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Helical Model of the Electron

Oliver Consa

Published Aug 1, 2014• 2 citations
OpenAlex

Abstract

A semiclassical model of the electron is presented based on the Principle of Helical Motion (“A free electron always moves at the speed of light following a helical motion, with a constant radius, and with the direction of movement perpendicular to the rotation plane”). This model interprets the Zitterbewegung as a real motion that causes rotation of the electron spin and its magnetic moment. Based on this model, the quantum magnetic flux and quantum Hall resistance are obtained as parameters of the electron and special relativity theory is derived from the helical motion of the electron. Finally, a fix is proposed for the De Broglie’s wavelength that questions the very validity of the Dirac equation.

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A Planck-Einstein-Dirac Wave Model of Electron

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The spherical model of electron cannot explain Bohr magneton, de Broglie wave, etc. To explain these properties physically, we propose that spin, one core concept of quantum theory, is a reality, and introduce this alternative solely based on experimental findings, Planck’s, Einstein’s, and Dirac’s insights, and follows classical laws, that an electron is an electrical wave with a literal $\frac{1}{2}$ spin (Dirac), i.e., spins two circles in one cycle, both angular momentum are ℏ. Its wavelength is determined by $E=h\nu_{m_0}=\frac{hc}{\lambda_{m_0}}=m_0c^2$ (Plank and Einstein), λ m 0 is the Compton wavelength. As E⃗ ⊥ c⃗ for a transverse wave, it has a net electricity due to the first spin, $f(E_q)=\frac{1}{2}E_0\left(1-\cos(2\omega_0t)\right)$, half of its cycle, i.e., the minimal meaningful unit as F = E q , is related with its charge, q = E q ϵ 0 . Moreover, an electron has a ring shape due to the second spin, 2 π R 0 = λ m 0 , λ m 0 is the electron-impact cross section. As $\hbar=\frac{h}{\lambda_{m_0}}\frac{\lambda_{m_0}}{2\pi}=m_0cR_0$, it has a quantized classical angular momentum, with a speed of c . The Bohr magneton, $\vec{\mu_B}=\vec{\frac{q}{2m_0}\hbar}$, g -factor is 2. Critically, the full Dirac’s equation of an electron with a constant velocity is $\vec{mc} = \vec{\hbar_\perp}+\vec{p_\perp } +\vec{\hbar_{//}} +\vec{p_{//}}$, and its electromagnetic properties due to charge are $\vec{f(E_q)} = \vec{f(E_q)\frac{\sqrt{c^2-v^{'2}}}{c}} + \vec{f(E_q) \frac{v\rq}{c}} = \vec{E_{net}^{//}} + \vec{E_q^\perp}$, $\vec{v\rq} // \vec{\hbar_\perp}$ or $\vec{v\rq} // \vec{\hbar_{//}}$, and $\vec{v\rq} // \vec{E_{net}}$. Assuming heavy charged particles have equivalent though shall not identical mechanisms, mathematically this model is equivalent to Special Relativity, and supports ALL its formulas, i.e., v < c , E = m c 2 , direction-insensitive m = γ m 0 and $t=\frac{t_0}{\gamma}$; Moreover, the net electricity $\vec{E_{net}^{//}}$ and Bohr magneton is orientation-quantized, direction-sensitive and follows Lorentz transformation, leading to Lorentz transformation of length contraction; while following Relativity, magnetism can be defined by $\vec{f(B)} =\vec{f(E_q)\frac{v\rq}{c^2}}$, which is not circularly symmetric and self-evidently does not follow Lorentz transformation. The temporospatial averaged magnetic intensity of a current is relativistic, $\bar{\vec{B}} =\sum\vec{{f(E_q)}\frac{\Delta v}{c^2 }}=\frac{\vec{I}}{2 \pi \epsilon_0 c^2}=\frac{\mu_0\vec{I}}{2\pi}$, and μ 0 ϵ 0 c 2 = 1, supporting Maxwell’s equations. Further implications include chirality therefore magnetic spatial quantization, de Broglie wave, etc. The spherical model should be just an observation due to orientation quantization of $\vec{E_{net}}$. We conclude this model of electron may provide one step further to the ultimate Quantum Theory in a nonrelativistic spacetime.

