Quantum Mechanical Relations for the Energy, Momentum and Velocity of Single Photons in Dispersive Media

Abstract

Attempts to explain the refraction of light in dispersive media in terms of a photon or "corpuscular" model have heretofore been unable to account for the observed decrease in the speed of light as it passes from air into a region of higher refractive index n such as water or glass. In the present work it is argued on the basis of the quantum mechanical relations p = k and E = h omega that the energy of photons satisfies the equation E = pc/n. It is possible to obtain an exact prediction of the observed speed of the photons in a given medium by application of Hamilton's equations of motion to the above formula, but at the same time to conclude, in agreement with the arguments of Newton and other classical physicists, that the photon momentum increases in direct proportion to n, thereby producing the well-known bending of light rays toward the normal when entering water from air. The corresponding relativistic particle theory of light indicates that the potential V encountered by the photons in a given medium is attractive for n > 1 and is momentum-dependent, which suggests the microscopic interactions responsible for the refraction of light are non-Coulombic in nature and are instead akin to the spin-orbit and orbit-orbit terms in the Breit-Pauli Hamiltonian for electrons moving in an external field. The present theory concludes that the reason photons are slowed down upon entering water from air is that their relativistic mass p/v increases faster with n than does their momentum, which in turn requires that Einstein's famous E = mc^2 formula does not hold for light dispersion because the energy of the photons is expected to be the same in both media.