An identity for functions of operators

[Identity]

Is there an easy way to prove the identities:

$$e^{\hat{A}}e^{\hat{B}}=e^{\hat{A}+\hat{B}}e^{[\hat{A},\hat{B}]/2}$$ and

$$e^{\hat{A}}\hat{B}e^{-\hat{A}}=\hat{B}+[\hat{A},\hat{B}]+\frac{1}{2!}[\hat{A},[\hat{A},\hat{B}]]+\frac{1}{3!}[\hat{A},[\hat{A},[\hat{A},\hat{B}]]]+...$$In Zettili they give that: $$e^{\hat{A}}=\sum_{n=0}^\infty \frac{1}{n!}\hat{A}^n$$

But I have no idea how I would expand a product of series to prove the identity... the algebra is horrible. Is there an easier way to go about proving it?
Thx

 

[arkajad]

Your first equation is, for general A,B, wrong.

As for the second one set

$$f(t)=e^{At}Be^{-At}$$

calculate derivatives $f^{(n)}(0)$ (calculate first three, and you will see how it goes), and then use Taylor's expansion:

$$f(1)=f(0)+f'(0)+\frac{1}{2!}f''(0)+...$$

 

[Identity]

Thanks :)