•Nov 14, 2025•0 citations

Helical Phase Geometry of Electrons: A Geometric

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A geometric interpretation of electron dynamics is presentedbased on phase-coherent helical motion constrained by relativistic kinematics.Building on de Broglie’s concept of internal periodicity and theHamilton–Jacobi formulation of mechanics, the electron is modeled asan intrinsic light-speed helical trajectory whose axial pitch correspondsto the de Broglie wavelength and whose transverse scale is set by thereduced Compton length. Within this framework, spin-1/2 symmetry,magnetic moment, rest energy, and bound-state quantization emerge asinterconnected consequences of phase closure and geometric constraints,rather than as independent postulates. The formal operator structure ofquantum mechanics is recovered as a statistical description of ensemblesof phase-coherent helical trajectories, with momentum and energy operatorsarising as generators of spatial and temporal phase evolution. The frameworkprovides a geometric visualization of interference, tunneling, and quantumtransitions, while remaining fully consistent with the standard probabilisticpredictions of quantum mechanics and the Lorentz invariance of specialrelativity. The helical phase interpretation is intended as a conceptualand pedagogical complement to orthodox formalisms, offering physicalintuition for otherwise abstract quantum structures without modifying theirmathematical foundations. Thus, this work does not propose a modificationof quantum theory, but offers a geometric interpretation of its formal structures aimed at clarifying their physical content.

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Helical Phase Geometry of Electrons: A Geometric

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A geometric interpretation of electron dynamics is presentedbased on phase-coherent helical motion constrained by relativistic kinematics.Building on de Broglie’s concept of internal periodicity and theHamilton–Jacobi formulation of mechanics, the electron is modeled asan intrinsic light-speed helical trajectory whose axial pitch correspondsto the de Broglie wavelength and whose transverse scale is set by thereduced Compton length. Within this framework, spin-1/2 symmetry,magnetic moment, rest energy, and bound-state quantization emerge asinterconnected consequences of phase closure and geometric constraints,rather than as independent postulates. The formal operator structure ofquantum mechanics is recovered as a statistical description of ensemblesof phase-coherent helical trajectories, with momentum and energy operatorsarising as generators of spatial and temporal phase evolution. The frameworkprovides a geometric visualization of interference, tunneling, and quantumtransitions, while remaining fully consistent with the standard probabilisticpredictions of quantum mechanics and the Lorentz invariance of specialrelativity. The helical phase interpretation is intended as a conceptualand pedagogical complement to orthodox formalisms, offering physicalintuition for otherwise abstract quantum structures without modifying theirmathematical foundations. Thus, this work does not propose a modificationof quantum theory, but offers a geometric interpretation of its formal structuresaimed at clarifying their physical content.

