Equation of motion for space and momentum of a $\textbf{coherent state}$ Given the coherent states
$$| \alpha \rangle\, e^{\textstyle -|\alpha|^2/2}\,\sum_{n = 0}^\infty \dfrac{\alpha^n}{\sqrt{n!}}\,|n\rangle$$
that satisfy the eigenvalue-equation: $A|\alpha\rangle=\alpha\,|\alpha\rangle$
where $A$ is the annihilation operator defined as $A|n\rangle = \sqrt{n}\,|n-1\rangle$
And knowing relations to position and momentum-operator:
$X = \sqrt{\dfrac{\hbar}{2\,m\,\omega_0}}\,(A+A^\dagger)$ and $P = \dfrac{1}{i}\,\sqrt{\dfrac{\hbar\,m\,\omega_0}{2}}(A-A^\dagger)$
My task is to derive equations of motion for the expectation value of $X$ and $P$ of a coherent state.
Therfore I am supposed to use the Ehrenfest Theorem:
$\dfrac{\mathrm{d}}{\mathrm{dt}}\,\langle X\rangle = \dfrac{1}{i\,h}\,\langle[X, H]\rangle + \langle\partial_tX\rangle$
getting equations $\langle X(t)\rangle$ and $\langle Y(t)\rangle$ to initial conditions $\langle X(0)\rangle = \langle\alpha|X|\alpha\rangle$ and $P(0)\rangle = \langle\alpha|P|\alpha\rangle$
However I don't know what "equation of motion for $\langle X\rangle $ and $\langle Y\rangle$ of a coherent state" does mean at all?
How coherent states are connected to this task actually?
I'd just need some kind of hint.
 A: Given any sufficiently regular time dependent quantity (or set of quantities) $\vec u_0(t)$, it is possible to interpret the dynamics as a solution to differential (or integral, or integro-differential) equations by applying the fundamental theorem of calculus, and comparing the derivatives or indefinite integrals of $\vec u_0(t)$ or functions (e.g. arithmetic expressions) thereof.  For example, a given orbit or trajectory $\vec u_0(t)$ might be thought of (somewhat simply) as the solution to the differential equation $\vec u'(t) = \vec f(t)$, where $\vec f(t) = \vec u_0'(t)$.  It could be that $\vec f(t)$ bears a close resemblance to $\vec u_0(t)$ itself, or some multivariable polynomial of it: for example, $\vec u_0'(t)$ might equal $\vec{\vec M} \vec u_0(t)+\vec \eta(t)$, where $\vec{\vec M}$ is a matrix and $\vec \eta(t)$ is a "noise" term with relatively small magnitude.  It could also be that $\eta(t)$ can be approximated further in terms of a bilinear of $\vec u_0(t)$: e.g. $\vec \eta(t) = \vec{\vec{\vec{\Gamma}}}(u_0(t), u_0(t)) + \vec\epsilon(t)$, where $\vec\epsilon(t)$ is even smaller than $\eta(t)$.  This process can be continued, incorporating or redefining (adding components to) $u_0(t)$ to include derivatives of itself as needed (increasing the order of the resulting differential equation.)  At the end of the day, you would end up with a (hopefully relatively simple or at least causal) differential equation with a small error term, but you might (justifiably) worry about its reliability or predictive power.  After all, coming up with a differential equation based on just one trajectory $u_0(t)$ is sort of like fitting a statistical model to just one data point.  If the trajectory $u_0(t)$ is relatively smooth or simple (e.g. undergoing a simple oscillation or gradual exponential decay) then you can be reasonably confident in the quality of your model (sort of like how you can be reasonably confident that you just met a major celebrity in New York City if you meet someone who looks and talks conspicuously like Donald Glover, even though there is a small but non-insignificant probability that the person you thought was Donald Glover was in fact a gifted Donald Glover impersonator), but less so if the trajectory $u_0(t)$ is complicated or chaotic.
