So, ok. Given a light wave (etc.) w(t), you can do a Fourier transform and decompose it into a sum of sine waves with different frequencies and amplitudes. And the fact that light is quantized means that for each wave of frequency 1/λ in the decomposition, its energy must be an integer multiple of hc/λ. And since energy is proportional to amplitude (idk what the constant is, call it x), the amplitude must be an integer multiple of xhc/λ. And this is what you would actually see in the Fourier transform of the wave, if you looked at F(w)(1/λ), its value would be nxhc/λ for some integer n.
And this means, if you want, you can decompose the wave even further by writing each F(w)(1/λ) as the sum of n copies of the wave with frequency λ and amplitude xhc/λ. Uh, n copies of (xhc/λ)sin(t/λ). And each of these guys is called “a photon”.
Obviously there’s some other shit, like wave functions are complex valued or whatever it is, but is this the basic idea? Do I get it?
Pretty much, though with the caveat that what you’re describing is a “theorist’s photon”. Real, physical photons aren’t infinite plane waves, they’re confined to some volume. That is, their wavefunction will look like a sinusoid multiplied by some “envelope function”. Of course, you can take the Fourier transform of that thing too, and you’ll find that it’s still peaked around the fundamental frequency, but not as sharply. The peak will be broadened. This envelope function can take basically arbitrary shapes, which makes the question of “what is a photon, really” a little fuzzy. I believe this is what led my analysis professor in undergrad to rant at me one time about how “you physicists talk about particles all the time, but what is a particle? You can’t define it! Particles don’t exist!”
(Also, in addition to the wavefunction being complex, there’s the fact that you described the situation in 1d when of course it should be 3d, and that photons have polarization because the EM-field is vector-valued, and there’s some slight subtleties when you go from non-relativistic QM to QFT. But I agree with you that none of that really changes the situation in an essential way.)
Two questions:
- if light has a polarization, doesn’t that make the 3d situation basically the 1d situation embedded in 3-space, with the embedding determined by the polarization?
- what is the value of the constant x, and what is it called?