Question about Notes on Operator Algebra

Question

This question is concerned with the last line of these notes by M. Fowler titled "Coherent States of the Simple Harmonic Oscillator."

I understand the things before that but I don't see how the last line comes about.

Thank you, George

Answer 1

You have that $e^{A+B}=e^Ae^Be^{-\frac12[A,B]}$. Exchanging the roles of $A$ and $B$ you get $$ e^Be^Ae^{-\frac12[B,A]}=e^Ae^Be^{-\frac12[A,B]}, $$ so $$ e^Be^A=e^Ae^Be^{-\frac12[A,B]}e^{\frac12[B,A]}=e^Ae^Be^{-\frac12[A,B]}e^{-\frac12[A,B]} =e^Ae^Be^{-[A,B]}. $$ This, of course, provided that the original requirement of $[A,B]$ commuting with both $A$ and $B$ holds (so that the original identities are valid).

Answer 2

So, assume we know $e^{A+B} = e^A e^B e^{-\frac12[A,B]}$, then using $A+B=B+A$, this also must hold: $$e^A e^B e^{-\frac12[A,B]} = e^{A+B} = e^{B+A} = e^B e^A e^{-\frac12[B,A]} $$