Ok, so I understand that it was a hypothesis, but still I have a couple of questions that have boggled my mind for the past few months.
1) How did de-Broglie obtain his formula? Did he guess it on experimental basis?
2) I have seen two types of ad-hoc derivations.
- Using mass-energy equivalence The main idea of this derivation is that you have $E = mc^2$ and $E = h\nu$ and equate them both to get $\lambda = \dfrac{h}{mc}$, for a photon, and for a particle moving with a velocity $v$, you have $\lambda = \dfrac{h}{mv}$.
To me this approach seems plain wrong as it messes up conceptual stuff. From what I know, $E = mc^2$ is the energy for a particle at rest, and $E = h\nu$ is for the energy of a photon of frequency $\nu$. Yes, they are energies, but not energies of the same kind. How can you equate the two? It feels wrong.
- This one is not quite messy as the last one, but -
You have $E = \sqrt{p^2c^2 + m^2c^4}$ and for a photon with zero rest mass, you have $E = pc$. Equating this with $E = h\nu$, you have $p = \dfrac{h}{\lambda}$ or $\lambda = \dfrac{h}{p}$. Now we extend the same logic to particles having mass they must also have a de-Broglie wavelength $\dfrac{h}{p}$. But we started with zero rest mass, then how does it apply to electrons and other particles having non-zero rest mass?
3) Am I stressing too much on derivations, and just not letting it be what it is? Since it was a hypothesis, and cannot be proven (like a theorem).
Anyways, could you elaborate on the mistakes that I have spoken about in 2)? Or is it that I am conceptually mistaken? And also answer 1) and 3)?
