A Planck-Einstein-Dirac Model of Electron                    

A Planck-Einstein-Dirac Model of Electron | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 1 August 2025 V5 View latest version Share on A Planck-Einstein-Dirac Model of Electron Author : ZhaoXin Wang 0009-0000-3226-8974 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.169778649.95152502/v5 779 views 252 downloads Contents Abstract 1 The Planck-Einstein-Dirac Model of an Electron 2 Changes of Mass, Radius, Electromagnetism and Length, and Time due to speed 2.2 Mass Increase and the reduced Compton wavelength is unchanged 2.3 Changes of Electromagnetism of moving particles lead to Length Contraction 2.4 Time Dilation 3 de Broglie wave 4 Relationships with Quantum Theory, Spheric Model of Electron, Special Relativity, and Predictions 5 Open questions 6 Conclusions Statement of competing interest Author contribution Supplementary materials Acknowledgments References Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract The spherical model of electron should be incorrect as it cannot explain Bohr magneton, de Broglie wave, and its total electrical energy is likely bigger than \(m_0c^2\) without correction, etc. Based on Planck and Einstein’s Formulas, here we provide an alternative by simply visualizing Dirac’s insights, that an electron is an electrical wave (as magnetism is just electricity, Einstein) with a literal \(\frac{1}{2}\) spin, i.e., spins two circles in one wavelength; its wavelength is determined by the combination of Planck’s Formula and Einstein’s Energy-Mass Formula, \(E=h\nu_{m_0}=\frac{hc}{\lambda_{m_0}}=m_0c^2\), \(\lambda_{m_0}\)is the well-accepted Compton wavelength. Therefore, an electron has a ring shape, \(2\pi R_0=\lambda_{m_0}\) . Consequently, as it has an experiment-proven quantized angular momentum, \(\hbar=\frac{h}{2\pi}=\frac{h}{\lambda_{m_0}}\frac{\lambda_{m_0}}{2\pi}=m_0R_0c\), it is a classical angular momentum, with a speed of c . Moreover, as \(\overline{E}\bot\overline{c}\) for a transverse wave, it has a net electricity, \(\vec{E_q}=\frac{1}{2}\vec{E_0}\left(1-\cos\left(2w_0t\right)\right)\) that should be related with its charge. Furthermore, as other particles with charge should have similar though not identical mechanism, this model is equivalent to Special Relativity mathematically, and supports ALL its formulas, i.e., \(v<c\) , \(E=mc^2\) , \(m=\gamma m_0\) ,\(t=\frac{t_0}{\gamma}\) ; especially, \(\left|\vec{E_q}\right|=\left|\vec{E_q\frac{\sqrt{c^2-v^2}}{c}}+\vec{E_q{\frac{v}{c}}}\right|=\left|\vec{E_q\frac{\sqrt{c^2-v^2}}{c}}+\vec{Bc}\right|\) , i.e., 1), magnetism is just electricity due to speed (therefore supports the Maxwell’s equations ), and 2) the net electricity follows Lorentz transformation, leading to Length Contraction , \(L=\frac{L_0}{\gamma}\) . Other implications include g -factor, chirality, strong force, etc. The spheric model should be just an observation, as besides a possible effect of chirality, its electric dipole moment should be determined by its nearly undetectable moment of inertia, \(I=mR_0^2=\gamma{\hbar}\frac{R_0}{c}\) . We conclude this model is the realization of Dirac’s insights, the only model available that can physically explain Bohr magneton and de Broglie wave. This model may provide one step further to the ultimate Quantum Theory. To de Broglie, Albert Einstein , Erwin Schrödinger, Isaac Newton, Max Planck , Otto Wiener, Paul Dirac, etc. Bohr magneton is the basis of nuclear magnetic resonance, yet its mechanism is unclear. This should be related with the nature of particles. In this respect, Louis de Broglie first proposed that particles behave like waves [1] with a wavelength of \(\lambda_m=\frac{h}{p}\) . Davisson and Germer confirmed the de Broglie wavelength quantitatively [2]. Erwin Schrödinger later formulated the matter wave function and Schrödinger equation, and Paul Dirac further introduced the Dirac equation [3] based on the Energy Momentum Formula of Special Relativity that explained the fine structure of the spectral line of hydrogen, and showed that an electron has to spin at the speed of \(\frac{1}{2}\) . Dirac further asserted that the speed of electrons is c. Yet, the meaning of the wave nature of electrons as well as the spin remains unclear. The Born Rule of the Copenhagen Interpretation claimed that Schrödinger’s wave function is related to the probability of finding a particle in a particular region in space, has no physical reality; the spin as well as the experiment-proven angular momentum were also believed to be inherent properties without classic physical meanings and visual reality; despite that all of them have classic real-life physical effects. These issues also cannot be addressed by another influential model, Special Relativity itself . Note Special Relativity provides mathematically impeccable explanations of Lorentz Transformations of mass increase, time dilation, length contraction, and the Energy Momentum Equation, i.e., \(E=mc^2=\sqrt{m_0^2c^4+p^2c^2}\), etc.; and demonstrates that magnetism is just electricity due to speed, \(B=E\frac{v}{c^2}\) . Evidently, magnetism is NOT found in Wiener’s experiment of a standing wave of light [4]. If so, this is incompatible with Maxwell’s equations , in which magnetism is independent from electricity. To