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Look at the following attached picture, where they prove the coherent states are eigenfunctions of the annihiliation operators by simply proving aexp(φa^{†})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 p158159 of http://nanotheoryou.wikispaces.com/file/view/Atland+And+Simons.pdf
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 p158159 of http://nanotheoryou.wikispaces.com/file/view/Atland+And+Simons.pdf
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