quantum mechanics - Coherent states of harmonic oscillator

Why are coherent states of the harmonic oscillator called coherent? Coherent in what sense? Why are these states so special/useful?

From Wikipedia:

In physics, two wave sources are perfectly coherent if they have a constant phase difference and the same frequency.

Answer

Coherent states are eigenvectors for the (bosonic) annihilator,

$$\hat a ~|\alpha\rangle = \alpha~|\alpha\rangle,$$ and if we define the position and momentum quadratures as $\hat x = \hat a^\dagger + \hat a,$ $\hat p = i \hat a^\dagger - i \hat a,$ we have $[\hat x, \hat p] = 2i$ and the dimensionless Hamiltonian $\hbar\omega ~ \hat a^\dagger \hat a = \frac12 \hbar\omega~x^2 + \frac12 \hbar\omega~p^2 + \text{const.}$ to guide us. We can immediately see that in the coherent state we have $\langle x \rangle = \alpha^* + \alpha = 2 ~\Re~{\alpha}$ whereas $\langle p \rangle = i~\alpha^* - i ~ \alpha = 2 ~\Im~\alpha,$ so the position and momentum plane is basically just the complex plane $\mathbb C$ that $\alpha$ lives on.

Now this Hamiltonian of course has an eigenbasis $\hat a^\dagger \hat a ~ |n\rangle = n ~|n\rangle$ and in terms of that basis we see a recurrence that if $|\alpha\rangle = \sum_n c_n |n\rangle$ then we can work out that $\alpha |\alpha\rangle = \hat a |\alpha\rangle$ implies

$$c_n \sqrt{n} = \alpha c_{n-1},\text { so } c_n = c_0 \frac{\alpha^n}{\sqrt{n!}}.$$ Then working out $1 = \langle \alpha|\alpha\rangle = c_0\sum_n \big(|\alpha|^2\big)^n/n! = c_0 \exp\big(|\alpha|^2\big)$ gives the normalization constant $c_0.$

However note that under this Hamiltonian, $|n\rangle \mapsto e^{-in\omega t} |n\rangle$ and therefore,

$$|\alpha\rangle \mapsto \exp\left(-|\alpha|^2\right) \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} ~ e^{-i~\omega t~n} |n\rangle,$$ which we see on the right hand side combines by normal exponent rules to form $(\alpha e^{-i\omega t})^n.$ In other words the time evolution is that $|\alpha(t)\rangle = |\alpha_0 e^{-i\omega t}\rangle,$ and our coherent state simply makes a circle on the complex plane as it evolves.

It is in this precise sense that I understand the word "coherent," it is the meaning "it stays perfectly together as it goes along its journey." It's the same way I would say "lasers are a coherent phenomenon; light by its nature wants to spread out in a $1/r^2$ law but in a laser, the different wave packets are all arranged with just the right phase differences so that they destructively interfere for the waves that are trying to get out of the beam proper, and constructively interfere in the next position of the beam."