summarize, Bohr magneton, the wave nature of particle and de Broglie wave, even the nature of magnetism, cannot be satisfying explained, even in no way can be explained, by known theories based on the spheric model of particles. Also, the total energy calculated by its electrical fields is bigger than \(m_0c^2\) without correction, for example, using a refuted cutoff radius; or Renormalization, in violation of energy conservation. These inability points to an alternative possibility, that the classic spheric model of particles is only an observation but incorrect; these so-called inherent properties indeed have physical realities. Here we explore this possibility by simply visualizing Dirac’s insights of the \(\frac{1}{2}\) spin, together with the Planck and Einstein’s formulas, 1) together with the energy-mass formula, the Planck’s Formula is applied, i.e., \(E=h\nu=\frac{hc}{\lambda}=mc^2\) , a widely-accepted, experiment-proven idea; 2) magnetism is nonexistent but just electricity according to Relativity. The classic laws of conservation (including momentum, energy, angular momentum, and particularly electricity), and the concept of moment of inertia, are also followed. 1 The Planck-Einstein-Dirac Model of an Electron As a particle has an experiment-and theory-proven quantized angular momentum, Dirac has already suggested a model of electrons, that an electron is a wave with a literal \(\frac{1}{2}\) spin , i.e., spins two circles that perpendicular to each other in one cycle, i.e., in one WAVELENGTH. As magnetism is just electricity, it is an electrical wave. This is a caption 1.1 The electrical intensity of an electron with a spin Supposing an electron is a transverse wave with a typical cosine function, which propagates along the z axis with its electricity in the y axis; angular frequency of \(\omega_0\), initial phase = \(\frac{\pi}{2}\), the function of its electrical intensities: \(E_{y,t}=E_0cos\left(\omega_0t+\frac{\pi}{2}\right)\) \(E_{x,z,t}=0\) When it spins 1 circle per cycle in the x-y plane, i.e., the first spin , as its intensities \(E_{x,t}=E_0cos\left(\omega_0t+\frac{\pi}{2}\right)cos\left(\omega_0t+\frac{\pi}{2}\right)=\frac{1}{2}E_0\left(1-cos\left(2\omega_0t\right)\right) \) \(E_{y,t}=E_0cos\left(\omega_0t+\frac{\pi}{2}\right)sin\left(\omega_0t+\frac{\pi}{2}\right)=-\frac{1}{2}E_0sin\left(2\omega_0t\right) \) \(E_{z,t}=0\) The second spin has to be perpendicular to the first spin, in the x-z plane, its intensities \(E_{x,t}=\frac{1}{2}E_0\left(1-cos\left(2\omega_0t\right)\right) \) \(E_{y,t}=-\frac{1}{2}E_0sin\left(2\omega_0t\right)cos\left(\omega_0t\right)\) \(E_{z,t}=-\frac{1}{2}E_0sin\left(2\omega_0t\right)sin\left(\omega_0t\right)\) 1.2 Angular momentum, Charge, Rest mass, Bohr magneton, and -factor (I). Due to the second spin, the wave propagates in a ring shape, \(2\pi R_0=\lambda_{m_0}\), \(\lambda_{m_0}\) is the Compton wavelength, determined by \(E=h\nu=\frac{hc}{\lambda_{m_0}}=m_0c^2\) . As the experiment-proven quantized angular momentum is \(\hbar\), we have \(L_{p,2}=\hbar=\frac{h}{\lambda_{m_0}}\frac{\lambda_{m_0}}{2\pi}=m_0cR_0\), it is a classic angular momentum and its speed of the wave is c , fully supporting Dirac’s insights. The ring shape of an electron has been proposed [5-7]. Note the speed of the wave is NOT \(v\) that used in the Energy Momentum Formula, see Section 2. (II) The first spin should be also quantized and likely \(\hbar\), which should be determined by the distribution of its electricity and \(\omega_0\), i.e., \(L_{p,1}\propto E_0\omega_0\), though details and its physical index are unclear. Due to these two quantized angular momentums, its total energy is fixed, i.e., rest mass, \(m_0\) . The underlying mechanisms of both spins are unclear, for example, the concept of centripetal force shall be inapplicable. It may be argued that nothing changes after one cycle for a single wave; and the law of conservation is followed at the system level. (III) \(\vec{E}\bot\vec{c}\) for a typical cosine wave, in line with Maxwell’s insights. The electron has a net electrical field, \(E_{x,t}=\frac{1}{2}E_0\left(1-cos\left(2\omega_0t\right)\right)\) . The net electricity of one cycle, the area under the curve is \( \pi{E}_{0}\), should be related with its net charge if no other factor is involved, \(E_q\) , Fig . 1, left. This is different from the classic spheric model, and will be discussed in Section 4.2. (IV) The charge is rotating, \(\vec{IR_0}=\vec{E_{x,t}cR_0}=\vec{\frac{q}{2m_0}m_0R_0c}=\vec{\frac{q}{2m_0}\hbar}=\vec{\mu_B}\), this is the Bohr magneton, i.e., magnetism is just (the rotating vector of) electricity due to speed, in line with Relativity; see Section 2.3 for the same definition of magnetism of moving electrons. are time-balanced and its net magnetism should be zero. (V) \(E_{x,t}=\frac{1}{2}E_0\left(1-cos\left(2\omega_0t\right)\right)\), its frequency is, 2 is the g -factor. (VI) An electron must have an anti-electron to follow the laws of conservation of angular momentum and charge, i.e., a positron, see Section 2.3.4 regarding chirality for the difference between them. 1.3 Other charged particles As the experiment-proven magnetic moment of particles with charge is \(\vec{M}=\vec{\frac{q}{2m_{0\prime}}\hbar}\), particles with charge should have similar mechanisms, though not identical (which leads to different charges) and may have sub-particle components (e.g., quarks, which are also supposed to have \(\frac{1}{2}\) spins with current understandings; yet the overall angular momentum is quantized, \(\hbar=m_{0\prime}rv=m_{0\prime}r_{0\prime}c\), i.e., an effective radius is \(r_{0\prime}\) ; and \(\vec{M}=\vec{\frac{q}{2m_{0\prime}}\hbar}=\vec{\frac{q}{2m_{0\prime}}m_{0\prime}r_{0\prime}c}\) ), pending further studies. Therefore, we assume all charged particles have similar mechanisms in the following discussion. Fig. 1. The Planck-Einstein-Dirac Model of Electron Left: An electron is an electrical wave with a literal Dirac’s \(\frac{1}{2}\) spin, i.e., spins 2 circles in two orthogonal directions in one wavelength. 