If I remember correctly, quantisation occurs because of the solutions to certain differential equations in the geometry (eg of an atom, of a lasing chamber, etc

If I remember correctly, quantisation occurs because of the solutions to certain differential equations in the geometry (eg of an atom, of a lasing chamber, etc). The quantised states are the different solutions to that equation (the different energy levels + spin states in an atom.) - this holds for electrons in atoms being quantised. The photoelectric effect, which has to do with incident photons, showed that you can't turn up the intensity of long wavelength light and ping electrons off things: the power input didn't change things, but a very low power of short wavelength photons did. Thus, Einstein concluded that there must be something specific about the energy of individual particles, not just the total energy stored in a wave. So it is observed in an interaction, but the information defining the final effect travels with the photon.