# Where am I doing mistake of concept in deriving the De Broglie equation?

When I am deriving de Broglie wavelength for a relativistic particle using $$E^2=m^2c^4+p^2c^2$$ and equating with $$E=\frac{hc}{\lambda}$$, and then putting $$p=kmv$$, $$k$$ being relativistic factor, I am getting $$\lambda =\frac{h}{kmc}$$ instead of $$\frac{h}{kmv}$$.

Is there any mistake that i am doing with equating those 2 energy equations ? Or something else ?

## 2 Answers

The equation $$E^2=m^2c^4+p^2c^2$$ is valid for all particles with mass $$m$$.

But equation $$E=\frac{hc}{\lambda}$$ is true only for massless particles (i.e. $$m=0$$), e.g. for photons.

I guess you got this wrong equation by putting together $$E=h\nu$$ (which is indeed correct for all particles) and $$\nu=\frac{c}{\lambda}$$ (which is valid only for massless particles).

The speed is $$\frac{kmvc^2}{kmc^2}=\frac{pc^2}{E}$$. Using this speed in place of $$c$$ to get frequency from a wavelength,$$E=h\nu=\frac{hpc^2}{E\lambda}\implies\lambda=\frac{hpc^2}{E^2}=\frac{hv}{kmc^2}.$$This is the correct relation for massive particles.