This study deals with the physical (not the mathematical) origin of the quantum conditions. The assumption of an absolute space together with the speed of 369.5 km/s of the sun leads to cycloidal or helical orbits of the electron around the proton. This kind of orbits is a previously unnoticed fact and not compatible with the model of a closed circular standing wave. It is the aim of this study to search for alternative models. The Alternating Field model (AFM) assumes the existence of alternating fields for both the proton and the electron. The Higgs-field provides the energy requirements. By Puthoff’s effect, adapted for the electrostatic field, the electron changes its Compton frequency according to its position in the field. The interference of the alternating fields of the proton and the electron causes beats that are integer multiples of the orbital frequencies when the electron moves on Bohr’s orbits. Important results of this study are: The electrostatic Schwarzschild-radius is exactly half the value of the classical electron radius. Bohr’s first three principal quantum numbers, 1, 2, 3, are reproduced as 1.000052, 2.000214, and 3.000685. A two-dimensional simulation shows that the AFM causes orbits with ribbon shape. A formula, using only the orbital frequencies of the two innermost circular orbits, can predict the shortest wavelength of the Lyman series. The AFM is still incomplete. E.g., it does not prevent illegal orbits, but only indicates them. Also, the AFM does not use the magnetic core field of the proton. On the other hand, the model has produced results showing that Newton’s absolute space and the QM are compatible.
This study deals with the physical (not the mathematical) origin of the quantum conditions. The assumption of an absolute space together with the speed of 369.5 km/s of the sun leads to cycloidal or helical orbits of the electron around the proton. This kind of orbits is a previously unnoticed fact and not compatible with the model of a closed circular standing wave. It is the aim of this study to search for alternative models. The Alternating Field model (AFM) assumes the existence of alternating fields for both the proton and the electron. The Higgs-field provides the energy requirements. By Puthoff’s effect, adapted for the electrostatic field, the electron changes its Compton frequency according to its position in the field. The interference of the alternating fields of the proton and the electron causes beats that are integer multiples of the orbital frequencies when the electron moves on Bohr’s orbits. Important results of this study are: The electrostatic Schwarzschild-radius is exactly half the value of the classical electron radius. Bohr’s first three principal quantum numbers, 1, 2, 3, are reproduced as 1.000052, 2.000214, and 3.000685. A two-dimensional simulation shows that the AFM causes orbits with ribbon shape. A formula, using only the orbital frequencies of the two innermost circular orbits, can predict the shortest wavelength of the Lyman series. The AFM is still incomplete. E.g., it does not prevent illegal orbits, but only indicates them. Also, the AFM does not use the magnetic core field of the proton. On the other hand, the model has produced results showing that Newton’s absolute space and the QM are compatible.
The classical electrodynamic conclusion that an accelerating point charge mustradiate has long been regarded as an iron law, yet the electron in the hydrogen atom does notradiate and fall into the nucleus. This paper points out that the root of this contradiction is not inclassical electromagnetic theory itself, but in the untested hidden assumption that “the electronis a point particle”. Compton scattering experiments have already shown that the electron, whencolliding with a photon, behaves as a particle with a finite size, and its intrinsic geometric radius isgiven by the reduced Compton wavelength: re = ℏ/(mec) . Based on this physical fact, andconsidering that while the electron revolves around the nucleus (orbital motion) it also has anintrinsic spin, the electron itself generates two homologous electromagnetic radiation fields:orbital radiation and spin radiation. These two real electromagnetic waves always superpose andinterfere in space. Under specific conditions (frequencies forming an integer ratio, oppositephases, and matching amplitudes), the two waves completely interfere destructively, resulting inzero net external radiation and forming a stable standing wave inside the system– this is thestationary state of the hydrogen atom. The destructive interference condition naturally leads toωs = nΩ (n integer) and re/R = α/n2 , where α is the fine-structure constant. Combined withthe Coulomb force (centripetal force) equation, one uniquely obtains the quantized orbital radiiRn = n2a0 (a0 is the Bohr radius) and the angular momentum quantization Ln = nℏ . Thefine-structure constant thus acquires a geometric meaning: α = re/a0 .This paper further reveals the classical nature of wave-particle duality: the electron is always aparticle (with a definite trajectory), and its wave-like behavior comes from the superposition andinterference of the electromagnetic waves of spin and orbital radiation. The “particle-like”behavior observed in experiments is essentially the point-like signal produced when the photonradiated by the electron is absorbed, while the wave-particle duality of the photon itself is an1intrinsic property of it being an