I'm currently trying to understand the following:
Consider a quantum harmonic oscillator with a coherent state $|\alpha\rangle$. Show that $\langle E^2\rangle=\hbar\omega (|\alpha|^4+2|\alpha|^2+\frac{1}{4})$.
The solution in my notes goes as follows:
$\langle E^2 \rangle = \langle\alpha | (\hat{a}^{\dagger}\hat{a}+\frac{1}{2})^2\hbar^2\omega^2|\alpha\rangle = \hbar\omega\langle\alpha| (\hat{a}^{\dagger}(\hat{a}^{\dagger}\hat{a}+1)\hat{a}+\hat{a}^{\dagger}\hat{a}+\frac{1}{4})|\alpha\rangle=\hbar\omega\langle\alpha|(\hat{a}^{\dagger 2}\hat{a}^2+2\hat{a}^{\dagger}\hat{a}+\frac{1}{4})|\alpha\rangle=\hbar\omega(|\alpha|^4 + 2|\alpha|^2+\frac{1}{4})$
Question: Why does $\hat{a}^{\dagger 2}\hat{a}^2|\alpha\rangle=|\alpha|^4|\alpha\rangle$ hold? As far as I know, $\hat{a}$ is not normal so I'm guessing this is some property of coherent states? (Please excuse if this is a trivial question for you but I really don't know why this holds.)