1) The second spin results in a ring shape, \(2\pi R_0=\lambda_{m_0}\), \(\lambda_{m_0}\)is the Compton wavelength, determined by \(E=h\nu=\frac{hc}{\lambda_{m_0}}=m_0c^2\) ; as \(\hbar=\frac{{h}}{{\lambda}_{{m}_\mathbf{0}}}\frac{{\lambda}_{{m}_\mathbf{0}}}{\mathbf{2}{\pi}}={m}_\mathbf{0}{c}{R}_\mathbf{0}\) , this is a classical angular momentum, its speed is c , i.e., spin and angular momentum are physical realities. 2) As \(\vec{{E}}\bot\vec{{c}}\) , the first spin results in a net electrical field that perpendicular to the ring, \(E_{x,t}=\frac{1}{2}E_0\left(1-cos\left(2\omega_0t\right)\right)\), the total electricity in one cycle is related with its charge. 3) The charge is rotating, \(\vec{IR_0}=\vec{E_{x,t}cR_0}=\vec{\frac{q}{2m_0}m_0R_0c}=\vec{\frac{q}{2m_0}\hbar}=\vec{{\mu}_{B}}\), it is the Bohr magneton . 4) Its frequency is \(2\omega_0\), i.e., its g -factor is 2. 5) Combined, its rest mass is fixed due to these two quantized spins, \(m_0\) . Right. The net electromagnetism is zero. 2 Changes of Mass, Radius, Electromagnetism and Length, and Time due to speed 2.1 The physical interpretation of the modified Energy Momentum Formula by Dirac and the Origin of the de Broglie wave The original Energy Momentum Formula of Special Relativity, \(E=mc^2=\sqrt{{m_0}^2c^4+p^2c^2}\), cannot be used to explain the behavior of high-speed charged particles. Paul Dirac modified the Formula to \(\frac{E}{c}=mc=\sqrt{{m_0}^2c^2+p^2}=\beta m_0c+\ \vec{\alpha}\cdot\vec{p}\) and successfully explained the fine structure of the spectral line of hydrogen. Indeed, there is a simple way to help to understand Dirac’s modification. As momentums are vectors, the modified formula can be rewritten to \(|\frac{\vec{E}}{c}|=|\vec{mc}|=\sqrt{{\vec{|m_0c}|}^2+{\vec{|p}|}^2}\), that’s \(\vec{mc}\ =\vec{m_0c}+\vec{p}\) only when \(\vec{m_0c}\bot\vec{p}\) , hereafter \(\vec{p_{//}}=\vec{p}\) . It is the conservation of momentum. Note has a complex structure of momentum, i.e., the \(\frac{1}{2}\) spin. Its physical meaning can be explored with a simple ideal example. For an elastic collision between a photon with a momentum of \(\vec{p_1}=\vec{\frac{h\nu_1}{c}}\) and a static particle ( \(m_0, \vec{v}=0\) ), Fig. 2 , there will be a reflected photon, \(\vec{p_2}=\vec{\frac{h\nu_2}{c}}\), and the particle will gain a momentum, \(\vec{p_{//}}=\vec{mv}\) . The laws of conservation, including momentum, angular momentum, and energy, are followed, we have .\(\vec{p_1}-\vec{p_2}=\vec{\frac{h\nu_1}{c}}-\vec{\frac{h\nu_2}{c}}=\vec{mv}=\ \vec{p_{//}}\) Combined with the modified Energy Momentum Formula, we have \(\vec{mc}=\vec{m_0c}+\vec{p_{//}}=\vec{m_0c}+\vec{\frac{h\nu_1}{c}}-\vec{\frac{h\nu_2}{c}}=\vec{m_0c}+\vec{m_{//}c}\) defining \(m_{//}\vec{c}\cdot\vec{c}=|\vec{h\nu_1}-\vec{h\nu_2}|=h(\nu_1+\nu_2)\) . Consequently, 1) both \(m_0\) and \(m\) shall have momentums and a speed of c , \(\vec{m_0c}\) and \(\vec{mc}\) . 2) As a particle has an angular momentum and \(\vec{m_0R_0c}\bot\vec{p}\), its longitudinal velocity, \(v<c\) . 3) Energy may be regarded as the scalar of its momentum, \(E=m\vec{c}\cdot\vec{c}=mc^2\) . Fig 2 . A physical interpretation of the Energy Momentum Formula modified by Dirac The original Energy Momentum Formula of Special Relativity, \(E=mc^2=\sqrt{{m_0}^2c^4+p^2c^2}\) , cannot be used to explain the behavior of high-speed charged particles. Paul Dirac modified the Formula to \( \frac{E}{c}=mc=\sqrt{{m_0}^2c^2+p^2}=\beta m_0c+\ \vec{\alpha}\cdot\vec{p}\) . To help to understand its physical meaning, this modification can be rewritten to \(\vec{mc}=\vec{m_0c}+\vec{p}\) as momentums are vectors. Its physical meaning can be explored by this simple simulation. During an elastic collision between a static electron and a photon, \(\vec{p_1}=\vec{\frac{hv_1}{c}}\), there will be a reflected photon, \(\vec{p_2}=\vec{\frac{hv_2}{c}}\) ; and \(\vec{p_1}-\vec{p_2}=\ \vec{p_{//}}\) ; \(\vec{mc}=\vec{m_0c}+\vec{mv}=\vec{m_0c}+\vec{p_{//}}\ and\ |\vec{mc|}=\sqrt{{|m_0c|}^2+{|\vec{p_{//}}|}^2}\) only when \(\vec{m_0c}\bot\vec{p_{//}}\) , i.e., the law of conservation of momentum. Note the de Broglie wavelength is \(\lambda_{matter}=\frac{h}{m_{//}c}\) defining \(m_{//}c^2\ =h{(\nu}_1+\nu_2)\) . Energy may be regarded as the scalar of its momentum, \(E=m\vec{c}\cdot\vec{c}=mc^2\) . This is the mass-energy equation of Special Relativity. Note this is a simple simulation and \(m_0c\) has a complicate structure of momentums, i.e., the \(\frac{1}{2}\) spin. 2.2 Mass Increase and the reduced Compton wavelength is unchanged As \(\vec{m_0R_0c}\bot\vec{p}\), the trajectory of a moving electron is a helix, and its unfolded trajectory is illustrated in Fig. 3 . As the transverse vector momentum ( \(\bot\vec{v}\) ) is conserved, \(m\sqrt{c^2-v^2}=m_0c\) , \(m={\gamma m}_0\) , it is a true mass increase, the laws of conservation of energy and momentum are strictly followed, Fig. 3A . Note \( \gamma=\frac{c}{\sqrt{c^2-v^2}}=\frac{1}{cos{\theta}}=\frac{m}{m_0},\ tan{\theta}=\frac{m_{//}}{m_0}, \theta\) is the helix angle. The law of conservation of angular momentum is followed, we have \(\left|L_{p,2}\right|=\left|m_0R_0c\right|=\left|mRv\prime\right|=\left|mR\sqrt{c^2-v^2}\right|=\hbar\) As \(m={\gamma m}_0, R=R_0\), the reduced Compton wavelength does not change. 