The possibility to describe the evolution of an electromagnetic field by means of the photon distribution function in the phase space (r, q-space) is studied. This function defined by analogy with the coarse-grained Mandel operator of photon density in the configuration space is used to characterize the local density of photons with a given momentum. Approximate eigenfunctions and eigenvalues of the distribution function, corresponding to one-photon localized states of the electromagnetic field, are obtained. It is shown that the photon transport is governed by the Newton mechanics if the "external force" acting on photons is a slowly varying function of spatial variables. It is shown that the distribution function at any time can be expressed via the initialdistribution and photon's trajectories.
The possibility to describe the evolution of an electromagnetic field by means of the photon distribution function in the phase space (r, q-space) is studied. This function defined by analogy with the coarse-grained Mandel operator of photon density in the configuration space is used to characterize the local density of photons with a given momentum. Approximate eigenfunctions and eigenvalues of the distribution function, corresponding to one-photon localized states of the electromagnetic field, are obtained. It is shown that the photon transport is governed by the Newton mechanics if the "external force" acting on photons is a slowly varying function of spatial variables. It is shown that the distribution function at any time can be expressed via the initialdistribution and photon's trajectories.
Recently, the author [Phys. Rev. A 49, 2839 (1994)] proposed a quantum-mechanical theory of a photon, in which negative energy states can be dismissed from physical photon states without causing any difficulties. In this Brief Report the physical meaning of a photon wave function is investigated more precisely. The interpretation of a photon wave function as a probability amplitude is guaranteed by requiring the coarse-grained condition that the linear dimensions of a volume or a surface concerned in the configuration space are large compared to the photon wavelengths. The calculation made here is essentially based on that presented by Mandel [Phys. Rev. 144, 1071 (1966)] in his analysis of the photon number operator. But our calculation includes that of Mandel as the zeroth-order approximation. As a result, it is shown that although the position operator ◯ in the ordinary quantum mechanics be considered a correct photon operator strictly cannot, it becomes meaningful to some extent under the coarse-grained condition. In connection with ◯, we also examine the velocity operator $[\mathbf{◯},\ifmmode \hat{H}\else \^{H}\fi{}]/i\ensuremath{\Elzxh}$ ($\ifmmode \hat{H}\else \^{H}\fi{}$ is the photon Hamiltonian), and show that the probability current density is definable under the coarse-grained condition.
This chapter describes the photon wave function, and explains the basic properties of a well-defined mathematical object—a six-component function of space-time variables—that describes the quantum state of the photon. The most essential property that does not hold for the photon wave function is that the argument of the wave function cannot be directly associated with the position operator of the photon. The position operator for the photon simply does not exist. However, one should remember that for massive particles also, the true position operator exists only in the nonrelativistic approximation. The concept of localization associated with the Newton-Wigner position operator is not relativistically invariant. The photon wave function is not restricted to the wave mechanics of photons. The same wave functions also appear as mode functions in the expansion of the electromagnetic field operators.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
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.
The expressions of the eigenfunctions of the Hawton photon position operator in configuration space are derived for several classes of wave function, including the Riemann--Silberstein and Landau--Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals K(κ) and E(κ) with modulus κ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value q of the position eigenvalue, but also on a plane containing q and that they decay as inverse powers of the distance from q.