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A semiclassical model of the electron is presented based on the Principle of Helical Motion (“A free electron always moves at the speed of light following a helical motion, with a constant radius, and with the direction of movement perpendicular to the rotation plane”). This model interprets the Zitterbewegung as a real motion that causes rotation of the electron spin and its magnetic moment. Based on this model, the quantum magnetic flux and quantum Hall resistance are obtained as parameters of the electron and special relativity theory is derived from the helical motion of the electron. Finally, a fix is proposed for the De Broglie’s wavelength that questions the very validity of the Dirac equation.
A spatial model of a free electron (or a positron) is formed by a proposed superluminally circulating point-like charged superluminal quantum. The model includes the Dirac equation’s electron spin 12 and magnetic moment / 2e m as well as three Dirac equation measures of the electron’s Zitterbewegung (“jittery motion”): a speed of light velocity c, a frequency of 2 202 / 2.5 10mc h hz, and a size of 131 2 / 1.9 10mc m. The electron’s superluminal quantum has a closed double-looped helical trajectory whose circular axis ’ double-looped length is one Compton wavelength h/mc. In the electron’s rest frame, the equations for the superluminal quantum’s position are:
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass MMM of the particle differs from the mass mmm which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of MMM is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass MMM and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.
The spherical model of electron cannot explain Bohr magneton, de Broglie wave, etc. To explain these properties physically, we propose that spin, one core concept of quantum theory, is a reality, and introduce this alternative solely based on experimental findings, Planck’s, Einstein’s, and Dirac’s insights, and follows classical laws, that an electron is an electrical wave with a literal 12\frac{1}{2}21​ spin (Dirac), i.e., spins two circles in one cycle, both angular momentum are ℏ. Its wavelength is determined by E=hνm0=hcλm0=m0c2E=h\nu_{m_0}=\frac{hc}{\lambda_{m_0}}=m_0c^2E=hνm0​​=λm0​​hc​=m0​c2 (Plank and Einstein), λ m 0 is the Compton wavelength. As E⃗ ⊥ c⃗ for a transverse wave, it has a net electricity due to the first spin, f(Eq)=12E0(1−cos⁡(2ω0t))f(E_q)=\frac{1}{2}E_0\left(1-\cos(2\omega_0t)\right)f(Eq​)=21​E0​(1−cos(2ω0​t)), half of its cycle, i.e., the minimal meaningful unit as F = E q , is related with its charge, q = E q ϵ 0 . Moreover, an electron has a ring shape due to the second spin, 2 π R 0 = λ m 0 , λ m 0 is the electron-impact cross section. As ℏ=hλm0λm02π=m0cR0\hbar=\frac{h}{\lambda_{m_0}}\frac{\lambda_{m_0}}{2\pi}=m_0cR_0ℏ=λm0​​h​2πλm0​​​=m0​cR0​, it has a quantized classical angular momentum, with a speed of c . The Bohr magneton, μB⃗=q2m0ℏ⃗\vec{\mu_B}=\vec{\frac{q}{2m_0}\hbar}μB​​=2m0​q​ℏ​, g -factor is 2. Critically, the full Dirac’s equation of an electron with a constant velocity is mc⃗=ℏ⊥⃗+p⊥⃗+ℏ//⃗+p//⃗\vec{mc} = \vec{\hbar_\perp}+\vec{p_\perp } +\vec{\hbar_{//}} +\vec{p_{//}}mc=ℏ⊥​​+p⊥​​+ℏ//​​+p//​​, and its electromagnetic properties due to charge are f(Eq)⃗=f(Eq)c2−v′2c⃗+f(Eq)v′c⃗=Enet//⃗+Eq⊥⃗\vec{f(E_q)} = \vec{f(E_q)\frac{\sqrt{c^2-v^{'2}}}{c}} + \vec{f(E_q) \frac{v\rq}{c}} = \vec{E_{net}^{//}} + \vec{E_q^\perp}f(Eq​)​=f(Eq​)cc2−v′2​​​+f(Eq​)cv′​​=Enet//​​+Eq⊥​​, v′⃗//ℏ⊥⃗\vec{v\rq} // \vec{\hbar_\perp}v′//ℏ⊥​​ or v′⃗//ℏ//⃗\vec{v\rq} // \vec{\hbar_{//}}v′//ℏ//​​, and v′⃗//Enet⃗\vec{v\rq} // \vec{E_{net}}v′//Enet​​. Assuming heavy charged particles have equivalent though shall not identical mechanisms, mathematically this model is equivalent to Special Relativity, and supports ALL its formulas, i.e., v < c , E = m c 2 , direction-insensitive m = γ m 0 and t=t0γt=\frac{t_0}{\gamma}t=γt0​​; Moreover, the net electricity Enet//⃗\vec{E_{net}^{//}}Enet//​​ and Bohr magneton is