2.3 Changes of Electromagnetism of moving particles lead to Length Contraction Length Contraction of an object shall be related with speed-related changes of electromagnetism due to spin. 2.3.1 Electromagnetic properties of of a free moving electron and Magnetism As the net electricity of \(m_{//}\) is time-balanced, the net electricity of a particle is determined by \(m_0\), and is conserved. However, as \(\vec{E} \bot\vec{c}\) , in the moving/longitudinal direction we have \(\vec{E_{net}}=\vec{E_q^{//}}=\frac{\vec{E_q}}{\gamma}\) The net electricity in the moving direction is reduced; interestingly, in the form of Lorentz Transformation, Fig. 3B . Crucially, it obtains a vector component in the transverse direction (\(\bot\vec{v}\) ), its intensity \(\vec{\ E_q^\bot}=\vec{E_q\frac{v}{c}}\) We can define \(|B|=|E_q\frac{v}{c^2}|\) [8] mathematically following Special Relativity. Therefore, the electrical force with another moving particle ( \(q_1\), with a speed of \(v_1\) ) due to their rotating components is \(|F|=|\vec{E_q\frac{v}{c}}\cdot\vec{E_{q_1}\frac{v_1}{c}}|=|\vec{B}q_1\times\vec{v_1}|\) as \(\vec{v}\bot\vec{B}\) . This is the definition formula of the Lorentz force, i.e., magnetism is just (the rotating vector of) electricity due to speed , the same definition as Bohr magneton, Section 2.2. Note the net \(\vec{B}\) cannot cancel out each other . Collectively, \(|E_q|=\left|\vec{E_q^{//}}+\vec{\ E_q^\bot}\right|=\left|\vec{\frac{E_q}{\gamma}}+\vec{E_q\frac{v}{c}}\right|=\left|\vec{E_{net}}+\vec{Bc}\right|\). Note 1) If an electron has a spheric shape, magnetism and Lorentz transformation of electricity cannot be physically explained by this model; the Relativity must be right; 2) Should magnetism do exist, as \(\vec{E}\bot\vec{B}\bot\vec{c}\) according to Maxwell’s equations, this model is not allowed; 3) It can be further demonstrated that mathematically the transverse vector component of \(\vec{B}\), if exists, does not change with speed in this model; which is incompatible with experimental findings. 2.3.2 Electromagnetic properties of a moving electron in an external electrical field In an external electrical field, the direction of the net electricity of a charged particle shall be allowed to change, and aligns with the external electrical field, along with the direction of the ring. Indeed, as its moment of inertia of an electron is \( I_m=m{R_0}^2=R_0\frac{mR_0c}{c}=\gamma\hbar\frac{R_0}{c}\) , it is extremely easy to change its direction. For an ideal simulation, the trajectory of \(m_0\) likes a wheel moving in various directions with the same radius to follow the Lorentz Transformation, with its unfolded trajectory as a line, Fig. 3C . Consequently, as \(\vec{E} \bot\vec{c}\), \(E_{net}=E_q\ \frac{\sqrt{c^2-{v\prime}^2}}{c}\), here \({v\prime}\) is the vector velocity in the direction of the external field, \(\vec{E_{net}}//\vec{E_{external}}//\vec{v\prime}\), and \(|E_q|=\left|\vec{E_{net}}+\vec{Bc}\right|\) . Therefore, \(E_{net,//}=\frac{E_q}{\gamma}\)in the moving direction when \( \vec{v\prime}=\vec{v}\), while \( E_{net,\bot}=E_q\) in the transverse direction when \(\vec{v\prime}=0\), i.e., the Lorentz Transformation. The Bohr magneton determined by its net electricity, and is changed accordingly. Admittedly, this part is a reverse-reasoning of Special Relativity as Lorentz Transformation should be followed. Further studies are desired for the complicate details of the changes, for example, 1) how the angular momentums are conserved; 2) the wavelength of the component may be changing, as its net intensity can be changed, so does its spin speed. 2.3.3 Length Contraction due to Changes of Electromagnetism For an ideal, rough simulation, it is assumed that the locations of particles of an object are determined by the same balanced electromagnetic force; i.e., particles are not free; therefore, magnetism due to between-particle speed difference can be neglected. The net electromagnetism of is balanced and is not considered. Therefore, we only need to consider the electricity of . The temperature-related changes are also not considered. As the effective electrical intensity of a charged particle is roughly inversely proportional to the distance due to the spin, as the electric field of is distributed in a fan-shaped pattern in one cycle, the location with balanced electromagnetic force is changed. We have \(r\prime=\frac{r}{\gamma} \) as \(E_{net,//}=\frac{E_q}{\gamma}\), therefore \(L_{//}=\frac{L_0}{\gamma}\) in the moving/longitudinal direction, and measured net