The radial velocity formula and the Planck-Einstein relation give us the zbw frequency (E = ħω = E/ħ) and zbw radius (a = c/ω = cħ/mc2 = ħ/mc) of the electron. We interpret this by noting that the c = aω identity gives us the E = mc2 = ma2ω2 equation, which suggests we should combine the total energy (kinetic and potential) of two harmonic oscillators to explain the electron mass. We do so by interpreting the elementary wavefunction as a two-dimensional (harmonic) electromagnetic oscillation in real space which drives the pointlike charge along the zbw current ring. This implies a dual view of the reality of the real and imaginary part of the wavefunction: 1.The x = a·cos(ωt) and y = a·sin(ωt) equations describe the motion of the pointlike charge. 2.As an electromagnetic oscillation, we write it as E = E·cos(ωt+π/2) + i·E·sin(ωt+π/2). The magnitudes of the oscillation a and E are expressed in distance (m) and force per unit charge (N/C) respectively and are related because the energy of both oscillations is one and the same. The model – which implies the energy of the oscillation and, therefore, the effective mass of the electron is spread over the zbw disk – offers an equally intuitive explanation for the angular momentum, magnetic moment and the g-factor of charged spin-1/2 particles. Most importantly, the model also offers us an intuitive interpretation of Einstein’s enigmatic mass-energy equivalence relation. Going from the stationary to the moving reference frame, we argue that the plane of the zbw oscillation should be parallel to the direction of motion so as to be consistent with the results of the Stern-Gerlach experiment.
The radial velocity formula and the Planck-Einstein relation give us the zbw frequency (E = ħω = E/ħ) and zbw radius (a = c/ω = cħ/mc2 = ħ/mc) of the electron. We interpret this by noting that the c = aω identity gives us the E = mc2 = ma2ω2 equation, which suggests we should combine the total energy (kinetic and potential) of two harmonic oscillators to explain the electron mass. We do so by interpreting the elementary wavefunction as a two-dimensional (harmonic) electromagnetic oscillation in real space which drives the pointlike charge along the zbw current ring. This implies a dual view of the reality of the real and imaginary part of the wavefunction: 1.The x = a·cos(ωt) and y = a·sin(ωt) equations describe the motion of the pointlike charge. 2.As an electromagnetic oscillation, we write it as E = E·cos(ωt+π/2) + i·E·sin(ωt+π/2). The magnitudes of the oscillation a and E are expressed in distance (m) and force per unit charge (N/C) respectively and are related because the energy of both oscillations is one and the same. The model – which implies the energy of the oscillation and, therefore, the effective mass of the electron is spread over the zbw disk – offers an equally intuitive explanation for the angular momentum, magnetic moment and the g-factor of charged spin-1/2 particles. Most importantly, the model also offers us an intuitive interpretation of Einstein’s enigmatic mass-energy equivalence relation. Going from the stationary to the moving reference frame, we argue that the plane of the zbw oscillation should be parallel to the direction of motion so as to be consistent with the results of the Stern-Gerlach experiment.
This paper further explores the structural similarities between the elementary quantum-mechanical wavefunction and circularly polarized electromagnetic waves to further tune a possible physical interpretation of the wavefunction. The interpretation that is offered analyzes the real and the imaginary part of the wavefunction as oscillations which each carry half of the total energy of the particle. These oscillations are perpendicular to each other, and the interplay between both may describe how energy propagates through space over time. The model is based on three fundamental premises: 1.The dimension of the matter-wave field vector is force per unit mass (N/kg), as opposed to the force per unit charge (N/C) dimension of the electric field vector. This dimension is an acceleration (m/s2), which is the dimension of the gravitational field. 2.This gravitational disturbance may cause a charged mass to move about some center, combining linear and circular motion. This interpretation may reconcile the wave-particle duality to some extent: fields interfere but if, at the same time, they do drive a pointlike particle, this may explain why, as Feynman puts it, “when you do find the electron some place, the entire charge is there.” This hybrid hypothesis is supported by an elegant yet simple derivation of the Compton radius of an electron. 3.In light of the direction of the magnetic moment of an electron in an inhomogeneous magnetic field, the plane which circumscribes the circulatory motion of the electron should also comprise the direction of its linear motion. Hence, unlike an electromagnetic wave, the plane of the two-dimensional oscillation cannot be perpendicular to the direction of motion of our electron. Finally, this paper addresses an issue which has hampered other physical interpretations of the wavefunction: amplitude transformations when going from one representation (or reference frame) to another. Indeed, the author re-visits the original mathematical arguments here and shows that the proposed model, and the related physical interpretation of the wavefunction, are not incompatible with the key results and mainstream logic in this regard.
