Solution to Operators Problem Using the Operator Expansion Theorem

[y35dp]

Homework Statement

Use the operator expansion theorem to show that

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

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

Homework Equations

Operator expansion theorem

Exp(A)$$\ast$$B$$\ast$$Exp(-A) = B + [A,B] (2)

The Attempt at a Solution

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

Exp(xA)$$\ast$$Exp(xB) = C(x)

differentiate wrt parameter x

C'(x) = A$$\ast$$Exp(xA)$$\ast$$Exp(xB) + Exp(xA)$$\ast$$B$$\ast$$Exp(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?

 

[arkajad]

What about writing the second term as

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

 

[y35dp]

Hero