They quantize because the underlying solution is a circular standing wave, which is stable if it consists of a whole number of periods. The wave itself is of course analog, quantization emerges through many hoops and provides stable states with certain potential well like any stable state, and can squiggle a little under perturbation or squiggle a lot under, say, Zeeman effect.
Can we really speak of the wave(let) "being" analog when it is impossible (according to our current understanding of physics) to see that experimentally ?