orientation-quantized, direction-sensitive and follows Lorentz transformation, leading to Lorentz transformation of length contraction; while following Relativity, magnetism can be defined by f(B)⃗=f(Eq)v′c2⃗\vec{f(B)} =\vec{f(E_q)\frac{v\rq}{c^2}}f(B)​=f(Eq​)c2v′​​, which is not circularly symmetric and self-evidently does not follow Lorentz transformation. The temporospatial averaged magnetic intensity of a current is relativistic, B⃗ˉ=∑f(Eq)Δvc2⃗=I⃗2πϵ0c2=μ0I⃗2π\bar{\vec{B}} =\sum\vec{{f(E_q)}\frac{\Delta v}{c^2 }}=\frac{\vec{I}}{2 \pi \epsilon_0 c^2}=\frac{\mu_0\vec{I}}{2\pi}Bˉ=∑f(Eq​)c2Δv​​=2πϵ0​c2I​=2πμ0​I​, and μ 0 ϵ 0 c 2 = 1, supporting Maxwell’s equations. Further implications include chirality therefore magnetic spatial quantization, de Broglie wave, etc. The spherical model should be just an observation due to orientation quantization of Enet⃗\vec{E_{net}}Enet​​. We conclude this model of electron may provide one step further to the ultimate Quantum Theory in a nonrelativistic spacetime.
A geometric interpretation of electron dynamics is presentedbased on phase-coherent helical motion constrained by relativistic kinematics.Building on de Broglie’s concept of internal periodicity and theHamilton–Jacobi formulation of mechanics, the electron is modeled asan intrinsic light-speed helical trajectory whose axial pitch correspondsto the de Broglie wavelength and whose transverse scale is set by thereduced Compton length. Within this framework, spin-1/2 symmetry,magnetic moment, rest energy, and bound-state quantization emerge asinterconnected consequences of phase closure and geometric constraints,rather than as independent postulates. The formal operator structure ofquantum mechanics is recovered as a statistical description of ensemblesof phase-coherent helical trajectories, with momentum and energy operatorsarising as generators of spatial and temporal phase evolution. The frameworkprovides a geometric visualization of interference, tunneling, and quantumtransitions, while remaining fully consistent with the standard probabilisticpredictions of quantum mechanics and the Lorentz invariance of specialrelativity. The helical phase interpretation is intended as a conceptualand pedagogical complement to orthodox formalisms, offering physicalintuition for otherwise abstract quantum structures without modifying theirmathematical foundations. Thus, this work does not propose a modificationof quantum theory, but offers a geometric interpretation of its formal structures aimed at clarifying their physical content.
A geometric interpretation of electron dynamics is presentedbased on phase-coherent helical motion constrained by relativistic kinematics.Building on de Broglie’s concept of internal periodicity and theHamilton–Jacobi formulation of mechanics, the electron is modeled asan intrinsic light-speed helical trajectory whose axial pitch correspondsto the de Broglie wavelength and whose transverse scale is set by thereduced Compton length. Within this framework, spin-1/2 symmetry,magnetic moment, rest energy, and bound-state quantization emerge asinterconnected consequences of phase closure and geometric constraints,rather than as independent postulates. The formal operator structure ofquantum mechanics is recovered as a statistical description of ensemblesof phase-coherent helical trajectories, with momentum and energy operatorsarising as generators of spatial and temporal phase evolution. The frameworkprovides a geometric visualization of interference, tunneling, and quantumtransitions, while remaining fully consistent with the standard probabilisticpredictions of quantum mechanics and the Lorentz invariance of specialrelativity. The helical phase interpretation is intended as a conceptualand pedagogical complement to orthodox formalisms, offering physicalintuition for otherwise abstract quantum structures without modifying theirmathematical foundations. Thus, this work does not propose a modificationof quantum theory, but offers a geometric interpretation of its formal structuresaimed at clarifying their physical content.