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Physical Review A

Photon dynamics

Richard J. Cook

Published Apr 1, 1982• 75 citations
Publisher

Abstract

A theory of the motion of photons in physical space is presented. Operators representing the photon density and the photon current density are defined in terms of a pair of vector field operators in much the same way as the energy density and the Poynting vector are defined in terms of the electric and magnetic fields. For polychromatic light, this theory predicts an ideal photon-counting rate which differs from that of the usual theory of photoelectron counting.

DOI
10.1103/physreva.25.2164

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A theory of the motion of photons in physical space is presented. Operators representing the photon density and the photon current density are defined in terms of a pair of vector field operators in much the same way as the energy density and the Poynting vector are defined in terms of the electric and magnetic fields. For polychromatic light, this theory predicts an ideal photon-counting rate which differs from that of the usual theory of photoelectron counting.
The photon dynamics proposed by Cook [Phys. Rev. A 25, 2164 (1982); 26, 2754 (1982)] is reformulated in terms of conventional quantum mechanics. Here, the subject is confined to a photon in vacuum. The photon density D(x,t) and photon current density C(x,t) in the Cook theory are here regarded as the probability density and probability current density of a single photon, respectively. A wave function \ensuremath{\Psi} for a photon is defined by a 6\ifmmode\times\else\texttimes\fi{}1 column matrix ${(}_{\mathrm{\ensuremath{\varphi}}}^{\mathrm{\ensuremath{\psi}}}$) where \ensuremath{\psi} and \ensuremath{\varphi} correspond to the photon vector fields of the Cook theory. The Schr\"odinger equation for \ensuremath{\Psi} is easily obtained from equations of photon fields \ensuremath{\psi} and \ensuremath{\varphi} and the Hamiltonian form is determined as the time-evolution operator. By this reformulation, negative-frequency solutions were shown not to be physical and need not be considered for a physical photon. This explains why, in the Cook theory, they are dismissed from plane-wave expansions of photon fields \ensuremath{\psi} and \ensuremath{\varphi}.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev. A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of quantum mechanics, this wave function gives the probability to count a photon at any position in space.
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.
A photon number density operator proportional to the dot product of the electric field and the vector potential is introduced as an alternative to the Mandel operator. In the Lorentz gauge it is the time component of the four-vector obtained by contracting the electromagnetic field tensor with the vector potential. Its other components can be interpreted as a current-density vector, and these number and current-density operators satisfy a continuity equation. The photon density operators introduced here are all products of operators that satisfy Maxwell's equations and whose Lorentz and gauge transformation properties are well known.