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Progress of Theoretical Physics

Derivation of the Photon Wave Equation in a Medium via a Poisson Process

Tomoshige Kudo, Ichiro Ohba

Published Nov 1, 2004• 0 citations
Publisher

Abstract

We construct a general photon wave equation in a nondispersive and homogeneous medium.Then, we demonstrate the relation between the photon wave equation and a Poisson process.Through this formalism, the passage time of photons in the medium can be defined.Furthermore, we derive a photon wave equation in a dispersive medium from the general formalism of the Poisson process by introducing a time shift generator.

DOI
10.1143/ptp.112.763

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We construct a general photon wave equation in a nondispersive and homogeneous medium.Then, we demonstrate the relation between the photon wave equation and a Poisson process.Through this formalism, the passage time of photons in the medium can be defined.Furthermore, we derive a photon wave equation in a dispersive medium from the general formalism of the Poisson process by introducing a time shift generator.
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Recent experiments have demonstrated that it is possible to alter the dispersion of a medium without significantly altering its absorption or refractive index and that this may be done while a wave propagates through the medium. This possibility opens up a new set of potential experiments to the field of nonstationary optics. We consider the basic kinetics of waves propagating through a medium whose group and phase velocities are a function of position and time. We compare the dynamics of waves propagating through two homogeneous media, one a nondispersive medium with a time dependent phase velocity, and one a dispersive medium with a time dependent group velocity and show that new dynamic effects accompany new kinetic ones.
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Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.
The theory of non-dissipative diffusion is constructed on an example of diffusion of a light particle in a dilute medium of heavy particles and it is shown that in low dissipative systems there are specific effects of non-dissipativity similar to quantum effects.In a non-dissipativity region mean energy of the light particle is conserved and processes are described by two non-linear diffusion equations with forward and backward time derivatives.Then these two diffusion equations are linearized and give one linear Schrödinger equation for complex amplitudes of probability.As a result, in the non-dissipative classical diffusion should be added probability amplitudes and there holds the superposition principle for these amplitudes.A mean square length of free passage and a mean square momentum define an elementary phase volume and a diffusion constant and they obey the uncertainty relations.It is shown that the formalism of quantum mechanics describes the classical non-dissipative diffusion with a homogeneous diffusion constant and that quantum mechanics is only a particular case when the elementary phase volume of free passage is universal and equal to the Planck constant.