electromagnetic wave packet. Based on this picture, the paperprovides a classical causal explanation for the electron double-slit experiment– the electron asa particle barely passes through the slits; the interference fringes originate from theself-interference of the radiation wave packet (photon) generated by the electron interactingwith the slit plate, and the bright spots on the screen come from the discrete absorption eventsof photons, with the accumulation of many events forming the fringes. Observation inevitablydisturbs the electron because any measurement disrupts the destructive interference condition,thereby eliminating the mystery of “wavefunction collapse”.The paper also systematically expounds the electromagnetic nature of the photon: the photon isan electromagnetic wave packet radiated when the electron transitions between stationarystates due to the imbalance of interference; its energy E = hλ is determined solely by thefrequency, with constant amplitude; the photon is born at the speed of light because it originatesfrom a pre-existing electromagnetic wave. Finally, the paper discusses the essence of theelectromagnetic field: the electromagnetic field is not matter but energy; the vacuum is theground state electromagnetic field (filled with photons), and the cosmic microwave backgroundradiation is its observational evidence. Macroscopic electromagnetic fields (such as radio waves,visible light) are perturbations superimposed on the ground state field, acting as carriers ofenergy and information. The paper also links the background photon field to the cosmicmicrowave background radiation, providing a new causal picture for the interaction mechanismof charged particles and the inverse-square law of forces.This model is completely based on classical electromagnetism and the physical fact that theelectron has a finite size, providing a causally clear and physically intuitive unified picture for thestability of the hydrogen atom, quantized orbits, wave-particle duality, the nature of the photon,the double-slit experiment, the measurement problem, and the energy origin of theelectromagnetic field.The model further suggests that the proton may also be a similar self-sustaining electromagneticbound state, and the origin of cosmic matter can be attributed to the process of energy lockingby the condensation of electrons and positrons, offering a unified perspective for physicalphenomena from the atom tothe cosmos.
In 1925 the concept of electron spin was introduced to explain two experimental observations: the splitting of silver atom beams in an inhomogeneous magnetic field (Stern–Gerlach, 1922), and the fine structure of atomic spectra. The physical picture proposed — an electron spinning on its own axis — was immediately shown by Hendrik Lorentz to require surface velocities exceeding the speed of light. The paper was published anyway. Dirac subsequently rescued the two-valuedness mathematically, whereupon the physical question was declared irrelevant. For one hundred years the field has been unable to state what spin physically is. This paper traces the sequence of observations, inferences, and decisions that produced this situation and identifies two distinct wrong turns. The first wrong turn was publishing a model known to be physically impossible. Lorentz's calculation was correct — at the classical electron radius (2.818 fm), the required surface velocity is 274 times the speed of light. But the classical radius is not the electron's physical rotation radius. The correct rotation radius, derived from the Sagnac closure condition, is 285.5 fm — 101 times larger. At that radius, the rotation velocity is 0.822c, the bare magnetic moment is 0.608 μB, and Lorentz's objection does not arise. Both values are computed explicitly with no free parameters. The second wrong turn was a conflation. The closure geometry of the electron involves two independent numerical facts, both related to the number ½: (1) the least-work oscillation path that closes the Sagnac phase condition is a two-lobe ellipse, giving a winding number of n = ½ in the mass equation; (2) the electron's S³ closure surface presents two poles to the external field, giving a topological factor of 2 in the g-factor. In 1925, both facts were numerically present in the spectroscopic data, but they were collapsed into a single postulate: intrinsic angular momentum ℏ/2. That collapse created a half-integer angular momentum with no classical analogue, forced Dirac to construct the SU(2) spinor formalism to accommodate it, and caused the spinor to appear as a new kind of mathematical object requiring new physics — when it was always a description of closure surface topology. Every experimental result attributed to spin-½ as an intrinsic quantum property — two spots in Stern–Gerlach, the electron magnetic moment, Pauli exclusion, and fine structure — is reproduced by real rotation with two stable orientations, without superluminal motion, without abstract mathematical structure disconnected from physical content, and without the hundred-year admission of ignorance that has accompanied the standard treatment. The mathematical apparatus of quantum mechanics — the Dirac equation, the Pauli matrices, the SU(2) structure — remains valid. What changes is the interpretation. The spinor is not a mysterious new object. It is the natural representation of the two-pole topology of the S³ closure surface. The conflation of 1925 prevented this from being seen. Related work: The Sagnac closure condition and the γcause invariant used in this paper are derived in: D. J. Hallman, "The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure," Zenodo (2026). doi:10.5281/zenodo.20193279
