# Campbell Baker Hausdorff proof

1. Sep 3, 2012

### center o bass

1. The problem statement, all variables and given/known data
I'm trying to prove the formula by taking step b) in problem 2.1 in

http://www.uio.no/studier/emner/matnat/fys/FYS4110/h12/undervisningsmateriale/problemset2.pdf

2. Relevant equations
All the relevant equations is stated in the above link.
3. The attempt at a solution
Since

$$\frac{dG}{d\lambda} = (A+ F)G$$
and
$$\frac{dF}{d\lambda} = [A,F]$$
then
$$\frac{d^2G}{d\lambda^2} = ( [A,F] + (A+F)^2)G$$
and when doing a mclauren expansion of G I then get
$$G(\lambda) = G(0) + \frac{dG}{d\lambda}|_{\lambda = 0} + \frac{\lambda^2}{2} \frac{d^2G}{d\lambda^2}|_{\lambda = 0} +\ldots = I + \lambda (A+F)G|_{\lambda = 0} + \frac{\lambda^2}{2}([A,F] + (A+F)^2)G|_{\lambda = 0} + \ldots = I + \lambda A + \lambda B + \frac{\lambda^2}{2} ([A,B] + (A+B)^2) + \ldots$$

Now I essentially got it except for the irritating (A+B)^2 term which gives more contributions to the λ^2 term. I'm frustrated in that I don't see how this vanishes.. And I persume there is no reason it should be neglected. Can someone help me out here?