Why do we replace $c$ (speed of light) with $v$ in de Broglie's equation?

Question

Deriving de Broglie's equation (as per my text and teacher) involves equating $E = mc^2$ with $E = h\nu$, where $\nu$ is the frequency. It goes like :

$$mc^2 = h\nu$$ $$mc^2 = \frac{hc}{\lambda}$$ $$mc = \frac{h}{\lambda}.$$

Then we replace $c$ with the velocity of the particle to apply it generally

i.e. $$mv = \frac{h}{\lambda}.$$

My doubt is exactly about this step. As far as I have read before and after learning this equation, I understood that $c$ in $E = mc^2$ was mainly used as a constant which can equate energy and mass rather than something relating energy, mass and velocity of the particle. Thus replacing $c$ with $v$ makes no sense as it would have contradicted the equivalence of $E=mc^2$ in the first place.

Can someone explain how $c$ can be replaced with $v$ without contradicting $E = mc^2$, or just simply point out what is wrong with my thought process?

Answer 1

Since I see this is your first post here and if you intend on keeping your contributions to the forum, I suggest you read this link to learn about MathJax.

Second of all, I see the problem you're having here. The famous equation which relates wave-like properties such as the wavelength $\lambda$ and a mechanical property (present to all objects), the momentum, $p$, wasn't really derived from the energy-mass equivalence $$E=mc^2$$

In fact, you have two fundamental relations, called the De-Broglie-Einstein relations which state $$E=h\nu \ ; \ \lambda=\frac{h}{p}$$ where as you should know, $h$ is the Planck's constant. These are rather general formulas, and at first De-Broglie was criticized (by his thesis evaluators) that his idea of everything having a related wavelength was original, but sounded "magical". Actually, the energy-mass equivalence in the way I wrote it shall only be used for particles at rest, the "complete" formula is $$E=\sqrt{m^2c^4+p^2c^2}$$ so that it reduces to the above one when the particle has null velocity. Coming back to the De-Broglie-Einstein relations, you can substitute $p$ for whatever expression you have for your system. For non-relativistic, "normal" systems, the usual expression is $p=mv$ where $v$ is the speed of the particle.

I hope this clears your question a bit.