Laser power and coherent state amplitude How it is possible to relate the power $P$ of a laser light with frequency $\omega$, to the amplitude $\alpha$ of its description as a coherent state $\vert \alpha \rangle$ ?
For a massive particle in an harmonic potential one has
$$
A = \sqrt{\dfrac{2 \hbar}{m \omega}} \vert\alpha\vert
$$
where $A=\sqrt{P}$ is the oscillation amplitude. But what about a masses particle?
Is it possible to use the mean photon number during time $T$, $\dfrac{P T}{\hbar \omega}=\bar{n}(T)$ ?
 A: You can look at the details in the Wikipedia article on the quantization of the electromagnetic field, but I will sum up some relevant bits below.
First, you seem to take the analogy with the massive harmonic oscillator too literally. The analogy is pretty strong, but purely formal, so as you guessed, the mass as no meaning here. Furthermore, if you speak of amplitude, you may think of the electromagnetic field, the exact prefactor will depend whether you look at the electric field $\left(\frac{\hbar}{2\omega V \epsilon_0}\right)$ or the magnetic field $\left(\frac{\hbar\omega}{2 V \epsilon_0}\right)$ (if I made no mistake).
Since you are interested in power, the question is simpler: the average energy of a coherent state is $\lvert\alpha\rvert^2\hbar\omega$ in Joules (or $\lvert\alpha\rvert^2$ in photons). If this is a coherent state trapped in a cavity, it is the total energy of the coherent state. For a propagating laser, this information is also sufficient if $\left|\alpha\right>$ describes the coherent state of a pulse, since it gives the total (average) energy of the pulse. For a square pulse of duration $\Delta t$, just divide by $\Delta t$ to have the power. If the pulse is not square, you should use the pulse profile to have the correct evolution of the power over time.
For a continuous wave (CW) laser, it is slightly more subtle. $\left|\alpha\right>$ describes the state in a single temporal mode and is associated with an energy, not a power. You can chose to see a CW laser of power $P$ as a succession of square pulses of duration $\Delta t$. In which case, each of these pulse has an energy $P\Delta t$ and would be described by a coherent state $\left|\sqrt{\frac{P\Delta t}{\hbar \omega}} e^{i \phi(t)}\right>$. The choice of $\Delta t$ is arbitrary, but the different description of the state it leads are in fact equivalent.