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 21 spin (Dirac), i.e., spins two circles in one cycle, both angular momentum are ℏ. Its wavelength is determined by E=hνm0=λm0hc=m0c2 (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)=21E0(1−cos(2ω0t)), 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=λm0h2πλm0=m0cR0, it has a quantized classical angular momentum, with a speed of c . The Bohr magneton, μB=2m0qℏ, g -factor is 2. Critically, the full Dirac’s equation of an electron with a constant velocity is mc=ℏ⊥+p⊥+ℏ//+p//, and its electromagnetic properties due to charge are f(Eq)=f(Eq)cc2−v′2+f(Eq)cv′=Enet//+Eq⊥, v′//ℏ⊥ or v′//ℏ//, and 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=γt0; Moreover, the net electricity 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)c2v′, which is not circularly symmetric and self-evidently does not follow Lorentz transformation. The temporospatial averaged magnetic intensity of a current is relativistic, Bˉ=∑f(Eq)c2Δv=2πϵ0c2I=2πμ0I, 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 Enet. We conclude this model of electron may provide one step further to the ultimate Quantum Theory in a nonrelativistic spacetime.
In Parts I and II we noted that a quantum oscillator time-independent Schrodinger equation may be scaled such that x→y with yy=sqrt(km)xx and E→ Esqrt(m/k). Given that the LHS side is a function strictly of y which may be integrated from -infinite to infinite, the RHS must be independent of k and m. Thus E is proportional to sqrt(k/m) which “ happens” to be the oscillator classical angular frequency. In Part II we noted that the scaling occurs in the Schrodinger equation because one uses -id/dx as the momentum operator which follows from the exp(ipx) form of a free particle wavefunction. In this note, we show that for any equation: ad/dx d/dx W + b xx W = E W scaling of the form x→y such that yy =xx sqrt(b/a) and E→ E 1/sqrt(ba) occurs. Furthermore if d/dx is associated with momentum then writing conservation of energy using m/2 vv leads to 1/a www cos(wt)cos(wt) + b sin(wt)sin(wt) = b and w is proportional to sqrt(ab). Thus E must be proportional to w, the classical frequency. (An equation of the form a df/dt df/dt + b f f = constant must have a cos(wt) solution as is already known.) We try to extend these ideas by considering a wave on a string with both ends fastened. The string produces sinusoidal waves, but sound is created if there is air. One may consider sound to come in identical units of energy proportional to the angular frequency w. This is similar to the Planck blackbody radiation assumption about light. It may be shown that the string follows an oscillator solution because d/dx dy/dx which represents the change in the sin(theta) multiplied by the constant tension from x to x+dx is set equal to y i.e Force is proportional to y which is the oscillator result. If one sets energy proportional to w as in the case of light and wishes to have an equation strictly in x describing this scenario and also linked to energy conservation then the momentum squared term should have dimensions 1/xx. This suggests d/dx d/dx as a solution because E must be proportional to w. Thus we argue that one may work “backwards” modeling phonons like photons in a classical problem to obtain the quantum operator p = -id/dx which is linked to the free particle wavefunction exp(ipx).
