the above needs to be in a similar form to (2) but I can't seem to get it to work. Are there any operator rules that can help?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

[arkajad]

What about writing the second term as

e^{xA}Be^{xB}=e^{xA}Be^{-xA}e^{xA}e^{xB}

[y35dp]

Hero