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I have a few questions related to coherent states. I use this source https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf.

Using standart inner product $\langle\cdot|\cdot\rangle$ in Hilbert space $L^2(\mathbb{R})$ we can find the value of coherent states and Fock states at a point, if I understand correctly, as shown in $(5.126)-(5.128)$ by the link above. But directly calculating, I get a different result $$\langle x|m\rangle=\exp^{-|x|^2/2}\sum^\infty_{n=0}\frac{x^n}{\sqrt{n!}}\,\langle n|m\rangle=\exp^{-|x|^2/2}\frac{x^m}{\sqrt{m!}}.$$ Where am I mistaken? This is the first question.

It is also interesting to me, Is the set of coherent states dense in $L^2(\mathbb{R})$? Or maybe it forms some Hilbert subspace? Is it possible to build a reproducing kernal in it?

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  • $\begingroup$ You should consult the canonical textbook by Perelomov on coherent states for your questions. $\endgroup$ – ZeroTheHero Feb 4 at 14:40
  • $\begingroup$ Note that (126) and (127) are NOT coherent states, but the standard h.o. wavefunctions in the position representation. $\endgroup$ – ZeroTheHero Feb 4 at 14:43
  • $\begingroup$ This book? "Generalized coherent states and some of their applications A M Perelomov" $\endgroup$ – Иван Петров Feb 4 at 15:54
  • $\begingroup$ That is, it is not a inner product between the coherent state and wavefunction in 126-127? $\endgroup$ – Иван Петров Feb 4 at 16:04

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