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3.9: Appendix- Some Exponential Operator Algebra
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Fowler)/03%3A_Mostly_1-D_Quantum_Mechanics/3.09%3A_Appendix-_Some_Exponential_Operator_Algebra
\[\begin{align} [A,e^{\lambda B}] &= \left[A,1+\lambda B+ \left(\dfrac{\lambda^2}{2!} \right)B^2+ \left(\dfrac{\lambda^3}{3!}\right)B^3+\dots\right] \\[5pt] &= \lambda c+ \left(\dfrac{\lambda^2}{2!}\r... $\begin{align} [A,e^{\lambda B}] &= \left[A,1+\lambda B+ \left(\dfrac{\lambda^2}{2!} \right)B^2+ \left(\dfrac{\lambda^3}{3!}\right)B^3+\dots\right] \\[5pt] &= \lambda c+ \left(\dfrac{\lambda^2}{2!}\right)2Bc+ \left(\dfrac{\lambda^3}{3!}\right)3B^2c+\dots \\[5pt] &= \lambda ce^{\lambda B}. \label{3.6.51} \end{align}$ $Ae^{\lambda B}|a\rangle=e^{\lambda B}A|a\rangle+\lambda ce^{\lambda B}|a\rangle=(a+\lambda c)|a\rangle. \label{3.6.53}$

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