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Quantization in Classical Electrodynamics
Abstract
The theory of a system consisting of two electrically charged particles is deduced using classical electrodynamics. This theory is applied to hydrogen and hydrogen-like atoms.
... where z 0 and E 0 represent the Rayleigh range and amplitude of the beam, with r = x 2 + y 2 1/2 as radial coordinate. Evidently, such a field distribution does not fulfill the transverse constraint of Maxwell's equations -Gauss's law in vacuum [27] -∇ · E = 0. However, it is possible to revise Eq. (2) accordingly by introducing a longitudinal field component [28]. ...
... Following this line of arguments, we can derive a similar expression for the focal distribution of the magnetic field of the described Gaussian beam. By starting with a y-polarized magnetic field -perpendicular to the xpolarized electric field -and by applying Gauss's law of the magnetic field [27], ∇ · H = 0, we result in ...
... To calculate the far field of the magnetic components of the dipole emitter, a second rotation matrixR is introduced, which is required due to interchanging electric and magnetic field vectors [27,28]: ...
When a beam of light is laterally confined, its field distribution can exhibit points where the local magnetic and electric field vectors spin in a plane containing the propagation direction of the electromagnetic wave. The phenomenon indicates the presence of a non-zero transverse spin density. Here, we experimentally investigate this transverse spin density of both magnetic and electric fields, occurring in highly-confined structured fields of light. Our scheme relies on the utilization of a high-refractive-index nano-particle as local field probe, exhibiting magnetic and electric dipole resonances in the visible spectral range. Because of the directional emission of dipole moments which spin around an axis parallel to a nearby dielectric interface, such a probe particle is capable of locally sensing the magnetic and electric transverse spin density of a tightly focused beam impinging under normal incidence with respect to said interface. We exploit the achieved experimental results to emphasize the difference between magnetic and electric transverse spin densities.
... Afterwards the numerous experiments of other physicists [11]- [16] pointed to the conclusion that the ra- ification of theory in this way is also present" [17]. My statistical analysis of the results of said experiments has shown that the ratio em E ν is not constant, i.e., Planck's h is not constant [18]- [24]. Harmonious results between theory and measurements are obtained using relativity and one alternative law. ...
... Stable states and discretization appear in the atom as a result of the harmonization of two different periodical processes; electromagnetic oscillations and the circular motion of electrons [18] [19] and [24]. Equation (13) extended to other states of atoms (not just to the first state, i.e., not just 1 1 n ± = ); read [24], Equation (83): ...
... The quantization is only a consequence of harmonizing two continuous processes in the atom: the propagation of electromagnetic energy and the circular motion of electrons [18] [24]. ...
... where z 0 and E 0 represent the Rayleigh range and amplitude of the beam, with r = x 2 + y 2 1/2 as radial coordinate. Evidently, such a field distribution does not fulfill the transverse constraint of Maxwell's equations -Gauss's law in vacuum [27] -∇ · E = 0. However, it is possible to revise Eq. (2) accordingly by introducing a longitudinal field component [28]. ...
... Following this line of arguments, we can derive a similar expression for the focal distribution of the magnetic field of the described Gaussian beam. By starting with a y-polarized magnetic field -perpendicular to the xpolarized electric field -and by applying Gauss's law of the magnetic field [27], ∇ · H = 0, we result in ...
... To calculate the far field of the magnetic components of the dipole emitter, a second rotation matrixR is introduced, which is required due to interchanging electric and magnetic field vectors [27,28]: ...
When a beam of light is laterally confined, its field distribution can exhibit points where the local magnetic and electric field vectors spin in a plane containing the propagation direction of the electromagnetic wave. The phenomenon indicates the presence of a non-zero transverse spin density. Here, we experimentally investigate this transverse spin density of both magnetic and electric fields, occurring in highly-confined structured fields of light. Our scheme relies on the utilization of a high-refractive-index nano-particle as local field probe, exhibiting magnetic and electric dipole resonances in the visible spectral range. Because of the directional emission of dipole moments which spin around an axis parallel to a nearby dielectric interface, such a probe particle is capable of locally sensing the magnetic and electric transverse spin density of a tightly focused beam impinging under normal incidence with respect to said interface. We exploit the achieved experimental results to emphasize the difference between magnetic and electric transverse spin densities.
