Derivation of the De-Broglie wave relation

Question

I've just tried to make sense of the De-Broglie wave relation. That is what I came up with: $E=mc^2$, but $E=hf$ as well. $\lambda=vT$ so $f=\frac{v}{\lambda}$. Equating the two equations for energy you get $mc^2=h\frac{c}{\lambda}$ because for light the velocity is $c$. $mc$ is just the non-relativistic momentum so $p=\frac {h}{\lambda}$ which is the De-Broglie wave relation. But if I didn't replace $mc=p$, $m=\frac {hc}{\lambda}$ or $m=fh$, which would imply that light does have rest mass. What would be the more rigorous derivation of the De-Broglie wave relation? Do I need to use relativistic momentum $\gamma mv$? Or is the above reasoning just simply not applicable to light?

Answer 1

$p=mc$ is NEVER true.

Like you mentioned, for a particle with relativistic velocity (v is not negligible compared to $c$), the right equation is:

$$p = \gamma m v = v\cdot \frac{m}{\sqrt{1-(\frac{v}{c})^2}} $$ Now if you try to apply it to photons, $ m=0$ and $\sqrt{1-(\frac{v}{c})^2}=\sqrt{1-1}=0$ so you get a $\frac{0}{0}$ situation - which gives us no information at all! So you can't apply this formula for photons.

But, there is another, more accurate equation:

$$E^2 = (pc)^2 + (mc^2)^2 $$

So here, for photons, $m=0$ and we get $$E^2=(pc)^2 \Rightarrow E=pc$$ Side note: that was known before Einstein, you can get this result straight from Maxwell's equations and the Lorentz force.

Now, if $E=pc$, then $p=\frac{E}{c}$ , which for a single photon means $$ p = \frac{E}{c} = \frac{h\frac{c}{\lambda}}{c} = \frac{h}{\lambda} $$

I hope it helped :)

Answer 2

de Broglie suggested that a moving body behaves in certain ways as though it has a wave nature. His conjecture of dual nature of matter was based on two points : (1) dual nature of radiation, and (2) nature loves symmetry.

de Broglie deduced the connection between particle and wave properties from the Einstein-Planck expression for the energy of the electromagnetic wave and the classical result for the momentum of such a wave. The two expressions are,

$E=hf=\frac{hc}{\lambda}$

$and\space E^{2}=(m_\circ c^{2})^{2}+(Pc)^{2}=(Pc)^{2}$, because the rest mass of photon is $0$

From these equations we get,

$\lambda=\frac{h}{P}$

de Broglie said that this equation is a completely general one that is applied to a material particle as well as photons. The waves associated with the moving particle are called matter waves or de Broglie waves.

The theory of matter waves was used to justify quantization of angular momentum as proposed in Bohr's model. If we consider standing electron wave, then to maintain a standing wave over the circumference of a circular orbit, the wavelength must be an integral fraction of that circumference.

$2\pi r = n\lambda = \frac{nh}{mv}$,

And hence we get, $mvr=\frac{nh}{2\pi}$, this equation is an integral part of Bohr's atomic model.