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Homework Help: Campbell Baker Hausdorff proof

  1. Sep 3, 2012 #1
    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


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

    [tex] \frac{dG}{d\lambda} = (A+ F)G[/tex]
    [tex] \frac{dF}{d\lambda} = [A,F][/tex]
    [tex]\frac{d^2G}{d\lambda^2} = ( [A,F] + (A+F)^2)G[/tex]
    and when doing a mclauren expansion of G I then get
    [tex] 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[/tex]

    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?
  2. jcsd
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