I'm trying to prove that $\langle {S_{z}} \rangle =S \cos \theta$ using the notion of spin coherent states, so $|\theta,\phi \rangle = e ^{ -i \phi S_{z} } e ^{ -i \theta S_{y} } |S \rangle$, and i know that $\langle S|S_{y}|S \rangle$ = $\langle S|S_{x}|S \rangle $= 0 and $\langle S|S_{z}|S \rangle=0$.

But I'm stuck in the very beginning

$\langle \theta,\phi| e ^{ i \phi S_{y} } e ^{ i \theta S_{z} } \theta S_{z} e ^{ -i \phi S_{z} } e ^{ -i \theta S_{y} } | \theta,\phi \rangle = \langle S| e ^{ i \phi S_{y} } e ^{ i \theta S_{z} } \theta S_{z} e ^{ -i \phi S_{z} } e ^{ -i \theta S_{y} } |S \rangle$

I have the BCH formula $ e ^{ -X } Ye ^{ X } = Y + [X,Y].... $ but i don't see how to use it with this $i \theta $

  1. You forgot to interchange the order of rotations in your bra vector, i.e. if $\vert\theta,\phi\rangle=R_z(\phi)R_y(\theta)\vert SS\rangle$ then $\langle \theta,\phi\vert=\langle SS\vert R_y^{-1}(\theta) R_z^{-1}(\phi)$.
  2. You have a typo and there is no $\theta S_z$, just $S_z$, i.e you need to compute $$ \langle SS\vert R_y^{-1}(\theta)R_z^{-1}(\phi)S_zR_z(\phi)R_y(\theta)\vert SS\rangle. $$

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