Look at the following attached picture, where they prove the coherent states are eigenfunctions of the annihiliation operators by simply proving aexp(φa(adsbygoogle = window.adsbygoogle || []).push({}); ^{†})l0> = φexp(φa^{†})l0>. I understand the proof but does that also prove that:

a_{i}exp(Σφ_{i}a_{i}^{†})l0> = φ_{i}exp(Σφ_{i}a_{i}^{†})l0> ?

I can see that it would if you can use that:

exp(A+B+...) = exp(A)exp(B)exp(C)...

but does that identity hold for operators and how do you see that?

Because if you just taylor expand the operator sum you get cross terms between i and j and I'm not sure what to do with these.

Edit: the picture might be a bit too small, so you can also just look at p158-159 of http://nanotheoryou.wikispaces.com/file/view/Atland+And+Simons.pdf

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# Coherent states

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