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Pencilvester

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- TL;DR Summary
- I’m having trouble working out how ##e^{\hat A} e^{\hat B} = e^{\hat A + \hat B} e^{[\hat A , ~ \hat B ]/2}##

I am reading Zettili’s “Quantum Mechanics: Concepts and Applications” and I am in the section on functions of operators. It starts with how ##F(\hat A)## can be Taylor expanded and gives the particular and familiar example: $$e^{a \hat A} = \sum_{n=0}^\infty \frac{a^n}{n!} \hat A^n \tag{2.109}$$ Later it says how if ##[\hat A , ~ \hat B] \neq 0## then ##e^{\hat A} e^{\hat B} \neq e^{\hat A + \hat B}##. Then here’s where I am having trouble: It says, “using (2.109) we can ascertain that $$e^{\hat A} e^{\hat B} = e^{\hat A + \hat B} e^{[\hat A , ~ \hat B ]/2},\\

e^{\hat A} \hat B e^{- \hat A} = \hat B + [\hat A , ~ \hat B] + \frac{1}{2!} [\hat A , ~ [\hat A , ~ \hat B ]] + \frac{1}{3!} [\hat A , ~ [\hat A , ~ [\hat A , ~ \hat B ]]] + \cdots”$$ I guess I am not a savvy enough mathematician to figure out how they got either of these from eq. 2.109. Can someone help me out?

e^{\hat A} \hat B e^{- \hat A} = \hat B + [\hat A , ~ \hat B] + \frac{1}{2!} [\hat A , ~ [\hat A , ~ \hat B ]] + \frac{1}{3!} [\hat A , ~ [\hat A , ~ [\hat A , ~ \hat B ]]] + \cdots”$$ I guess I am not a savvy enough mathematician to figure out how they got either of these from eq. 2.109. Can someone help me out?