electrical intensity at the contracted length are the same, \(E_{//,measured}=E\) ; in the transverse direction, particle-wise \(E_{net,\bot}=E_q\), the length in the transverse direction is unchanged, \(L_\bot=L\) , while length-averaged \(\bar{E_{\bot,measured}}=\gamma E\) due to length contraction . Therefore, the length of an object follows Lorentz transformation, and the null result of the Michelson–Morley experiment [9] thus can be explained by this model, in which the authors found that lights travel with the same velocity in the various horizontal directions tested. This is a caption Fig. 3. Changes of an object due to speed A. (I). The momentum in the vertical direction is conserved, \(m\sqrt{c^2-v^2}=m_0c, m=\gamma m_0\) , i.e., mass increase. II) \(\left|m_0R_0c\right|=\left|mR\sqrt{c^2-v^2}\right|, R=R_0\), Its frequency is slowed, \(f=\frac{\sqrt{c^2-v^2}}{2\pi R_0} = \frac{f_0}{\gamma}\) . B . I) The electricity of is conserved. As \(\vec{E}\bot\vec{c}\), its net intensity in the moving direction is reduced, \(E_{net}=E_q^{//}=\frac{E_q}{\gamma}\) . Critically, it obtains a rotating vector component in the transverse direction, \(E_q^\bot=\ E_q\frac{v}{c}\) , it is magnetism following Relativity, \(|B|=|E_q\frac{v}{c^2}|\), \(|E_q|=|\vec{E_{net}}+\vec{Bc}|\). II) The locations of particles of an object are determined by the same balanced net electrical force. As the effective electrical intensity is inversely proportional to the distance due to the \(\frac{1}{2}\) spin, and \(E_{net,//}=\frac{E_q}{\gamma}\) in the moving direction, \(L_{//}=\frac{L_0}{\gamma}\), i.e., length contraction; \(E\) at the contracted length is the same , \(E_{//}=E_q\) . C. The net electricity of the particle will align with an external electrical field with a changed trajectory, as a wheel moving in various directions with the same radius. Its unfolded trajectory is a line, its starting and end points were added. As \(\vec{E}\bot\vec{c}\) , its electromagnetism is changed, \(E_{net}\ =E_q\ \frac{\sqrt{c^2-{v\prime}^2}}{c}\), \(\vec{v\prime}\ //\vec{E_{external}}\), \(B_{//}c=E_q\ \frac{v\prime}{c}\), in support of Relativity. When \(v\prime=0\), \(E_{net,\bot}=E_q\), therefore \(L_\bot=L_0\), and \(\bar{E_\bot}=\gamma E_q\) due to length contraction. Combined, measured time based on frequency is slowed given its radius is the same, \(t=\frac{t_0}{\gamma}\), i.e., time dilation. 2.3.4 Chirality As magnetism is its rotating electricity, according to the Biot-Savart Law , a moving charged particle has chirality, which is literally the direction of the second spin relative to its velocity. There are some implications. (I) Free moving charged particles, regardless its charge, have the same chirality ; a point may have large-scale implications, for example, on the spin of stars. Therefore, the only difference between electron and positron is the phase difference of the first spin , \( \Delta\varphi\ =\pi\) . (II) When free particles with the same charge move in different directions, the electrical fields between them are the same, they repel each other; while when free particles with opposite charge move in different directions, the electrical fields between them are the opposite, they attract each other. It should be mentioned that the \(\frac{1}{2}\)spin-related magnetism tends to maximize the force between particles by changing the direction of the net electrical force. (III) Counterintuitively, particles with the same charge can attract each other. For example, when an external electrical field (e.g., voltage) exists, the directions of the net electricity will align with the external electrical fields, they attract each other; though unstable in most cases due to their \(\frac{1}{2}\) spin-related magnetism that tends to change the direction of electrical force; and disturbances of Brownian movement. 2.4 Time Dilation As mentioned in section 2.2, the radius of a moving particle ring does not change, \(R=R_0\) ; and the direction of the ring shall be allowed to change due to external electrical fields and is supposed to have the same radius, Section 2.3. Combined, the frequency of a particle is changed from \(f_0=\frac{c}{2\pi R_0}\) to \(f=\frac{\sqrt{c^2-v^2}}{2\pi R_0}=\frac{f_0}{\gamma}\) as long as it has the same radius. Therefore, time measured based on frequency is slowed , \(t=\frac{t_0}{\gamma}\), which may be even regarded as kind of a true time dilation, as the wavelength of \( \lambda_{m_0}\) can be taken as conserved, Section 3. Clearly, time dilation has real-life implications . For example, the half-life of a moving decayable particle , which is presumed to be a combination of multiple waves, shall be slowed due to its slowed frequency. 3 de Broglie wave As stated in Section 2.1, Fig. 2 , \(\vec{mc}=\vec{m_0c}+\vec{mv}\ =\vec{m_0c}+\vec{\frac{h\nu_1}{c}}-\vec{\frac{h\nu_2}{c}}=\vec{m_0c}+\vec{m_{//}c}\), with the definition \(\vec{m_{//}c}=\vec{\frac{h{(\nu}_1+\nu_2)}{c}}\), note \(\vec{m_0c}\bot\vec{mv}\). Planck’s Formula is strictly followed, \(E=mc^2=h\nu_m=\frac{hc}{\lambda_m}=\gamma m_0c^2\) \( \gamma=\frac{c}{\sqrt{c^2-v^2}}=\frac{1}{cos{\theta}}=\frac{m}{m_0}\) \(Here\ \theta\ is\ the\ helix\ angle,tan{\theta}=\frac{m_{//}}{m_0}=\frac{\nu_{m_{//}}}{\nu_{m_0}}=\frac{\lambda_{m_0}}{\lambda_{m_{//}}}\), \(\lambda_m=\lambda_{m_0}cos\theta=\lambda_{m_{//}}sin\theta\), the de Broglie wavelength is the vector component of \(\lambda_m\) in the moving direction, Fig. 4, left. Also, \(\lambda_{m_{//}}=\lambda_{m_0}cot{\theta}=\lambda_{matter}\) . A moving electron can be taken as a polarized wave with two components, \(m_{//}\)and \(m_0\) . 