Energy as Frequency, Relative Velocity and Lorentz Transformations

 In quantum mechanics, energy is associated with frequency for both a particle with rest mass and a photon. In the case of the photon, we try to show that this result follows from special relativity (Lorentz transformation) which in the photon case gives rise to relative velocities.

   We consider a photon moving along the x axis and bouncing back and forth between two mirrors separated by L and moving to the right with speed v. We argue that Lorentz transformations for photons create the notion of relative velocity because E=|p|c, but there still exists a moving frame with velocity v. As a result, a relationship between the energies of the photon moving to the right and then to the left exists with each energy being multiplied by a respective time= 1/ relative speed. This suggests photon energy representing a frequency.

   . We also note that relative velocities from the Lorentz transformations are the opposite of those associated with the times the photon spends moving from one mirror to the other.. In other words, the photon moving to the right is associated with 1/c-v, but its time with 1/c-v because it looks as if the photon is moving more slowly when trying to hit a mirror that is moving  away from it.