Quantum-mechanical description of a photon in refractive mediums and momentum operator

Abstract

A quantum-mechanical description of a photon in refractive mediums is proposed. Here we assume that the medium is isotropic and linear, and the material constants (the dielectric constant and permeability) are both the functions of the position coordinates only and dispersionless. It is found that the momentum operator is generally ill defined as well as the position operator. The momentum operator, however, becomes approximately well defined when the extents of the appreciable changes of the material constants are very large compared to the photon wavelengths. Then, it is shown that the various expressions of the physical quantities in the classical electromagnetic theory, such as the energy and momentum, can be obtained from the expectation values of the corresponding operators. The Abraham-Minkowski controversy in the classical electromagnetic theory is glanced from the viewpoint of our quantum-mechanical approach.