Solution
= Solution
Orthogonality of the energy eigenstates gives
$$
\langle z|z\rangle=e^{-|z|^2}\sum_{n=0}^{\infty}\frac{|z|^{2n}}{n!}=1.
$$
Using the <creation and annihilation operators>[lowering-operator] action from part (a) and shifting the summation index,
$$
A|z\rangle=e^{-|z|^2/2}\sum_{n\geq1}
\frac{z^n\sqrt n}{\sqrt{n!}}|n-1\rangle
=z|z\rangle.
$$
\b[Thus this <coherent state> is a normalized eigenstate of $A$ with eigenvalue $z$.]