Spontaneous emission laser Why exactly can we not create a 'laser beam' from a medium undergoing spontaneous emission inside a cavity?
It seems like the dielectric mirrors are going to cause a standing wave to appear irrespective of whether the medium is doing stimulated or spontaneous emission, and that will produce something looking like a laser beam.
Why does it specifically have to be stimulated emission?
 A: For driven harmonic motion, there must be a resonant frequency and  an oscillating driver, and they must be somewhat close in frequency. You may think of a pendulum for example and not immediately realize we do have that situation in a way. One way to think of it is that driven oscillation situation is mimicked. We have no such mimicking here. With undriven SHM, the restoring force is proportional to displacement which is related, and the same in some ways, but not identical to driven SHM. You do not have that with standing waves in dielectric mirrors. Nothing equivalent to $F=-kx$.
Here’s just one example of actually driven harmonic motion: a hose hanging from the ceiling oscillates sometimes. There are two characteristic frequencies that can reinforce each other, the pendulum frequency $f_p$, and the flow length frequency $f_{uL}$, where $u$ is fluid velocity:
$$f_p= \frac{\sqrt{g}}{2\pi \sqrt{L}} \text{ , } f_{uL}=\frac{u}{L}$$
To drive oscillation: $$ \sqrt{gL}=2\pi u \text{ , eg: } L=1m, u=\tfrac{1}{2}\tfrac{m}{s}$$
But what’s happening is that the driver is oscillating.
A: If it were spontaneous emission, even if they matched the wavelength, the photons would have a random direction and phase, there would be no amplified standing wave. From wiki
"The emitted photon exactly matches the original photon in wavelength, phase, and direction. This process is called stimulated emission."
https://en.m.wikipedia.org/wiki/Laser
Edit: FYI all emission is stimulated actually,  spontaneous emission is stimulated by vacuum fluctuations,  but that's not necessary info here.
