A common identity in Quantum Mechanics is relation between the momentum of a photon and its wavelength:
$$p = \frac{h}{\lambda}$$
The identity is discussed here, for example:
https://en.wikipedia.org/wiki/Matter_wave
Apparently, this is the identity rearranged by de Broglie to give the wavelength of the wave nature of a particle. But where does this identity come from in the first place? I have seen some quite "hand-wavy" ways of deriving this using $E=mc^2$, but it seems quite strange having to rely on relativity to obtain this identity. Or is it exactly what we must do? This seems to be a quite fundamental identity in Quantum Mechanics, so I would like to understand its justification as well as possible. I've been told that light having momentum is an idea present in classical mechanics as well and was known much before quantization of light and photons were discovered.
Best Answer
It seems like your are not satisfied by answers involving axioms. I think that you instead want to know the motivation behind the axiom beyond just saying that it works. I am not sure if my answer is the original motivation, but I think it can be viewed as a good motivation for the validity of $p= \frac{h}{\lambda}$. While other answers do a great job at going into the theory, I will tackle the question using more of an experimental motivation.
We will first start with the double slit experiment. This experiment is usually first introduced as evidence of the wave-like nature of light, where light emanating from one slit interferes with light emanating from the other (of course a different interpretation is found if we send single photons through the slits and the same interference pattern arises, but I digress). However, this experiment also works with electrons. You get an interference pattern consistent with treating the electrons as waves with wavelength $$\lambda=\frac hp$$
You get maxima in intensity such that $$\sin\theta_n=\frac{n\lambda}{d}$$
Where $\theta$ is the angle formed by the central maximum, the slit, and the maximum in question, $d$ is the slit separation, and $n$ is an integer.
This would then be a way to experimentally motivate/verify this relationship between momentum and wavelength for matter, but what about photons? The double slit experiment does not give us a way to validate $p=\frac h\lambda$ (that I know of. Maybe you could determine the radiation pressure on the detector?). Let's look at a different experiment.
We know that the energy of a photon from special relativity is $$E=pc$$
So, if our momentum relation is true, it must be that $$E=\frac{hc}{\lambda}=hf$$ which is something that can be verified experimentally to be true. The photoelectric effect is one such experiment we could do, where shining light onto a material causes electrons or other charge carriers to become emitted from that material. The higher the frequency of the light, the more energetic the electrons coming from the material are, and the maximum kinetic energy of an electron can be shown to follow $K_{max}=h(f-f_0)$ where $f$ is the frequency of the light and $f_0$ is the material-dependent threshold frequency (i.e. we need $f>f_0$).
I know that my answer does not get to a fundamental explanation of this relation in question, but I hope it shows why one would want it to be a fundamental idea that holds true when formulating QM. If you want a more fundamental explanation, then I will edit or remove this answer due to some pretty good fundamental answers already here.