Hence, in general you might hope to be given an ensemble of orbits, $\vec u_i(t)$, $1\leq i\leq N$ (sort of like an "ensemble" of encounters with Donald Glover) in order to really understand the heart or essence of the underlying invariant, unchanging laws of motion.  In your example, the ensemble would be furnished by samples over the initial conditions, $(\langle X_0\rangle, \langle P_0\rangle) = (x_i, p_i)$, $1\leq i\leq N$, say (or at a purely theoretical level, the samples could cover a continuum.)  Given such an ensemble, you would consider differential equations that produce orbits that most closely match those of the entire set uniformly. Even then, however, there can be inherent uncertainties or ambiguities (there may be two or more "best fit" candidates for the entire ensemble), which are usually a sign that the system is potentially mathematically interesting (e.g. lives on some kind of weird variety in joint $(k,x)$-space.)
If you are given a set of differential equations that induce the differential equations that you are interested in, for example a matrix differential equation for one or more matrices $\vec A(t)$, of the form
\begin{align*}
\vec A' = \vec f(\vec A)
\end{align*}
where $\vec f$ is some function of the matrix elements of $A$ and you are asked to determine a differential equation for some induced quantity, like a linear function of $A$ (e.g. $u^T A v$, where $u$ and $v$ are both column vectors with the same dimension as $A$), then things can be a bit more complicated depending on the function $\vec f$ and the induced quantity.  That is, there isn't always a simpler way to describe the dynamics of a (simpler) induced quantity than the original equation.  An example of this phenomenon is the dynamics of non-dissipative or weakly-dissipative open quantum systems, where the full (coherent, system-bath) density matrix can be thought of as a quantum generalization of the classical idea of a "memory kernel" or "memory matrix" from non-Markovian stochastic processes, but it is extremely difficult to simplify the equations of motion without invoking some kind of Markovian (or rapid dissipation) approximation and preserving the (uniquely quantum) memory effects.
However, in situations where $\vec f$ is linear function of $\vec A$, and the induced quantity is linear, then linearity can be invoked to obtain a linear equation for the induced quantities (possibly with a few "extra" variables) from the original differential equation: for example, if we're interested in the dynamics of $u^T_\alpha A(t)v_\alpha$ where $u_\alpha$ and $v_\alpha$ are a set of pairs of (constant, Heisenberg-picture) vectors, then, by linearity, if $\dot A_{ij}(t) = M_{ijk\ell}A_{k\ell}(t)$, then
\begin{align*}
\sum_{i, j} u_{\alpha, i}\dot A_{ij}(t)v_{\alpha,j} &= \sum_{i,j,k,\ell} u_{\alpha, i}M_{ijk\ell}v_{\alpha, j} A_{k\ell},\\
\langle \dot A(t)\rangle_\alpha &=\sum_{k\ell} M_{\alpha,k\ell} A_{k\ell},\quad M_{\alpha,k\ell} \equiv \sum_{i,j} u_{\alpha i} M_{ijk\ell} v_{\alpha j}
\end{align*}
We can then try to expand $M_{\alpha, k\ell} A_{k\ell}(t)$ in terms of the quantities $\langle A(t)\rangle_\alpha \equiv \sum_{i,j} u_{\alpha i} A(t)_{ij} v_{\alpha j} = u^T A(t) v$, as well as possibly auxiliary quantities $\langle A(t)\rangle_\beta = u_{\beta i}A_{ij}(t) v_{\beta j}$, where the $u_{\beta}$ and $v_\beta$ are chosen to ensure that equality is maintained for all time (in some cases, this could require foregoing the use of $\langle A(t)\rangle_\alpha$ on the right hand side entirely.)  The dynamics of the $u_{\beta}$ and $v_\beta$ variables, if any, would be induced by the original equations of motion entirely analogously to the $\langle A(t)\rangle_\alpha$ variables (in your example, it turns out that such "auxiliary states" aren't necessary.)
