second quantization
I am trying to remember which physical phenomena require that we invoke photons as an explanation--which phenomena can't be explained by describing light as a wave. My memory is that there are three phenomena. Google is failing me, because except for, like, two hits, it's mostly insisting that one of these phenomena is the photoelectric effect, which is false.
Of course, the idea of photons was championed/semi-invented by Einstein in the first place* because of the photoelectric effect. But a few decades later, it was shown that there are other ways to explain the photoelectric effect that don't involve photons at all. Of course, just because you can explain it without photons doesn't mean it isn't equally valid to explain it with photons. That's the weird thing about quantum mechanics--multiple contradictory things can be true simultaneously. But that you can explain it without photons does mean that it isn't empirical proof of photons--for that you need something that can only be explained with photons.
So here's what I'm coming up with so far, based on memory and Google and Wikipedia.
1) Spontaneous decay. When a simple quantum mechanical object spontaneously falls from a higher energy state to a lower energy state, the energy has to go somewhere, and that's in the emission of a single photon. If you have a single photon, you can count how many photons you have; therefore light is quantized, therefore photons exist. Examples of this Include certain types of fluorescence (including phosphorescence and bio-luminescence), where a valence electron in an atom spontaneously falls from a higher energy level to a lower energy level, thus causing the atom to emit light (glow), and some types of radioactive decay where an atom changes from one element to another and the energy/mass difference is made up by the creation of a gamma ray.
2) Photon bunching/number squeezing/Fock states. There are ways to show that, statistically, in certain circumstances, you can get photons to correlate. That is, you can say sometimes that if you have a photon in location A you probably won't have a photon in location B (or alternatively you probably will have a photon in location B) and likewise if you have a photon in location B you probably won't (or alternatively probably will) have one in location A. Furthermore, in the extreme case of Fock states (perfect correlation) you're going to end up with a well-defined number of photons with no phase coherence (the opposite of a laser, which is defined as light with perfect phase coherence), and if you're just talking about waves, phase coherence and amplitude (i.e. amount of light) wouldn't be related like that. The mathematical theory showing you can't have correlations like that if you describe light as a wave--that you need to describe it as discrete particles--is more than a little involved. Well, from the particle direction it's not that bad, but from the wave direction it can get nasty. And the experiment isn't very easy, either. But both the theory and the experiment have been done successfully.
3) Entanglement. Entanglement can only be mathematically described in a quantized system. Entanglement has been experimentally demonstrated with light, so therefore there must be such thing as photons.
So that's three, and I think those are the right three, but I'm kind of coming up with more?
4) Matter/anti-matter annihilation and maybe some other Feynman-diagram-type interactions. I don't know, maybe this falls under the heading of (1), but it seems to me like a distinct thing. In this case, instead of one photon being created, two (usually gamma rays) are created going in opposite directions. I don't know, maybe I'm wrong about needing the photon interpretation to explain the light that comes out. But I kind of thought you needed second quantization to even prove the existence of antimatter. Or maybe talking about quantum electrodynamic phenomena is cheating and I'm supposed to just stick to quantum optics--which, unlike quantum electrodynamics, doesn't have second quantization as quite such a fundamental postulate--in which case there really are only three.
And maybe there are more still that I'm just not thinking about.
*I won't say the concept of photons was quite invented by Einstein because he got the idea from Planck, who needed it to mathematically explain one of his own theories but didn't really believe it was a true physical phenomenon--much the way people didn't think quarks were a real thing at first, but rather just a mathematical artifact. But Einstein was, at least, the first person to believe in photons.
