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.
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 geometry of the elementary quantum-mechanical wavefunction and a linearly polarized electromagnetic wave consist of two plane waves that are perpendicular to the direction of propagation: their components only differ in magnitude and – more importantly – in their relative phase (0 and 90° respectively). The physical dimension of the electric field vector is force per unit charge (N/C). It is, therefore, tempting to associate the real and imaginary component of the wavefunction with a similar physical dimension: force per unit mass (N/kg). This is, of course, the dimension of the gravitational field, which reduces to the dimension of acceleration (1 N/kg = 1 m/s2). The results and implications are remarkably elegant and intuitive: - Schrodinger’s wave equation, for example, can now be interpreted as an energy diffusion equation, and the wavefunction itself can be interpreted as a propagating gravitational wave. - The energy conservation principle then gives us a physical normalization condition, as probabilities (P = |ψ|2) are then, effectively, proportional to energy densities (u). - We also get a more intuitive explanation of spin angular momentum, the boson-fermion dichotomy, and the Compton scattering radius for a particle. - Finally, this physical interpretation of the wavefunction may also give us some clues in regard to the mechanism of relativistic length contraction. The interpretation does not challenge the Copenhagen interpretation of quantum mechanics: interpreting probability amplitudes as traveling field disturbances does not explain why a particle hits a detector as a particle (not as a wave). As such, this interpretation respects the complementarity principle.
If mathematics is the queen of science, then physics might well be the king. It successes are obvious. However, as a science, physics may have failed in one regard, and that is to explain what its basic concepts – such as state vectors, wavefunctions, and transformation matrices – actually represent. When studying quantum mechanics, it is, effectively, hard to keep up the initial enthusiasm, and those who branch out to other fields – which is most of us – quickly end up going through the motions only: we regurgitate models and equations and know how to solve the standard problems, so as to pass the exam, but then forget about them as soon as possible. This paper explores a very intuitive sentiment about the issue: the wavefunction is a rather ‘flat’ mathematical object – it is two-dimensional, basically – so it can’t do the trick, perhaps. In contrast, Maxwell’s equations have real vectors in them, which is why a deeper or more intuitive understanding of electromagnetism comes relatively easily. Indeed, when everything is said and done, we are just human beings living in three-dimensional space, and that is why vector equations (or systems of vector equations), as a mathematical tool, make sense to us. This paper further explores this sentiment. It also offers a way out by, predictably, presenting yet another possible physical interpretation of the wavefunction. More importantly (for the reviewer of this paper, at least), this paper offers a sensible response to the mainstream view that three-dimensional physical interpretations of the wavefunction cannot make any sense because of the weird 720° symmetry of the wavefunction when describing spin-1/2 particles (fermions or – for all practical purposes – electrons). The author does so by analyzing (1) Dirac’s belt trick more in detail – and what it implies in terms of the interaction between the observer and the object – as well as (2) Feynman’s derivation of the transformation matrices for spin-1/2 two-state systems.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
This article explores a possible physical interpretation of the wavefunction. It zooms in on the physical model it implies for an electron in free space (further building on Schroedinger's ideas in regard to the apparent Zitterbewegung of an electron), but then also discusses the wavefunction for particles (including non-charged particles) in general. While it basically concludes that the mainstream interpretation of quantum physics (the Copenhagen interpretation) is and remains the most parsimonious explanation, it also argues that one or two extra assumptions – the wavefunction as a two-dimensional self-sustaining electromagnetic or gravitational oscillation in space – would make more frivolous explanations (many worlds, pilot wave, etcetera) redundant.
It is shown that a wave mechanical quantum theory can be derived from relativistic classical electrodynamics, as a feature of the magnetic interaction of Dirac particles modeled as relativistically circulating point charges. The magnetic force between two classical point charges, each undergoing relativistic circulatory motion of small radius compared to the separation between their centers of circulation, and assuming a time-symmetric electromagnetic interaction, is modulated by a factor that behaves similarly to the Schrödinger wavefunction. The magnetic force between relativistically-circulating charges has been shown previously to have a radially-directed inverse-square part of similar strength to the Coulomb force, and sinusoidally modulated by the phase difference of the charges' circulatory motions. The magnetic force modulation in the case of relatively moving centers of charge circulation solves an equation formally identical to the time-dependent free-particle Schrödinger equation, apart from a factor of two on the partial time derivative term. Considering motion in a time-independent potential obtains that the modulation also satisfies an equation formally similar to the time-independent Schrodinger equation. Using a formula for relativistic rest energy advanced by Osiak, the time-independent Schrödinger equation is solved exactly by the resulting modulation function. The significance of the quantum mechanical wavefunction follows straightforwardly from these observations. After considering the modification of Wheeler-Feynman absorber theory required by the adoption of Minkowski-Osiak relativity, the model is extended to obtain the full complex Schrödinger wavefunction.
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.