In Part 1, we tried to understand why spin arises. The basic conclusion was that it is linked with an intrinsic force which in turn is associated with E,p (energy momentum) and geometry (rotational symmetry and other symmetries). In practical terms, this means finding an equation linear in intrinsic force (as force appears linearly in nature), but still tied to E,p and geometry. Interestingly, this leads to two very different results for particles with rest mass (spin ½) and photons, although both satisfy the same special relativistic equations (albeit the photon with rest mass mo=0) and both have exp(-iEt+ipx) free wavefunctions. In the photon case, the Electric field El is the intrinsic force (together with the magnetic B) which gives rise to both energy and momentum density. Specifically, energy density= .5eo El dot El + .5/uo B dot B and momentum density = 1/uo El x B. In the photon case, however, there is no rest frame and so the classical vector El is directly associated with E (energy) and p (momentum) without one needing to consider special relativity. Instead one directly uses Maxwell’s electromagnetic equations in order to find a linear equation describing the classical vector El. This equation involves d/dt partial and grad partial which are linked to the quantum operators for E and p (id/dt, -igrad) and El itself is proportional on exp(-iEt+ipx) (actually cos(-wt+kx)), also from Maxwell’s equations. Given this scenario, one has a classical vector El which is perpendicular to B and both lie in a plane perpendicular to motion. For circular polarization, El moves in a circle and given that it gives rise to energy, this energy circulates like classical angular momentum. For the 3-vector for El, one says that one has spin 1, but for circular polarized light, this is classical angular momentum. As a result, spin in the photon case follows from the behaviour of the intrinsic force (a linear equation in this force as shown in Part 1), but one must also have a clear link between the intrinsic force and the consequential energy and force, because it is the energy which actually has to rotate. It is not enough to have El rotate. In the case of a particle with rest mass, we argued in Part 1 that mo (Newton’s inertia) is the intrinsic force which together with velocity gives rise to the consequential energy E and momentum p. The situation is that one actually has two physical frames, the rest frame with the intrinsic force mo and the moving frame in which one has E,p. In order to link the intrinsic force mo to E, p, one must use a Lorentz invariant. We suggested in Part 1 that -Et+px = -mocc t1 is associated with -E dt + p dx = -mocc dt1 + (p=0) * (dx1->inifite) =0. Then dx=hbar/p, dt=hbar/E and E/dx and p/dt are average forces as is mocc/dt1. Tthe idea of force is very much the cornerstone of exp(-iEt+ipx) we argue. In order to see spin arise, one needs an equation linear in mo (the intrinsic force), but linked to E,p and showing geometrical invariances, according to the recipe of Part 1. -EE + cc p dot p = -momocc ((1)) links E,p, mo and shows invariances, but is not linear in mo. One must then linearize it. This means that one compares to Lorentz frames, the rest frame containing mo and the moving containing E,p. Both mo and p are linked to forces, but mo is the intrinsic force and p, the consequential. As a result, they must somehow be separated as they are in ((1)), in particular they are perpendicular if one considers: EE = pz pz + momo (c=1, p->px). In order to linearize without using unit vectors, one cannot use numbers, but must use matrices (as Dirac showed). The operator linked with pz must anticommute with that linked with mo. The operators associated with the consequential and intrinsic forces differ and in fact anticommute in the rest mass case. One may show that one requires at least a 2x2 matrix (the full Dirac treatment uses a 4-vector made of 2 spinors), which means that there is a two vector state. This is presumably associated with motion, if exp(ipz z) is linked to motion as well. In order to analyze this 2 vector state (spinor), we first compare the situation with the photon case. There, one had a 3-vector El which means that one knows an exact position in the plane. The second piece of information must be the direction in which this vector is moving in a circle (for circularly polarized light). There are two a priori given pieces of information which together with the link between El and energy density yields classical angular motion. In the rest mass case, one again expects rotational motion in a plane perpendicular to the z axis (because pz describes all motion along z). (1,0) and (0,1), however, describe classical rotational direction, clockwise and counterclockwise, but this spin ½ case cannot be classical rotation because the physical vector in the plane is not given (as it is by El the intrinsic force). Thus, one has only the direction of rotation. We argue that if exp(ipx) means that one has p hits at different x within hbar/p with probability exp(ipx), one may imagine different x,y positions of energy in the plane (with probabilities) as long as the rotation for (1,0) for each is in the clockwise direction (for example). This means that a physical vector may appear at some x,y in the plane and then at an x1,t1 which would be considered a classical time in the past as long as the motion is clockwise. In this way, one may have an angle of 720 degrees (instead of 360 degrees) describe the motion. In terms of interaction with a B field, one has the clear two directions (spin up clockwise and spin down counterclockwise as examples). Thus, spin must exist in the rest mass case, because one must have mo perpendicular to p if special relativity is to hold. If one does not want to have spin, then one must disregard the equivalence of frames moving at constant speeds with respect to each other and that seems unphysical. (Note: There are spin 0 objects (Higgs particle), but we do not consider that here.