1) As the first angular momentum of \(m_0\) is conserved, i.e., \({\ L_{p,1}\propto E}_0\omega_0=E_q^{//}\omega\prime\), here \(\vec{E_q^{//}}=\frac{\vec{E_q}}{\gamma}\), the net electricity in the moving direction, section 2.3. That’s, \(\omega\prime=\gamma\omega_0\), the component of \(m_0\) is spinning faster, resulting in a wavelength of \(\lambda_{m_0}cos\theta\), i.e., \(m_0\) still spins \(2\pi\) per cycle for the first spin. 2) As stated earlier, the second angular momentum is also conserved; and the radius of the particle ring is unchanged, the reduced Compton radius. The de Broglie wavelength is the combined effects of the helix. For a simple mathematical demonstration, the wavelength of the moving particle is \(\lambda_{m_0}\cos\theta\) along its trajectory, and its projection to the plane of the ring is \(\lambda_{m_0}\cos^2\theta\), and its projection in the moving direction is \(\lambda_{m_0}\cos\theta\sin\theta\) ; note the radius of the projection is unchanged, \(R_0\), the reduced Compton wavelength; its perimeters is \(\lambda_{m_0}\) . Therefore, the phase difference between \(\lambda_{m_0}\cos^2\theta\) and \(\lambda_{m_0}\) is \(\Delta\varphi=\frac{\lambda_{m_0}-\lambda_{m_0}\cos^2\theta}{2\pi}=\frac{\lambda_{m_0}\sin^2\theta}{2\pi}\), cycles needed to compensate this phase difference is \(\frac{\lambda_{m_0}}{\lambda_{m_0}{sin}^2\theta}=\frac{1}{{sin}^2\theta}\) . Its corresponding distance in the moving direction is \(\lambda_{m_0}cos{\theta}sin{\theta}\frac{1}{{sin}^2\theta}=\lambda_{m_0}cot{\theta}=\lambda_{matter}\) . For a further simple mathematical demonstration and simulation, we use a typical cosine wave along the unfolded trajectory, or the \(m_{//}\) component, its wavelength is \(\lambda_{m_0}cos{\theta}\). The wavelength of the projection of this wave in \(\vec{v}\) is \(\lambda_{m_0}cos{\theta}sin{\theta}\), in the \(x\) direction ( \(\vec{x}\bot\vec{v}\) ), its intensity is \(Acos\left(\frac{2\pi vt}{\lambda_{m_0}cos{\theta}sin{\theta}}\right)\), \(A=Esin{\theta}\). This intensity is modulated by the particle ring. As the radius of the particle ring is the same, its wavelength in the unfolded trajectory is \(\frac{\lambda_{m_0}}{cos{\theta}}\), and its projection in the moving direction is \(\frac{\lambda_{m_0}}{cos{\theta}}sin{\theta}\) . Therefore, \[E_{x,t}=Acos(\frac{2πvt}{λ_{m_0}cosθsinθ})cos\frac{2\pi vt}{\frac{\lambda_{m_0}sin{\theta}}{cos{\theta}}}=\frac{1}{2}A(cos(\frac{2πvt}{λ_{m_0}cosθsinθ}+\frac{2πvtcosθ}{λ_{m_0}sinθ})+cos(\frac{2πvt}{λ_{m_0}cosθsinθ}-\frac{2πvtcosθ}{λ_{m_0}sinθ})=\frac{1}{2}A(cos\frac{(2πvt(1+cos^2θ)}{λ_{m_0}cosθsinθ)}+cos(\frac{2πvt(1-cos^2θ)}{λ_{m_0}cosθsinθ)})=\frac{1}{2}A(cos(\frac{2πvt(1+cos^2θ)}{λ_{m_0}cosθsinθ)}+cos(\frac{2πvt}{λ_{m_0}cotθ)})\] The wavelength of the second component, \(cos(\frac{2\pi vt}{\lambda_{m_0}cot{\theta}})\)is , \(\lambda_{m_0}cot{\theta}=\lambda_{matter}\) i.e., the de Broglie wavelength. We further marked all peaks of \(cos\left(\frac{2\pi vt}{\lambda_{m_0}cos{\theta}sin{\theta}}\right)\)in Fig. 4, right. Though its absolute value is the same, its intensity in a given direction ( \(\bot\vec{v}\) ) is changing due to the spin, with a combined wavelength of \(\lambda_{matter}\). This demonstration also shows that the combined wavelength of is also de Broglie wavelength. This is a caption Fig. 4. de Broglie wavelength and a simple mathematical demonstration and simulation Left. The Planck’s Formula is strictly followed, \(E=mc^2=h\nu_m=\frac{hc}{{\lambda}_{m}}=\gamma m_0c^2\), \(\gamma=\frac{c}{\sqrt{c^2-v^2}}=\frac{1}{cos{\theta}}\), the wavelength of a moving particle \(\lambda_m=\lambda_{m_0}cos{\theta}=\lambda_{m_{//}}sin{\theta}\) . \(\theta\) is the helix angle. The de Broglie wavelength can be regarded as the vector component of \(\lambda_m\) in the moving direction, \(\lambda_{matter}=\lambda_{m_{//}}=\lambda_mcsc{\theta}\) . Right. The de Broglie wavelength is the combined effects of the helix. \(R_{m_0}\) is the radius of the particle, i.e., the reduced Compton wavelength. I) A simple cosine wave function along the unfolded trajectory is used for a simple demonstration and simulation. Its electrical intensity in the axis (\(\vec{v}\bot \vec{v}\)) is \(E_{x,t}=\frac{1}{2}A(cos(\frac{2πvt(1+cos^2θ)}{λ_{m_0}cosθsinθ)}+cos(\frac{2πvt}{λ_{m_0}cotθ)})\) . The wavelength of the second component is \(\lambda_{m_0}cot{\theta}=\lambda_{matter}\), i.e., the de Broglie wavelength . II), All peaks of are marked as green circles, its intensity in a given direction ( \(\bot \vec{c}\) ) is changing due to the spin, with a combined wavelength of \(\lambda_{matter}\) . In this example \(\theta=30°\). The script is provided in the supplementary materials. 