The first realistically photon-like Schrödinger solution of Maxwell's classical equations in dispersive media is presented. Classical modes of transverse electric or transverse magnetic fields with angular frequency ω propagating along an axis are shown to be able to be enveloped with counter-rotating helical modulations which have a different angular frequency Ω. These helical rotations, called distributed spin rotations, propagate at the group velocity. The formation of a completely closed packet of electromagnetic energy requires that the axial fields and transverse fields have a common axial length of envelope. This forces Ω to take quantized values in terms of ω with Ω related to the Schrödinger frequencies of a harmonic oscillator. The spin rotations permit flexible transverse confinement allowing for localization of the photon wave-packet over different spatial areas. It is argued that the energy of this packet is not related to its volume but depends on the quantized helical frequency Ω. Such photon-like packets possess classical phase and group velocities in keeping with experimental evidence. A single photon-like packet does not disperse in dispersive media. Incrementing or decrementing the rate of helical rotation promotes or demotes the packet energy in keeping with standard photon theory. The model offers explanations for self-interference and entanglement.
The first realistically photon-like Schrödinger solution of Maxwell's classical equations in dispersive media is presented. Classical modes of transverse electric or transverse magnetic fields with angular frequency ω propagating along an axis are shown to be able to be enveloped with counter-rotating helical modulations which have a different angular frequency Ω. These helical rotations, called distributed spin rotations, propagate at the group velocity. The formation of a completely closed packet of electromagnetic energy requires that the axial fields and transverse fields have a common axial length of envelope. This forces Ω to take quantized values in terms of ω with Ω related to the Schrödinger frequencies of a harmonic oscillator. The spin rotations permit flexible transverse confinement allowing for localization of the photon wave-packet over different spatial areas. It is argued that the energy of this packet is not related to its volume but depends on the quantized helical frequency Ω. Such photon-like packets possess classical phase and group velocities in keeping with experimental evidence. A single photon-like packet does not disperse in dispersive media. Incrementing or decrementing the rate of helical rotation promotes or demotes the packet energy in keeping with standard photon theory. The model offers explanations for self-interference and entanglement.
A photon-like wavepacket based on novel solutions of Maxwell's equations is proposed. It is believed to be the first 'classical' model that contains so many of the accepted quantum features. In this new work, novel solutions to Maxwell's classical equations in dispersive guides are considered where local helical twists with an arbitrary angular frequency W modulate a classical mode (angular frequency w, group velocity vg). The modal field patterns are unchanged, apart from the twist, provided that the helical velocity vh equals vg. Pairs of resonating retarded and advanced waves with modal and helical frequencies (w,W) and (w,-W)respectively, trap one temporal period of the underlying classical mode forming a photon-like packet provided W = (M+1/2)w: 'Schrodinger' frequencies. This theory supports experimental evidence that the photon velocity does not change with M in dispersive systems. Promotion and demotion increase or decrease the helical frequencies in units of w. An energy of interaction between retarded and advanced waves in the wave-packet is also proportional to these helical frequencies W = (M+1/2)w similar to Planck's law. Group velocity and polarisation are unaffected by the value of M. Advanced waves enable phase and polarisation to be predicted along all future paths and may help to explain the outcomes of experiments on delayed-choice interference and entanglement, without causality being violated.
From a combination of Maxwell’s electromagnetism with Planck’s law and the de Broglie hypothesis, we arrive at quantized photonic wave groups whose constant phase velocity is equal to the speed of light c = ω/k and to their group velocity dω/dk. When we include special relativity expressed in simplest units, we find that, for particulate matter, the square of rest mass , i.e., angular frequency squared minus wave vector squared. This equation separates into a conservative part and a uniform responsive part. A wave function is derived in manifold rank 4, and from it are derived uncertainties and internal motion. The function solves four anomalies in quantum physics: the point particle with prescribed uncertainties; spooky action at a distance; time dependence that is consistent with the uncertainties; and resonant reduction of the wave packet by localization during measurement. A comparison between contradictory mathematical and physical theories leads to similar empirical conclusions because probability amplitudes express hidden variables. The comparison supplies orthodox postulates that are compared to physical principles that formalize the difference. The method is verified by dual harmonics found in quantized quasi-Bloch waves, where the quantum is physical; not axiomatic.
Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.
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.
Schrödingers equation with boundary conditions gives quantized energy states for electron waves, but Maxwell’s wave equations have quantized states only by analogies with harmonic oscillators. This problem is addressed by a novel theory of wave‐packets using diffracting Transverse Electric and Transverse Magnetic fields defined by axial H‐ and E‐fields. All transverse fields and gradient operators can together be rotated about the propagation axis at frequencies, independent of the modal frequency. Without altering the axial fields, any helical motion propagates at the group velocity. This is quite different from single frequency helical modes (e.g. Laguerre Gaussian) travelling at the phase velocity. Reversing time and frequency, allows counter rotating helical solutions. These are referred to as adjoint or a fields that may interact and propagate with the classical causal reference or r fields. Overlapping and counter rotating r and a fields with slightly different frequencies interfere, leaving circular polarization states unaltered and creating a nodal structure in the transverse fields distinct from the nodal structure in the axial fields. Number states arise from requiring that transverse and axial nodes co‐locate with integral spacings to form a wave‐packet,. The a fields act as pilot waves for future potential positions of a quantized interaction between r and a fields. Uncertainty in the position of the overlap leads to conventional probabilistic quantum interpretations. The a fields are not fully determined until their detection with the r wave and this late determination can offer explanations for non‐local entanglement.
Several important questions regarding the photon wave and particle concepts have so far not been fully settled by means of conventional theory which is based on Maxwell's equations. This concerns the problems of the angular momentum and the divergence of the obtained solutions in vacuum space. In attempts to tackle these and other unclear points, a number of extended non-Maxwellian theories have been elaborated. The present analysis is due to one of these approaches, as being based on a nonzero electric field divergence in the vacuum state, under the condition of Lorentz invariance. This theory leads to models for the individual photon being based on axisymmetric wave packet solutions with imposed relevant quantum conditions. Such solutions can be confined to a limited volume in vacuum space, thereby giving rise to a nonzero angular momentum, as required for a boson particle. Two wave packed modes are obtained which have limited characteristic diameters in the direction being perpendicular to that of the propagation. For one of the modes the diameter even becomes comparable to atomic dimensions. This provides the possibility for an efficient photon-electron interaction in the widely discussed but not fully understood photoionization effect. Both modes further have features in common with an earlier theory by de Broglie, where the photon is represented by a propagating pilot wave with a superimposed particle-like part having a very small rest mass. A proposal is finally made for the photon to appear in different quantum states, and where rapid transitions can take place between these states in the form of "photon oscillations".