The atom model of Niels Bohr is commonly considered as antiquated even if it describes the atomic spectrum of hydrogen quite accurately. The later published relation of Louis De Broglie could arithmetically be implemented into Bohr’s formulation, leading to the concept of standing waves as the existence cause of excited electron states. However, at that time it was not possible to find well defined electron trajectories being classically describable. As a consequence, the »quantum mechanics« were developed by several authors, particularly by Schrodinger and by Heisenberg, but delivering a hardly under¬standable formalism where the classical physical laws appear being abrogated while abstract terms replace concrete and imaginable ones, abandoning the particle perception of mass points. In particular, Heisenberg’s »uncertainty principle« and the assumption of state probabilities seem to be in a striking variance to the idea of standing waves. In contrast, a formulation is given here which exactly describes the electron trajectories in the exited states solely by applying classical physical laws. Firstly, the original Bohr-model - in combination with De Broglie’s relation - is rolled up. From this formula system a vibration frequency - corresponding to the De-Broglie frequency – is deduced which is n-times larger than the rotation frequency of the Bohr-model. Furthermore, a direct coherence between that vibration-frequency of the electron and the frequency of the involved light is evident being explainable as a resonance effect. Then, a three-dimensional model is proposed where the electron oscillates and pulses perpendicularly to a virtual rotation plane i.e. rotating around a vertical axis, accompanied by a perpetual energy exchange between potential and kinetic energy. This leads at the excited, metastable energy states to well-defined, three-dimensional and wavy trajectories winding up on a surface similar to the one of a hyperboloid, whereas at the ground state the trajectory is planar and stable.
This article is a continuation of Part(I) to demonstrate more aspects for a successful merge of classical mechanics to the quantum theory. The article starts first with the deduction of the general form of the complimentary energy, next a robust firm mathematical proof is given indicates that the complimentary energy has a wave nature and it is a potential energy, then the radiation condition has been determined, a vector formula expresses the corresponding force was deduced and this force was called adhesion force. A mathematical convergence process has proved the duality of the electromagnetic wave parameters and particle parameters, which results in the existence of a “wave-particle” object. While finding the relationship between the electromagnetic wave parameters and particle parameters, a mathematical relationship was found indicating that not every moving charged mass generates a magnetic field. This relationship for the mass was linked to the effective mass tensor, The geometric representation of the constant energy surface and its relationship to the energy levels of both the orbital model (Bohr model) and the cloud model for different geometric 3D shapes were investigated as a sample of the possible geometric interpretations of the energy equation, that is by employing the triangle method, it is possible to create a geometric representation of the constant energy surfaces. The relation between the energy levels, quantum numbers, and band width is represented geometrically. As a result of proving the relationship between the motion of an electromagnetic wave and the motion of a particle associated with the wave, two new, unprecedented mathematical formulas were developed. The first formula specifies a moving charged mass and its relationship to the resulting wavelength. Thus, the radius of the charged mass is obtained in terms of its wavelength and mass density. The second formula specifies the kinetic energy of a photon. The grand potentials are the potential energy of the Newton’s Law of Universal Gravitation, Coulomb’s Law, and Inverse Square Law of Electromagnetic Waves Intensity. From the potential energy of Newton’s law of universal gravitation, the wavelength of the gravitational wave, as well as the wavelength of quantum gravity were deduced. A mathematical relationship was then deduced that determines the lengths of the functions of conic sections, and in particular the length of perimeter of ellipses. From the potential energy of Coulomb’s law of attraction, a mathematical formulation of the energy levels in general for any atom of an element was obtained. This mathematical formulation was compared with the results obtained by Bohr and Schrödinger for the hydrogen atom. As for the inverse square law of electromagnetic radiation intensity, several innovative mathematical equations have been deduced that link different parameters together. Following the contents of this article requires to be quite familiar with the contents of the first part with the same title, as well as a complete understanding of the concepts of intermediate parameters derived and used in the various equations. Otherwise, it will be extremely difficult to follow the contents of this article.
