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
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
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).
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]} $