... However, after the explanation of radiation [1][2][3][4], Maxwell's equations can now contribute to modern physics much more than before. From Maxwell's equations we derive Schrödinger's equation, which is the basis of quantum physics. ...
... A minimum of two separate oscillating processes are performed simultaneously within an atom, i.e., the circular motion of electrons around the nucleus and oscillation of electromagnetic wave energy [3]. The time period of one circular tour of electrons around the nucleus is T e = 2rπ/v = 1/f, where f is the frequency of circulation of ...
... Long term existence of the rotation of electrons and long term existence of the electromagnetic wave in the atom (stationary state) is only possible if there is synchronism between them (synchronously stationary state) [3,4]. Namely, to be coherent with the active power of the electromagnetic wave in an atom, the electron needs to oscillate (i.e., rotate) with dual frequency of the wave, because the active power of wave oscillates with dual frequency 2ω = 2(2πν), (this will be further discussed in Sub-Heading 4.5). ...
... This LC oscillator I use after I read Planck's review on the Bohr model of the atom [4]. Complaints about the collapse of atoms in such theories are eliminated through full access to the atom which includes emission and absorption of the radiation [5,6], where it is possible to absorb and its own radiation emission. Electromagnetic radiation, which we observe in an area outside of the atom, has its source in the atom. ...
... is the impedances ratio [5], and when we put the result from Eq. (24) in Eq. (23) we get: ...
... (1) A vacuum (a space empty of matter, which has no substance), (2) A particle (electron, proton, neutron), (3) An element (a basic substance that can't be simplified; hydrogen, oxygen, etc.), (4) An atom (the smallest unit of an element, having all the characteristics of that element and consisting of a very small and dense central nucleus containing protons and neutrons, surrounded by one or more shells of orbiting electrons; atoms remain undivided in chemical reactions except for the donation, acceptance, or exchange of valence electrons), (5) A nuclide (which may be isotopic nuclides, with the same atomic number Z, isobaric nuclides, with the same mass number M or isotonic nuclides, with the same difference between the mass number and the atomic number, M-Z), (6) A molecule (which has no more properties of its constituent atomic parts), (7) A compound (a molecule that contains more than one element), and (8) An ion (it is an element, atom, nuclide, molecule or compound, in which the total number of electrons is not equal to the total number of protons, giving the element, atom or molecule a net positive or negative electrical charge). ...
... With the now known more than 3180 nuclides, it can be said that quantity Type of substance is about 0.0023 percent filled. oscillator in an atom, and Z is atomic number [6,7,8,9] (do not replace the structural coefficient with the structural constant; those are two related but different things; σ =s0 2 /Z ). b. ...
The International System of Units (SI) is founded on 7 SI base units (meter, kilogram, second, ampere, kelvin, mole, candela) for 7 base quantities (length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity) assumed to be mutually independent. There is a need for a new quantity that describes the quality or type of the substance. The processing and introduction of this quantity is investigated. The method of the maximum possible atomic number of elements in Mendeleev’s Periodic Table was used. Here is a proposal of the 8th base quantity called type of substance and its unit boscovich, B. This quantity cannot be derived from 7 existing quantities and therefore should be introduced as a separate new one. It is also explained why the unit boscovich for type of substance is suggested.
... In order to emphasize the fundamental difference between a spinning (circularly polarized) dipole and the σ-dipoles on which we will focus mainly in this study, we plot corresponding radiation patterns in Fig. 1 (a)-(f). The color-code refers to the helicity σ defined by the normalized far-field Stokes parameter S 3 = (I rhc − I lhc )/(I rhc + I lhc ) [16], with I rhc and I lhc the intensities of the emitted right-and left-handed circularly polarized fields, respectively. In Fig. 1 (a)-(c) the dipoles are emitting in free-space, where (a) shows the circularly polarized electric dipole with p = p 0 (1, −ı, 0). ...
... For this purpose, we first design an excitation field capable of inducing electric and magnetic dipole moments, which are oriented parallel and have a phase difference of π/2. For paraxial beams this is not possible because there electric and magnetic fields are always perpendicular to each other [16]. One possible solution can be realized by tightly focusing a superposition of radially and azimuthally polarized vector beams, which exhibit purely z-polarized electric and magnetic fields on the optical axis, respectively [5,21]. ...
