It has been found that, if a dispersion-free primary wave packet being identical with a material particle in the four-dimensional space is introduced, then the Fourier central component of the primary wave packet has a factor which is the de Broglie wave function. Thus it is infered that the electromagnetic wave is related to the Fourier central component of the photon as a primary wave packet. A light pulse like the Schrodinger wave packet is a secondary wave packet which may be even infinitely short in time, even infinitely small in space and may contain even infinitely small average number of photons.
It has been found that, if a dispersion-free primary wave packet being identical with a material particle in the four-dimensional space is introduced, then the Fourier central component of the primary wave packet has a factor which is the de Broglie wave function. Thus it is infered that the electromagnetic wave is related to the Fourier central component of the photon as a primary wave packet. A light pulse like the Schrodinger wave packet is a secondary wave packet which may be even infinitely short in time, even infinitely small in space and may contain even infinitely small average number of photons.
The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrödinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.
A physics-oriented review is first given of wave equations: examples, the Cauchy problem, solutions in various coordinates, their symmetries, how to build a wave packet. Fourier analysis and dispersion relations are then introduced, and here the key tool is the symbol of differential operators, which makes it possible to study them in terms of algebraic polynomials involving cotangent-bundle variables. Further basic material deals with geometrical optics from the wave equation, phase and group velocity (and their dual relationship in momentum space), the Helmholtz equation and the eikonal approximation for the scalar wave equation. The Schrödinger equation is then derived with emphasis on the wave packet and its relation to classical behaviour in the light of the Einstein–de Broglie relation. For this purpose, it is shown that it is possible to build a wave packet for which the wave-like properties manifest themselves for distances of the order of atomic dimensions. The Fourier transform with respect to time of the wave packet obeys the stationary Schrödinger equation, while the wave function is found to obey a partial differential equation which is of first order in time. The wave equation The wave equation occurs in several branches of classical physics, e.g. the theory of sound, electromagnetic phenomena in vacuum and in material media, and elastic vibrations of material bodies. We consider first the simplest situation, i.e. the propagation of waves in a homogeneous, isotropic and stationary medium. Propagation is characterized by the refractive index n , a quantity that is independent of the point in space-time.
On the base of the photon’s wave function (wave packet) in Schrodinger coordinate representation developed in previous papers the modeling of the propagation in space and in time of a wave packet of the photon corresponding to a separate laser impulse of radiation of femtosecond diapason is carried out. The development is proceeded from single-particle approach with the usage of the basis eigenfunctions of energy, momentum and helicity operators in continuous momentum spectrum case. The wave packet is constructed in the form of superposition of these eigenfunctions as the integral over the all momentum space. In the case of Gaussian momentum distribution the character of an extension of a form of this wave packet is established. The essence of wave-particle duality of light and microparticles is specified. The hypothesis is formulated, however, that the photon is a quasi-particle corresponding to the propagation of spin wave in physical vacuum. It is claimed that this question is related to the structure of the leptons and other particles on the Planck distances.
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).
This letter proposes an explanation for the century-old puzzle of the wave-particle-duality of light, which thousands of physicists, including Einstein, Plank, Feynman …, have been trying to resolve. Since a ‘particle ’ is localized in space, it is mathematically characterized here as an impulse-function in space. Then it is argued that if this ‘particle’ has anything to do with waves then we can know it by taking Fourier-transform of the impulse-function in space. When Fourier-transformed into wave-number-domain; we find that a ‘particle ’ should contain a wide ‘set ’ of waves, and not just a single frequency. Then we show that in the experiments performed so far [1] the red lasers had significantly wide line-width, means the sources have been producing a wide set of waves, and not just a pure single frequency. Similarly, in the single-particle interference-experiments incandescent filament-lamps were used with green filters inserted to isolate single photons; but it is obvious that at the frequencies of light very narrow-band-filters are not yet technically feasible, so the green filters used allowed significantly wide band of waves. So in the experiments performed so far we got ‘particles ’ localized in space. And in the double-slit-interference-experiments this wide band of waves passed from both the slits, interfered like waves, and whenever and wherever they got coherently added, a ‘particle ’ called ‘photon ’ got detected. Atoms emit short-duration pulses of radiation, so their Fourier transform contains a wide band of frequencies. And at the time of detection, an atom needs only a small piece of continuous ‘wave’, of a suitable duration, amplitude and frequency, for ejection of an electron. So it is concluded here that: emission and detection of light is in the form of ‘photons’; whereas its propagation in space is in the form of a wide band of ‘waves’. 2
The electromagnetic component waves, comprising together with their generating oscillatory massless charge a material particle, will be Doppler shifted when the charge hence particle is in motion, with a velocity v, as a mere mechanical consequence of the source motion. We illustrate here that two such component waves generated in opposite directions and propagating at speed c between walls in a one-dimensional box, superpose into a traveling beat wave of wavelength ${\mitΛ}_d$$=(\frac{v}{c}){\mitΛ}$ and phase velocity c2/v+v which resembles directly L. de Broglie's hypothetic phase wave. This phase wave in terms of transporting the particle mass at the speed v and angular frequency ${\mitΩ}_d=2πv /{\mitΛ}_d$, with ${\mitΛ}_d$ and ${\mitΩ}_d$ obeying the de Broglie relations, represents a de Broglie wave. The standing-wave function of the de Broglie (phase) wave and its variables for particle dynamics in small geometries are equivalent to the eigen-state solutions to Schrödinger equation of an identical system.