1. The analysis of single photon behaving in coordinate systems of various geometries has shown the following: · In Cartesian coordinate system the components of single photon eigen-vector are progressive and regressive plane waves. In other words, the photon behaving is characterized by coordinates and momentum, only. · In cylindrical coordinate system the components of single photon eigen-vector of are progressive and regressive plane waves, but only along z axis. In comparison to the results of analysis in Cartesian coordinates the fact that in x y planes photon oscillations are attenuated according to 1/2 law represents a generalization. This conclusion has been based on asymptotic behaving of Bessel functions. · The most authentic result is obtained by the analysis in spherical coordinates. Photon states do not depend on angles, and they are damped according to r1 law, where r is radius of the sphere. It has also turned out that orbital momentum of a free photon is equal to zero (this is understandable considering the fact that it moves along a straight line). The important element determining the behavior of a free photon is spin. One components of photon eigen-vector corresponds to spin Sz = 1/2 projection, while the other component corresponds to spin Sz = 1/2 projection. This was concluded on the basis of the fact that its eigen-vector components are Bessel functions having indices +1/2 and 1/2. The last result shows that linearization of photon Hamiltonian gives more complete picture of single photon than Kline-Gordon's approach. Concluding the exposed analysis we shall try to connect the results obtained in series of experimental investigation of photon orbital momentum (Beth, 1936; Leach, et al., 2002; Allen, et al., 1992; Allen, 1966; He, et al., 1995; Friese, et al., 1996; Markoski, et al., 2008; van Enk & Nienhuis, 2007; Santamato, et al., 1988; O'Neil, et al., 2002; Volke-Sepulveda, et al., 2002). We shall not describe all quoted experiments. Instead of it we shall describe the essential idea: the orbital momentum of photon was determines from the changes of
The analysis of single photon behaving in coordinate systems of various geometries has shown the following: · In Cartesian coordinate system the components of single photon eigen-vector are progressive and regressive plane waves. In other words, the photon behaving is characterized by coordinates and momentum, only. · In cylindrical coordinate system the components of single photon eigen-vector of are progressive and regressive plane waves, but only along z axis. In comparison to the results of analysis in Cartesian coordinates the fact that in x y planes photon oscillations are attenuated according to 1/2 law represents a generalization. This conclusion has been based on asymptotic behaving of Bessel functions. · The most authentic result is obtained by the analysis in spherical coordinates. Photon states do not depend on angles, and they are damped according to r1 law, where r is radius of the sphere. It has also turned out that orbital momentum of a free photon is equal to zero (this is understandable considering the fact that it moves along a straight line). The important element determining the behavior of a free photon is spin. One components of photon eigen-vector corresponds to spin Sz = 1/2 projection, while the other component corresponds to spin Sz = 1/2 projection. This was concluded on the basis of the fact that its eigen-vector components are Bessel functions having indices +1/2 and 1/2. The last result shows that linearization of photon Hamiltonian gives more complete picture of single photon than Kline-Gordon's approach. Concluding the exposed analysis we shall try to connect the results obtained in series of experimental investigation of photon orbital momentum (Beth, 1936; Leach, et al., 2002; Allen, et al., 1992; Allen, 1966; He, et al., 1995; Friese, et al., 1996; Markoski, et al., 2008; van Enk & Nienhuis, 2007; Santamato, et al., 1988; O'Neil, et al., 2002; Volke-Sepulveda, et al., 2002). We shall not describe all quoted experiments. Instead of it we shall describe the essential idea: the orbital momentum of photon was determines from the changes of
Introduction: In the rapidly developing areas of photonics associated with quantum teleportation of photons, quantum cryptography, and quantum computing, ideas are becoming increasingly relevant about the certain localized states of single or several entangled photons moving in space and time. As is known, in quantum mechanics, localized states of particles can only be described using wave functions in coordinate representation – the corresponding wave packets. Aims: To explain Young's one- and two-photon experiments using coordinate 6-component quantum mechanical and 1-component quasi-classical photon wave functions. Methods: The article outlines the method developed in authors' previous work in constructing the 6-component coordinate wave function of free photons in the form of the wave packet (an integral over the entire momentum space and sum over two possible values +1 and -1 of the helicity). The basis functions are the circularly polarized monochromatic waves, corresponding to certain values of the momentum, helicity, and photon energy operators and forming a complete orthonormal set of generalized eigenfunctions of these operators. This photon wave function (PWF) is normalized to the unit probability of finding the photon anywhere in space. Both the basis functions and the coordinate PWF satisfy a Schrödinger-type equation derived from the Maxwell-Lorentz equations written in quantum mechanical form first introduced by Majorana. Then the results of modeling and consideration of the space-time evolution of the wave packet with Gaussian momentum distribution corresponding to the directed photon emission for 80 fs are briefly presented. Further, in view of the possibility of constructing the coordinate PWF, the basic formula for the waveparticle duality of photons and particles is formulated. It is emphasized that the photon is fundamentally different from “ordinary” particles, since, according to the authors, the photon is the result of the propagation of a certain complex spin wave in a physical vacuum, the details of which can only be considered simultaneously with the study of the structure of the physical vacuum at Planckian distances. Results: Young`s one- and two-photon experiments are explained using various options for modeling 6-component coordinate PWFs, including approximate functions corresponding to spherically symmetric and electric dipole photon radiation. Conclusion: This explanation is illustrated using specific examples of radiation and, at the same time, is compared with numerical simulations for spherically symmetric radiation and the use of 1-component quasi-classical PWFs.