We discuss the excitation of a chiral dipolar mode in an achiral silicon nanoparticle. In particular, we make use of the electric and magnetic polarizabilities of the silicon nanoparticle to construct this chiral electromagnetic mode which is conceptually similar to the fundamental modes of 3D chiral nanostructures or molecules. We describe the chosen tailored excitation with a beam carrying neither spin nor orbital angular momentum and investigate the emission characteristics of the chiral dipolar mode in the helicity basis, consisting of parallel electric and magnetic dipole moments, phase shifted by . We demonstrate the wavelength dependence and measure the spin and orbital angular momentum in the emission of the excited chiral mode.
... shall be called the action of the electromagnetic oscillator, [6,11,12], and ...
... . Each oscillator has its own action constant. Equation (11) is then: ...
Model of an atom by analogy with the transmission line is derived using Maxwell’s equations and Lorentz’ theory of electrons. To be realistic such a model requires that the product of the structural coefficient of Lecher’s transmission lines σ and atomic number Z is constant. It was calculated that this electromechanical constant is 8.27756, and we call it structural constant. This constant builds the fine-structure constant 1/α = 137.036, and with permeability μ, permittivity e and elementary charge e builds Plank’s constant h. This suggests the electromagnetic character of Planck’s constant. The relations of energy, frequency, wavelength and momentum of electromagnetic wave in an atom are also derived. Finally, an equation, similar to Schrodinger’s equation, was derived, with a clear meaning of the wave function, which represents the electric or magnetic field strength of the observed electromagnetic wave.
... Elliptically polarized dipoles in free-space.-The farfield emission pattern of an elliptically polarized dipole in free space, whose dipole moment is, without loss of generality, parallel to the y-z-plane, p = p y e y + p z e z ≡ |p y | e y + exp (ı∆ϕ) |p z | e z , with ∆ϕ the relative phase between the dipole components, is given by [2,18] ...
We investigate points of circular polarization in the far field of elliptically polarized dipoles and establish a relation between the angular position and helicity of these C points and the dipole moment. In the case of highly eccentric dipoles, the C points of opposite handedness exhibit only a small angular separation and occur in the low intensity region of the emission pattern. In this regard, we introduce an optical weak measurement approach that utilizes the transverse electric (azimuthal) and transverse magnetic (radial) far-field polarization basis. Projecting the far field onto a spatially varying post-selected polarization state reveals the angular separation and the helicity of the C points. We demonstrate the applicability of this approach and determine the elliptical dipole moment of a particle sitting on an interface by measuring the C points in its far field.
... Elliptically polarized dipoles in free-space.-The farfield emission pattern of an elliptically polarized dipole in free space, whose dipole moment is, without loss of generality, parallel to the y-z-plane, p = p y e y + p z e z ≡ |p y | e y + exp (ı∆ϕ) |p z | e z , with ∆ϕ the relative phase between the dipole components, is given by [2,18] ...
We investigate points of circular polarization in the far field of elliptically polarized dipoles and establish a relation between the angular position and helicity of these C points and the dipole moment. In the case of highly eccentric dipoles, the C points of opposite handedness exhibit only a small angular separation and occur in the low intensity region of the emission pattern. In this regard, we introduce an optical weak measurement approach that utilizes the transverse electric (azimuthal) and transverse magnetic (radial) far-field polarization basis. Projecting the far field onto a spatially varying postselected polarization state reveals the angular separation and the helicity of the C points. We demonstrate the applicability of this approach and determine the elliptical dipole moment of a particle sitting on an interface by measuring the C points in its far field.
In the study of experiments of laser spectroscopy, there appears a convergence of the methods of quantum electrodynamics and classical optics: for instance stochastic electrodynamics used for the study of "squeezed states" is common to both theories, and the quantum coherent states are almost classical states. The author shows that this convergence allows to explain the paradoxes of quantum mechanics. The interaction of ultrashort laser pulses with ordinary matter is equivalent to the interaction of incoherent light with extremely dilute gases. Thus, the interaction of light from stars with cosmic gas produces a redshift similar to the Doppler redshift. In a very low pressure gas, the absorption of incoherent light disappears completely, so that the "black matter" could be simply H2 and its products of decomposition by high-frequency radiation.