I came across the derivation, present all across the web, which utilized Einstein's energy mass equivalence equation and energy of a photon.
It goes like this:
$$
E = mc^2,\;\;E = h f \;\;[f = \text{frequency} ]\;\;\Rightarrow \;\;hf = mc^2\\
\frac{h c}{\lambda} = mc^2 \;\;[\lambda = \text{wavelength}]\\
\frac{h}{\lambda} = p, \;\;\;\frac{h}{p} = \lambda,\;\;\;\frac{h}{mv} = \lambda
$$
With this, I have a problem with every step
(like converting $mc$ to $p$ and then to $mv$)? IS this really correct? How?
Supposing we use, $E/c = p$ for a photon, then isn't it still wrong?
Aren't we using EM radiation to find an associated wave? Aren't these completely different?
Could someone please help with the real one?
Best Answer
When de Broglie published his proposed relationship he attempted to show that it was compatible with the Planck relation and Special Relativity; his arguments are quite detailed, and heuristic.
His goal was to show convincingly that if waves had particle properties, then particles must have wave properties --and he invoked Special Relativity as a principle in a variety of ways.
As you have noted, the de Broglie relation is trivially valid for the momentum of light; his arguments try to show that this relationship is the only possibility for a matter wave. But in the end one cannot derive this relationship: it is a physical hypothesis, and has to be shown experimentally.
So ultimately these "demonstrations" don't matter; even if they were to give all of de Broglie's arguments they would still be flawed. For more of the flavor of the original argument, see https://en.m.wikipedia.org/wiki/Matter_wave