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According to this answer, treating a photon as a particle, according to special relativity, the 4-vector of a photon defined as the momentum 4-vector with the energy component replaced with it divided by $c^2$ should undergo the same Lorentz transformation as the position of an event in space-time with a change in frame of reference. I have 2 questions.

  1. For a continuous electromagnetic wave travelling in the same direction with a wavelength equal to the De-Broglie wavelength of that photon, if you make the same change in frame of reference and compute wavelength and the total energy of that wave according to classical physics and then compute the energy of the photons that wave is composed of using the De-Broglie relation, do you get the exact same energy as when you compute the energy of the photon in the new frame of reference treating it like a particle in special relativity?

  2. After you compute the number of photons the wave is composed of in the old frame of reference by dividing the total energy of the wave by the energy of each photon the wave is composed of, if you then compute the total energy of the wave in the new frame of reference and the energy of each photon in that frame of reference and divide the former by the latter, do you compute that the number of photons in the new frame of reference is the exact same number?

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The answer to both questions is yes. Einstein addresses this in section 8 of his original paper on special relativity. He talks about a "light complex," and he shows that its energy and frequency both transform according to the same Doppler factor. Even though this paper is classical, 1905 was the same year in which he published his paper on the photoelectric effect, so he probably realized this would be an important issue to check for that reason.

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