Schr\"odinger's work on the Zitterbewegung of the free electron is reexamined. His proposed "microscopic momentum" vector for the Zitterbewegung is rejected in favor of a "relative momentum" vector, with the value $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}=mc\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}$ in the rest frame of the center of mass. His oscillatory "microscopic coordinate" vector is retained. In the rest frame, it takes the form $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}=\ensuremath{-}i(\frac{\ensuremath{\hbar}}{2mc})\ensuremath{\beta}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}$, and the Zitterbewegung is described in this frame in terms of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$, and the Hamiltonian $m{c}^{2}\ensuremath{\beta}$, as a finite three-dimensional harmonic oscillator with a compact phase space. The Lie algebra generated by $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$ is that of SO(5), and in particular $[{Q}_{i},{P}_{j}]=\ensuremath{-}i\ensuremath{\hbar}{\ensuremath{\delta}}_{\mathrm{ij}}\ensuremath{\beta}$. It is argued that the simplest possible finite, three-dimensional, isotropic, quantum-mechanical system requires such an SO(5) structure, incorporates a fundamental length, and has harmonic-oscillator dynamics. Dirac's equation is derived as the wave equation appropriate to the description of such a finite quantum system in an arbitrary moving frame of reference, using a dynamical group SO(3,2) which can be extended to SO(4,2). Spin appears here as the orbital angular momentum associated with the internal system, and rest-mass energy appears as the internal energy in the rest frame. Possible generalizations of these ideas are indicated, in particular those involving higher-dimensional representations of SO(5).
Quantum mechanics is defined here as the theoretical framework crystallised at the fifth Solvay Conference of 1927, comprising five postulates: the wavefunction, the canonical commutation relation [x, p] = iℏ, the Born rule, the Schrödinger equation, and the transition probability rule. We present a complete derivation of all five Solvay postulates from purely classical foundations, starting from the directly observable Coulomb field of the electron. The single empirical input is ℏ—measured from pre-Solvay experiments (Planck 1900–1911, Einstein 1905, Casimir 1948/Lamoreaux 1997) without quantum mechanical interpretation, in the same way G was measured by Cavendish (1798) without general relativity. By Einstein’s photoelectric theory (1905) generalized to all electromagnetic fields, the Coulomb field energy is carried in electromagnetic quanta, each carrying both electric and magnetic fields with energy density ϵ0E^2. Lorentz invariance uniquely determines the ZPF spectral energy density u(ω) ∝ ω^3 (Boyer 1969, 1975). The Casimir force measurement gives the total ZPF energy per mode B = ℏ; classical E/B equipartition from Maxwell’s equations gives the electric energy per mode ℏω/2. The central original contribution is the physical justification of the Compton frequency cutoff ωc = mc^2/ℏ: an electron cannot emit an electromagnetic quantum with energy exceeding mc^2, because the residual electron would require negative mass, never observed in nature. By equating two independent expressions for the momentum variance of the ZPF-driven harmonic oscillator—one from the Abraham–Lorentz equation, one from Nelson’s osmotic velocity formula—D = ℏ/2m is derived without any quantum assumption. The ZPF-driven motion is then proved to be Brownian motion in the precise sense of Wiener (1923): all four Wiener conditions are verified from the properties of the ZPF established in Sections 3–4. This answers the question Nelson (1966) explicitly left open: “We do not attempt to give a physical explanation of why the electron should undergo Brownian motion.” The physical explanation is that the electron cannot escape its own Coulomb field. By Itô’s stochastic calculus and Nelson’s theorem, this gives the Schrödinger equation as an effective description valid for timescales Δt ≫ ℏ/mc^2. The transformation ψ = √ρe^(iS/ℏ) linearises the nonlinear Fokker–Planck equation, explaining why superposition holds. All five Solvay postulates emerge as consequences of classical physics plus the single measured constant ℏ. No Solvay postulate is assumed anywhere.