electromagnetism - Energy in an EM wave should depend on frequency

I just finished reading Feynman's Lectures on Physics vol.I, §34-9: "The momentum of light". The author explains that there is a relation between the wave 4-vector $k^{\mu}$ and the energy-momentum 4-vector $p^{\mu}$ of an EM wave, namely

$$p^{\mu}=\hbar k^{\mu}, $$ or equivalently $$\tag{deB}W=\hbar \omega, \mathbf{p}=\hbar \mathbf{k},$$

and those equations are called de Broglie relations.

However, as I learned in my classical electromagnetism course, flux of energy in such a wave is quantified by Poynting's vector, yielding formulas such as the following:

$$\tag{1} I=\frac{1}{2 \mu_0 c} E_0^2, $$

where $I$ stands for "average intensity" of the wave and $E_0$ for "maximum amplitude of electric field".

Question Where is $\omega$? It does not appear in formula (1) nor in any other formula based on Poynting's vector. But as of equations (deB) it should do so. Am I wrong?

Thank you.

Answer

The section you are referring to clearly states that those equations do not apply to the wave, but to the "particles" of light, the photons. The resolution is that two waves of the same amplitude but different frequencies contain different numbers of photons. This has interesting consequences, for instance it means it is possible to communicate via radio waves carrying miniscule amounts of power while an optical signal of similar intensity would be drowned out by shot noise.