For a given plane monochromatic electromagnetic wave in a vacuum, one can calculate the volumic momentum via $$\text{momentum density} = \epsilon_0 \,\vec{E} \times \vec{B}$$ and volumic energy via $$\text{energy density} = \frac{\epsilon_0 E^2}{2} + \frac{B^2}{2 \mu_0} \, ,$$ finding $p = E/c$. Then, using the the Planck-Einstein relation $E = h \nu$, one arrives at de Broglie's relation $p = h/\lambda$. Why is it that no book I've read mentions this derivation, but rather treats Planck-Einstein and de Broglie as two separate postulates?

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    $\begingroup$ 1) When you say "calculates the volumic momentum and volumic energy", are you referring to a calculation using Maxwell's equations? If so, please make that clear. 2) What is the "Planck-Einstein relation"? There are so many equations in physics, many of them discovered by the same famous people, so naming equations like that can be confusing. It's better to just write the equation. 3) Similarly, what do you mean by "de Broglie's relation"? $\endgroup$ – DanielSank May 24 '18 at 17:19
  • $\begingroup$ I'll be honest, I was trying to avoid having to write latex. I meant using the formulas p = epsilon_0*E x B, and u = 1/2epsilon_0 E^2 + 1/2*mu_0 * B^2; Planck-Einstein is E=hbar times omega and de Broglie's is p=hbar times k. Sorry for the inconvenience. $\endgroup$ – user115153 May 24 '18 at 17:36
  • $\begingroup$ Ok, well I did it for you. Mathjax is pretty easy to use. See this link. Or just hit the "edit" button to see what I did in your post. $\endgroup$ – DanielSank May 24 '18 at 17:45

Yes, starting from $E=hf$ and $E=pc$ and, of course, $c=f\lambda$ you can deduce $p=h/\lambda$, which is true for light quanta (photons) and also for matter-waves (de Broglie).

But you can also, playing around with the algebra, come up with lots of equations like $p=hf/c, E=h\lambda/c, \lambda=Ec/h$ and so on, which are true for photons but are not true for matter waves.

de Broglie was clever enough to pick out, amongst all the possibilities, the relation for photons that also works for particles with non-zero rest mass.


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