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

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Discussion Overview

The discussion revolves around the application of Glauber's formula and the Baker–Campbell–Hausdorff formula, focusing on specific mathematical manipulations and substitutions within the context of quantum mechanics. Participants seek clarification on the steps involved in deriving results from these formulas.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses confusion about how to transition from one equation (51) to another (66) and requests assistance.
  • Another participant suggests making the substitution ##F(B) = e^{At}## and setting ##A=B## as a potential approach.
  • A participant questions the implications of the substitution, particularly regarding the appearance of the variable t and the order of the commutator, suggesting it may need to be negative.
  • Clarification is provided that the derivative of ##e^{At}## with respect to A is ##(e^{At})' = \frac{d}{dA}e^{At}##, and that the order of the commutator is indeed reversed, leading to a cancellation of negative signs.
  • Participants acknowledge the oversight regarding the order of the commutator and express gratitude for the assistance provided.
  • One participant inquires about the source of the material being discussed.
  • Another participant identifies the source as "C. Cohen-Tannoudji et al. - Quantum Mechanics."

Areas of Agreement / Disagreement

Participants generally agree on the mathematical manipulations discussed, but there is some uncertainty regarding the implications of the substitutions and the order of the commutator. The discussion remains somewhat unresolved as participants clarify their understanding without reaching a definitive conclusion.

Contextual Notes

There are limitations related to the assumptions made in the substitutions and the specific definitions of the variables involved, which have not been fully explored in the discussion.

Who May Find This Useful

This discussion may be useful for students or individuals studying quantum mechanics, particularly those interested in the mathematical foundations of Glauber's formula and the Baker–Campbell–Hausdorff formula.

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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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