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.
In modern areas of photonics, the physical description of the interaction of photons with matter in the control, transmission and registration of single-photon and two-photon states implemented in practice is of great importance. An appropriate acceptable description may be faced with the need to take into account various kinds of interference effects associated with these states. Meanwhile, even the most “simple” case of single-photon interference in Young's experiment requires the use of a rather complex apparatus of quantum electrodynamics. This article explains one- and two-photon interference in Young's thought experiment based on the photon wave function (PWF) in coordinate representation. This explanation is illustrated by two examples of wavelengths: 10.6 μm and 1.5 cm. For both examples, two approaches to PWF modeling are used: “purely quantum-mechanical” and “quasi-classical”. In the first approach, a 6-component coordinate PWF is constructed using a spherically symmetric momentum distribution in a wave packet, followed by numerical and approximate analytical calculations. In the second approach, a one-component “quasi-classical” PWF is constructed, which corresponds to either electric dipole radiation or simulated spherically symmetric radiation. In all cases, the same pronounced interference pattern was obtained, which allows us to conclude that not only the quantum-mechanical coordinate PWF is able to explain the phenomena of one- and two-photon interference, but also a much simpler “quasi-classical” PWF. This conclusion sheds light on the theoretical interpretation of the measurement of the coordinate PWF in some recent experiments.
The main principles of photon quantum mechanics describing its single-particle states by means of the wave function (wave packet) in the coordinate representation are given. The results of modeling of the propagation in space-time of wave packet with a Gaussian distribution of the momentum of a photon corresponding to femtosecond laser radiation are discussed. On the basis of the general ideas about the evolution of this packet and within the framework of constructed photon quantum mechanics, the quantum-mechanical explanation of the experimentally registered (previously) reduction of speed of the free photons, “structured” in the same way as the light in Bessel and Gaussian beams, is proposed. It is emphasized that in fact the photon is not a some formed quantum corpuscle, but it is a quasi-particle, which is the result of propagation of spin wave in physical vacuum, the nature of which should be considered at the Planck distances simultaneously with structure of fundamental particles.
The description of photon-matter interaction upon control, transmission, and detection of single-photon, two-photon, and multiphoton states, including the entangled ones, will play an ever-increasing role in many areas of photonics. An appropriate description requires taking into consideration various types of interference effects associated with these states. However, the relatively complex apparatus of second quantization of the electromagnetic field is used even in the simplest single-photon experiments equivalent to the Young’s one, e.g., the experiments with the Mach–Zehnder interferometer. In the present work, the Young’s single-photon mental experiment is explained using the coordinate photon wave function (PWF). The explanation is illustrated by specific examples of the single-photon states at certain wavelengths and different duration of radiation within the framework of two approaches: the quantum mechanical and the “quasi-classical.” In the first approach, a 6-component coordinate PWF is constructed based on the spherically symmetric momentum distribution in a wave packet, followed by approximate analytical calculations. In the second approach, a one-component quasi-classical PWF corresponding to the electric-dipole radiation is constructed. The same pronounced interference pattern was obtained in both cases, which makes is possible to draw the conclusion that the coordinate PWF allows explaining the one- and two-photon interference phenomena. This conclusion sheds the light on theoretical interpretation of the measurement of the coordinate PWF carried out in some of the recent experiments.