In standard quantum mechanics, the electron shell filling rule is explained using four quantum numbers (), the Pauli exclusion principle, and spin as a fundamental property of the electron. This is not an explanation – this is a description. The question remains: why exactly ? In the TVM framework, the rule ceases to be an empirical rule or a consequence of quantum postulates. It becomes a geometric necessity – a direct consequence of: The radius of the -th level: (derived in the main work) The wavelength at the fundamental level: (derived in the First supplementary work) The condition of wave stationarity – so that destructive interference does not occur Starting from these TVM results, the work derives the rule as follows: The circumference of the orbit at level is The minimum distance between two waves on the same orbit is The maximum number of waves that can fit on the circumference is The work also derives the geometric origin of the subshells, showing that they are not spatial orientations, but places where extrema (maxima and minima) of the stationary wave on a circular orbit are realized. The number of possible configurations (arrangements of extremum points on the circle, not spatial orientations) relative to zero points gives the capacities 2, 6, 10, 14..., and their sum yields . This demonstrates that: Spin does not exist as a phenomenon – the factor 2 originates from geometry, not from spin The Pauli principle is not a separate law – it is a description of a geometric constraint Orbital quantum numbers are not necessary – at the fundamental level, comes directly The periodic system is not empirical – it is geometrically necessary The rule is not a hypothesis – it is recognized in the TVM structure. It has always been there, in the geometry of the circle, the wave nature of the electron, and the condition of wave stationarity to avoid destructive interference. Keywords: TVM, shell filling, , geometry, wavelength, interference, Pauli principle, periodic system, s, p, d, f, stationary state, standing wave. NOTE: This is the second in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the results of the First supplementary work from the series (de Broglie relation), this work derives the electron shell filling rule from pure geometry, without introducing spin, the Pauli principle, or orbital quantum numbers. It is recommended that the reader first study the First supplementary work [2], as well as Chapters 3 and 4 of the main work [1].
Niels Bohr’s model of the hydrogen and hydrogen-like atoms, in which the electron is allowed to revolve in certain circular orbits around the nucleus, was extended by Arnold Sommerfeld to allow for the possibility of elliptical orbits. Each elliptical orbit was constrained by two quantization conditions specified by a positive integer nφ and a non-negative integer nr. If, in the presence of the attractive Coulomb force between the electron and the nucleus, the dynamics of the atom were taken to be governed by the Newtonian laws of motion, the energy Ɛ of the two-particle system was found to be a function of the positive integer n = nr + nφ. The atomic energy levels thus exhibited an n-fold degeneracy, which would be lifted in the presence of external electric or magnetic fields, thus accounting for the fine-structure of the spectral lines emitted by the atom—phenomena observed, respectively, in the Stark effect and the Zeeman effect. Sommerfeld also examined the behavior of his elliptical orbits by allowing the revolving electron to obey Einstein’s relativistic dynamics. In this case, the energy-level degeneracies were lifted (even in the absence of external perturbations), and the elliptical orbits were found to precess around the nucleus—similarly to the precession of, say, the orbit of the planet Mercury around the Sun. What is astonishing about Sommerfeld’s relativistic expression of the energy Ɛ as a function of nφ and nr is that it is precisely the same expression as obtained for hydrogen and hydrogen-like atoms from Dirac’s equation, which came more than a decade later. Recall that the spin of the electron was not yet discovered when Sommerfeld developed his model, and that in Dirac’s fully-relativistic quantum mechanics, the spectral fine-structure arises from the coupling of the electron’s spin to its orbital motion. The present paper provides a streamlined yet detailed account of Sommerfeld’s elliptical orbits in hopes of bringing about a deeper appreciation for the miraculous coincidence of Sommerfeld’s result with that derived from Dirac’s equation.