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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

A complete description and applications for a photon

Zhixin Yao, Jing Zhong, Bailiang Pan

Published Aug 26, 2007• 2 citations
Publisher

Abstract

On the basis of comparing Maxwell equation for a light wave to Schroedinger equation for a photon, we conclude that the former solution that must be a real function is the real part of the latter solution that must be a complex function. Using the state-vector function that is the general solution of Schroedinger equation for a photon, we solve some true but strange optical problems including the simple demonstration of Malus's Law and the orthogonal decomposition of a natural light.

DOI
10.1117/12.732251

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On the basis of comparing Maxwell equation for a light wave to Schroedinger equation for a photon, we conclude that the former solution that must be a real function is the real part of the latter solution that must be a complex function. Using the state-vector function that is the general solution of Schroedinger equation for a photon, we solve some true but strange optical problems including the simple demonstration of Malus's Law and the orthogonal decomposition of a natural light.
The optical eigenmode representation of light fields defines a natural orthogonal basis of solutions of Maxwell's equations taking into account the geometry and interactions involved in a problem. Formally, the optical eigenmodes are similar to the wave functions in quantum mechanics. Here, I put forward that, in a linear interaction at the single photon level, the electromagnetic field collapses into an optical eigenmode of the interaction in question. This is to satisfy the principle that no linear optical system can distinguish between pure states and their superposition for single photons. To exemplify this statement, we will consider the case of angular momentum and linear momentum transfer in optical scattering.
We propose to treat Maxwell equations as a wave equation of motion for a photon (a quantum of light) and to develop the photon mechanics on this basis. Consider vectors of electric and magnetic fields E and H as a single object (a bivector) F , which corresponds to the function ψ in the wave theory of the electron. Multiplying the vacuum Maxwell equations − rotH+ 1 c ∂E ∂t = 0, rot E + 1 c ∂H ∂t = 0 (1) by hc 2πi , it is possible to rewrite them in the form HF + h 2πi ∂F ∂t = 0, (2) where H is the operator H = hc 2πi { 0 − rot rot 0 } ; (3) its meaning follows from the comparison of equation (2) with (1). Equation (2) can be interpreted as a wave equation for a photon.
On the basis of the generalized Schr?dinger equation, a novel state-vector function that obeys photon motion equation under three quantum conditions has been constructed for a photon in one dimension, which can describe completely the properties of a photon as a relativistic free particle, including its energy and momentum and spin angular momentum. The state-vector function not only defines microscopic parameters such as the probability amplitude and the phase for a photon, but also relates them with the macroscopic polarization of a light beam. As an example, it successfully explains the puzzling polarization problem of a light beam.
A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics . Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now. The Maxwell–Schrödinger equations We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.
This tutorial begins with the relationship of source-free classical electromagnetism to ultra-relativistic free-photon quantum mechanics. The linear transformation of the source-free classical-electromagnetic real-valued transverse vector potential to its corresponding free-photon Schroedinger-equation complex-valued transverse vector wave function is obtained. It is then pointed out that despite the free-photon Klein-Gordon equation's being formally identical to the source-free classical-electromagnetic vector-potential wave equation, it yields not only free-photon Schroedinger-equation wave functions but also their complex conjugates, which don't satisfy the free-photon Schroedinger equation. This is a consequence of admitting complex-valued solutions of the Klein-Gordon equation -- of course only its real-valued solutions apply to the classical vector potential. It is pointed out that solutions of the free-particle Dirac equation likewise occur in conjugate pairs, and that its Hamiltonian operator implies a variety of unphysical consequences, e.g., any Dirac free particle's speed is that of light times the square root of three.
Classical equations of motion that are first-order in time and conserve energy can only be quantized after their variables have been transformed to canonical ones, i.e., variables in which the energy is the system's Hamiltonian. The source-free version of Maxwell's equations is purely dynamical, first-order in time and has a well-defined nonnegative conserved field energy, but is decidedly noncanonical, an issue that has attracted little attention. The opposite parities of the electric and magnetic fields and consequent curl operations that typify Maxwell's equations are especially at odds with their being canonical fields. Transformation of the magnetic field into the transverse part of the vector potential helps but is not sufficient; further simple nonnegative symmetric integral transforms, which commute with all differential operators, are needed for both fields; such transforms also supplant the curls in the equations of motion. The canonical replacements of the source-free electromagnetic fields remain transverse-vector fields, but are more diffuse than their predecessors. Combined as the real and imaginary parts of a complex field, the canonical fields prove to be the transverse-vector wave function of a time-dependent Schroedinger equation whose Hamiltonian operator is the quantization of the free photon's square-root relativistic energy. Thus proper quantization of the source-free Maxwell equations is identical to second quantization of free photons that have normal square-root energy. There is no physical reason why first and second quantization of any relativistic free particle ought not to proceed in precise parallel, utilizing the square-root Hamiltonian operator. This natural procedure leaves no role for the completely artificial Klein-Gordon and Dirac equations, as accords with their grossly unphysical properties.