When your looking at basic Compton theory you find that if you shoot a stream of photons at a particle (usually atoms or electrons), then you have the basic laws of conservation of momentum. The photon acts like a particle, like a "billiard ball." The photon interacts with the said electron and the photon goes off in a new path described by $h/\lambda_2$ ($h$ being Planck's constant and $\lambda_1$ being the original wavelength of the photon). This wavelength is increased. Using a basic vector diagram with $h/\lambda_1$, $h/\lambda_2$ and $m\vec{v}$ for the particle you get basic conservation of momentum.
My question is basically what happens when you are only using a single photon? I'm not aware of any experiments done with a single photon, so far this concept (Compton experiments) have only been done with multiple photons. The reason a single photon is important is because the energy of a photon is inversely proportional to the wavelength. The problem that I have with this is that $E = h\nu$. This is how we get our inverse wavelength in the formula (wavelength and frequency being inversely proportional). With a single photon you have no frequency, since you only have 1 event/photon. Thus how can you place Planck's constant over $\lambda$ to represent the energy of the photon, since there is no frequency?
Best Answer
The frequency of light is not a property of many photons but a property of a single photon. (This is also strictly inaccurate, since we should think about a field with ripples in it; we then call a little clump of waves a photon.)
Anyway, let's imagine a photon/wavepacket as having a typical wavelength $\lambda$ given by the peak-peak spacing in this diagram:
This is a single photon, with the colour given by wavelength $\lambda$. Then this whizzes past you at the speed of light, $c$. But then you see the peaks go past at a rate of 1 per $\lambda/c$. Thus there is a sensible definition of frequency as $f=c/\lambda$. Then we let $E=hf$ and carry on happily.
This is all essentially an example of wave-particle duality. If you want to try to understand this better, try reading about the photoelectric effect. The key idea is that there is a wave-like phenomena (light) which we perceive as interacting with the rest of the world in discrete, particle-like events (photons).