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

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?

• Does this answer your question? Proof of de Broglie wavelength for electron Jun 10, 2020 at 13:38
• Jun 10, 2020 at 13:39
• I had already read about that question in my doubt clearing journey; The problem that I found was that the author was trying to avoid the alleged contradiction by using another way around the proof. I wanted to know whether my reasoning was correct or I was making a grave mistake. So all in all it does not serve my real purpose. Thank you for the suggestion ! Jun 10, 2020 at 13:54

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.
• So does this mean that $E = mc^2$ can't actually be used for proving de Broglie's equation as it involves a particle having velocity? Thank you for the suggestion about mathjax. Jun 10, 2020 at 14:01
• I don't know if this is your situation, but usually in high school you use $E=mc^2$ for this proof because it is intuitive, but yes. It's not entirely correct. This formula is only valid for particles at rest. Jun 10, 2020 at 14:56