4 Relationships with Quantum Theory, Spheric Model of Electron, Special Relativity, and Predictions 4.1 Quantum Theory and Bell’s Inequality This model strictly follows Planck’s Formula, and further extends the Quantum theory that a particle can have quantized angular momentum(s) by visualizing Dirac’s idea, in line with experimental findings. Clearly, a combination of waves behaves differently at different time or different directions, likely the basis of Quantum States. Consequently, the interactions between a medium and particles/photons with different phases/electrical directions are different, or electrical-intensity weighted, therefore the phase information can be stored by the medium and affects later incoming waves. These points should be related with experimental findings of single electron/photon interference. We believe that the Bell’s inequality [10] and Quantum Entanglement may be better explained as just further demonstrations of the wave nature of particles, which should be Bell’s original idea, instead of instantaneous effects; though it is plausible yet dubious that nature itself is absurd and based on probability, and surprisingly, affected by our observation. 4.2 Spheric Model of Electron and Strong force This model can explain many properties of electrons such as de Broglie wavelength that in no way can be explained by the spherical model. Moreover, the shape of the electricity of an electron is asymmetrical in this model. In the directions of \(\vec{E_0}\) and \(-\vec{E_0}\), it is literally strong force between two coupling particles. Note strong force cannot be explained by the spheric model itself, but must be complemented by other fancy theories. The most critical challenge of this model is Coulomb’s law as researchers were unable to detect the possible electric dipole moment of an electron [11,12], one of the experimental bases of the spheric model. Though further studies/simulations are desired, we argue that the electric dipole moment of an electron should be mainly determined by the moment of inertia of an electron. As \(I_m=m{R_0}^2=\gamma\hbar\frac{R_0}{c}\), the electric dipole moment of an electron is extremely difficult to detect. Though as a particle has chirality, changes in chirality may be also associated with energy exchange. Moreover, a heavy particle is believed to have a complex structure and a much smaller effective radius, \(r_{0\prime}\), as \(\hbar=m_{0\prime}rv=m_{0\prime}r_{0\prime}c\) ; and the directions of electromagnetic force between particles are constantly changing due to their \(\frac{1}{2}\) spins, resulting in complicated trajectories, while the laws of conservation of energy, momentum, and angular momentum should be followed. Also considering that the \( \frac{1}{2}\) spin-related magnetism tends to maximize the attractive force between particles with opposite charges, we suspect that the combined effect should be the time- and spatial-averaged reflection of the Gauss’ flux theorem, i.e., the well-established and experiment-proven Coulomb’s law, at least for free particles, with an effective surface area. A second possibility is that it is the requirement to follow the laws of conservation during interactions of particles. That’s, the classic spheric model of particles is probably just an observation, though further studies are desired. Note 1) the total electrical energy of an electron of this model is smaller than the spheric model as it has a de facto cutoff radius, i.e., the reduced Compton wavelength; though the relationships between its mass, charge and the area under the curve of the cosine electrical wave are unclear. 2) The between-particle interactions are likely affected by the moments of inertia of different particles, resulting in material-specific effective surface area that related with permittivity, \(\varepsilon\) ; It can be further reasoned that the interaction is affected by other factors, such as temperature, which may be associated with states of matter. Moreover, in ultracold conditions, the net electricity of particles could be perfectly aligned in some materials without the disturbance of Brownian movement; their interactions with non-aligned particles due to net electricity can be very weak even negligent, which is likely related with superfluidity, experimental findings of the needle-like shape of electron cloud of ultracold gas [13], even superconductivity in addition of Cooper’s theory [14], etc. 4.3 Special Relativity The measured average speed of light is c for any non-accelerating coordinate system due to Lorentz Transformations. Therefore, this model is equivalent with Special Relativity mathematically. However, physically the constancy of c is the consequence, not the cause; consequently, there are some differences. 1) It is overreaching to extend Relativity to two different objects. For example, for the long-debated twin paradox, if gravity is not involved, the one with slower time must has higher speed. 2) The Biot-Savart Law is followed. Lorentz Force can be attractive or repulsive regardless their charges; while Relativity predicts that particles with the same charge must repel each other. 