The formulation of the wave function of a single photon has always been in a con ict with the spatial localization of a photon. Lack of the correct treatment for the probability and current density of a single photon resisted self-consistent treatment of the wave function of a single photon. Here we proceed with the construction of the wave function for a single photon that allows us to introduce probability and current density of a single photon. The wave function of a single photon is constructed from the matrix elements of the electric and magnetic elds that couple vacuum state with the Fock state occupied by a single photon. Further we show that the spin of a photon, being projected on the direction of the photon ight propagation, de nes the time evolution of the photon wave function. As the result it can be identi ed with the Hamiltonian of a single photon that establishes Schrodinger equation for a single photon. The Schrodinger description naturally leads to the current and the probability density for a single photon that satis es the continuity equation. The Schrodinger formulation of the Maxwell equations provides a clear physical meaning for the spin of a photon in a similar way to the Dirac equation provides a natural explanation for the spin of an electron.
The formulation of the wave function of a single photon has always been in a con ict with the spatial localization of a photon. Lack of the correct treatment for the probability and current density of a single photon resisted self-consistent treatment of the wave function of a single photon. Here we proceed with the construction of the wave function for a single photon that allows us to introduce probability and current density of a single photon. The wave function of a single photon is constructed from the matrix elements of the electric and magnetic elds that couple vacuum state with the Fock state occupied by a single photon. Further we show that the spin of a photon, being projected on the direction of the photon ight propagation, de nes the time evolution of the photon wave function. As the result it can be identi ed with the Hamiltonian of a single photon that establishes Schrodinger equation for a single photon. The Schrodinger description naturally leads to the current and the probability density for a single photon that satis es the continuity equation. The Schrodinger formulation of the Maxwell equations provides a clear physical meaning for the spin of a photon in a similar way to the Dirac equation provides a natural explanation for the spin of an electron.
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}.
It is shown that a Dirac(-type) equation for a rank-two bi-spinor field ψph on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a quantum-mechanical photon wave equation. In particular, it implies that the photon wave function ψph yields conserved non-negative Born-rule-type quantum probabilities and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations E = ℏω and p = ℏk, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.
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.
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 present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.