About Glauber's formula(or special Baker–Campbell–Hausdorff

[Frost_Xue]

Hi, I don't understand how to use 51 get 66. Can you help me with that?

 

[blue_leaf77]

Make the following replacement: $F(B) = e^{At}$ and $A=B$.

 

[Frost_Xue]

Make the following replacement: $F(B) = e^{At}$ and $A=B$.

ok if I replace F(B) = e^At then should it be [A,B](e^At)' ? how does the t come from? and what do you mean by A=B? also the order of commutator seems reversed shouldn't it be negative then?
Sorry I am stupid.

 

[blue_leaf77]

$(e^{At})' = \frac{d}{dA}e^{At}$.

Yes, the order is reversed on both sides of the equation. So there will be negative sign in each side and they can cancel out.

 

[Frost_Xue]

$(e^{At})' = \frac{d}{dA}e^{At}$.

Yes, the order is reversed on both sides of the equation. So there will be negative sign in each side and they can cancel out.

oh I didn't pay attention on the order of the commutator on the right hand side. Thank you so much for the help and time. And sorry for bothering you again.

 

[blue_leaf77]

No worry, it's not a big deal.

 

[annaphys]

Which book is this from?

 

[dextercioby]

C. Cohen-Tannoudji et al. - Quantum Mechanics.