3) \(|\vec{E_q}|=|\vec{E_{net}}+\vec{Bc}|=|\vec{E_q\frac{\sqrt{c^2-{v\prime}^2}}{c}}+\vec{E_q\frac{\Delta v\prime}{c}}|\), only the net electricity follows Lorentz Transformation. It appears that magnetism of a current also does NOT follow Lorentz Transformation. As \(|B|={|E}_q\frac{\vec{v\prime}}{{c}^{2}}|\) for the magnetism elicited by a charged particle, the net magnetism of a current is determined by the relative speed between particles with opposite charges, \(|B_\bot|=|\vec{E_{q-}\frac{v\prime+\Delta v\prime}{c^2}}\ +\vec{E_{q+}\frac{v\prime}{c^2}}|=|\vec{E_{q-}\frac{\Delta v\prime}{c^2}}|\), where \(\vec{voltage}//\vec{v\prime}//\vec{\mathrm{\Delta v}\prime}\bot\vec{B_\bot}\), supporting Special Relativity mathematically. If \(\bar{B}=\gamma B_0\) in the transverse direction following the Lorentz Transformation, the relative speed \(\Delta\vec{v\prime}\) should be the same. However, as \(m\ =\gamma m_0\) while \(E_{//}=E\) in the moving direction, i.e., the electrical force is the same, \(\Delta v\prime\) should be decreased, i.e., \(\bar{B}<\gamma B_0\) , though the exact relationship deserves further investigation (possibly \(m\mathrm{\Delta v}\prime\prime=m_0\mathrm{\Delta v}\prime\) ). We believe this is a self-evident implication of Relativity that against the Lorentz transformation of magnetism. Note the intensity of magnetism is still subject to length contraction of the conductor. 5 Open questions There are many open questions. 1) This model is based on literal and classical explanations of concepts, facts and theories, and may be oversimplified in its current form. For example, typical cosine waves are used but may be incorrect; the measured g -factor is not exactly 2, note in this model g -factor should be affected by its speed as the Planck’s Formula is followed, \(g\prime=g\gamma\) ; and potentially gravity, as gravity can be regarded as acceleration according to General Relativity. 2) How the quantized angular momentums are reserved; the physical index of the first spin and whether the rotating speeds are different at different phase, this point may affect its charge; a particle wave with only the first spin is likely allowed; how many quantized angular momentums are allowed; how the angular momentums of a moving electron in an external electrical field are conserved. 3) The temporospatial property of electrical fields is unclear, as the speed of electrical field shall exceed c ; note this point can be applied to typical photons, and has been de facto demonstrated by the magnetic fields of electrons with extremely high speed of a collider. 4) The nature of heavy particles is not addressed, though is presumed to have similar but not identical mechanisms. 5) Its relationships with some other main theories that supposedly based on the spheric model of electrons, such as Field Theory, String theory, etc., are unclear; etc. 6 Conclusions We provide a Model of Electron based on Planck, Einstein, and Dirac’s insights. This model should be the visualization of Dirac’s insights, and likely the only model that can physically, mathematically, and visually explain de Broglie wave that in no way can be explained by classic spheric models, together with charge, g -factor, chirality, etc.; and is equivalent to Special Relativity mathematically as other charged particles should have similar but not identical mechanisms, confirming ALL its real-life implications, especially that magnetism This model may extend Quantum Theory in that a wave can have quantized angular momentum(s) somehow, and provide one step further to the ultimate Quantum theory of particles. Statement of competing interest The author declares no competing interest. Author contribution This is a sole work by WZX. Supplementary materials A MATLAB script for the simulation of de Broglie wave is provided. Acknowledgments I am indebted to Prof Shi Xi from Dept of Physics of Shanghai University and Prof Liu Jinming and Prof Wang Jiaxiang from Dept of Physics of East China Normal University for their critical instructions. References [1] L. D. Broglie, butsuri (1924). [2] C. Davisson and L. H. Germer, Nature 119 , 558 (1927). [3] P. A. M. Dirac, Proceedings of the Royal Society of London Series A 117 (1928). [4] O. Wiener, Annalen der Physik 276 , 203 (1890). [5] Q.-h. Hu, Physics Essays 17 (2006). [6] R. Gauthier, (2015). [7] d. Froedge, The Dirac Equation and the Two Photon Model of the Electron revised 2021). [8] A. Einstein, Annalen der Physik 17 , 891 (1905). [9] A. A. Michelson and E. W. Morley, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24 , 449 (1887). [10] J. S. Bell, Physics Physique Fizika 1 , 195 (1964). [11] V. Andreev et al. , Nature 562 , 355 (2018). [12] T. S. Roussy et al. , Science 381 , 46 (2023). [13] B. Mukherjee, A. Shaffer, P. B. Patel, Z. Yan, C. C. Wilson, V. Crépel, R. J. Fletcher, and M. Zwierlein, Nature 601 , 58 (2022). [14] L. N. Cooper, Physical Review 104 , 1189 (1956). Supplementary Material File (debrogliewave.m) Download 4.88 KB Information & Authors Information Version history V1 Version 1 20 October 2023 V2 Version 2 21 June 2024 V3 Version 3 01 July 2024 V4 Version 4 11 September 2024 V5 Version 5 01 August 2025 V6 Version 6 21 August 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords bohr magneton charge chirality debroglie waves dirac equation g-factor lorentz transformation magnetism max-planck theory special relativity spin wave Authors Affiliations ZhaoXin Wang 0009-0000-3226-8974 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 779 views 252 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation ZhaoXin Wang. A Planck-Einstein-Dirac Model of Electron . Authorea . 01 August 2025. 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