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If the laser mode is in a coherent state ##|\alpha \exp(-i\omega_L t)\rangle## with ##\alpha## being real, Then the average value

##\langle\alpha \exp(-i\omega_L t)|E(R)|\alpha \exp(-i\omega_L t)\rangle=\mathscr{E}_0 \cos(\omega_L t)##

with

##\mathscr{E}_0=2 \epsilon \sqrt{\frac{\hbar \omega_L}{2 \epsilon V}} \sqrt{\langle N \rangle}##

##\langle N \rangle = \alpha^2##

and

##E(R)=\sqrt{\frac{\hbar \omega_L}{2 \epsilon_0 V}}\epsilon_L(a+a^{\dagger})##

I do not understand it at all. I do know some thing about the coherent state.

such as

##a |\alpha\rangle = \alpha |\alpha\rangle##

and

##|\alpha\rangle = e^{-|\alpha|^2/2} \sum_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle##

But I don't understand what's going on here.

(1) Why the coherent state is ##|\alpha \exp(-i\omega_L t)\rangle##?

(2) ##\langle N \rangle = \alpha^2## means ##\langle \alpha \exp(-i\omega_L t)| N |\alpha \exp(-i\omega_L t)\rangle = \alpha^2##?

(3) Where does the ##\cos(\omega_L t)## come from in the first equation?