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

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SUMMARY

The discussion revolves around the application of Glauber's formula and the Baker–Campbell–Hausdorff (BCH) formula in quantum mechanics, specifically in the context of the equation F(B) = eAt. Participants clarify the importance of the order of operators in the commutator, noting that a negative sign arises when the order is reversed. The derivative of the exponential operator is also discussed, with the expression (eAt)' = d/dA eAt being highlighted as crucial for understanding the manipulation of these equations. The reference text for this discussion is "Quantum Mechanics" by C. Cohen-Tannoudji et al.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with operator algebra
  • Knowledge of the Baker–Campbell–Hausdorff formula
  • Basic calculus, particularly differentiation of exponential functions
NEXT STEPS
  • Study the Baker–Campbell–Hausdorff formula in detail
  • Learn about commutation relations in quantum mechanics
  • Explore the implications of operator ordering in quantum systems
  • Read "Quantum Mechanics" by C. Cohen-Tannoudji et al. for deeper insights
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Students and professionals in quantum mechanics, physicists working with operator algebra, and anyone seeking to understand the implications of Glauber's formula and the Baker–Campbell–Hausdorff formula in theoretical physics.

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Hi, I don't understand how to use 51 get 66. Can you help me with that?
 
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Make the following replacement: ##F(B) = e^{At}## and ##A=B##.
 
blue_leaf77 said:
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.
 
##(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.
 
blue_leaf77 said:
##(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.
 
No worry, it's not a big deal.
 
Which book is this from?
 
C. Cohen-Tannoudji et al. - Quantum Mechanics.
 
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