Wave Functions for the Electron and Positron

The Wave/Particle duality of particles in Physics is well known. Particles have properties that uniquely characterize them from one another, such as mass, charge and spin. Charged particles have associated Electric and Magnetic fields. Also, every moving particle has a De Broglie wavelength determined by its mass and velocity. This paper shows that all of these properties of a particle can be derived from a single wave function equation for that particle. Wave functions for the Electron and the Positron are presented and principles are provided that can be used to calculate the wave functions of all the fundamental particles in Physics. Fundamental particles such as electrons and positrons are considered to be point particles in the Standard Model of Physics and not considered to have a structure. This paper demonstrates that they do indeed have structure and that this structure extends into the space around the particle‟s center (in fact, they have infinite extent), but with rapidly diminishing energy density with the distance from that center. The particles are formed from Electromagnetic standing waves, which are stable solutions to the Schrödinger and Classical wave equations. This stable structure therefore accounts for both the wave and particle nature of these particles. In fact, all of their properties such as mass, spin and electric charge, can be accounted for from this structure. These particle properties appear to originate from a single point at the center of the wave function structure, in the same sort of way that the Shell theorem of gravity causes the gravity of a body to appear to all originate from a central point. This paper represents the first two fully characterized fundamental particles, with a complete description of their structure and properties, built up from the underlying Electromagnetic waves that comprise these and all fundamental particles.


Introduction
In order to provide the foundations of a link between Classical Physics field concepts and the wave/particle duality in Quantum Mechanics it is necessary to demonstrate how particles can be modeled both from a Classical Wave perspective while also satisfying the requirements of Quantum Mechanics, in particular the Schrödinger wave equation and the De Broglie equations.
There is already evidence of this connection in the energy sum of the Electric and Magnetic fields in the Hamiltonian function that expresses the total energy of an atomic system: (Unifiedphysics 2015) "In 1926 Schrödinger used energy conservation to obtain a quantum mechanical equation in a variable called the wave function that accurately described single-electron states such as the hydrogen atom. The wave function depended on a Hamiltonian function and the total energy of an atomic system, and was compatible with Hertz's potential formulation. The wave function depends on the sum of the squares of E-and H-fields as is seen by examining the energy density function of the electromagnetic field." In order to satisfy both the wave and particle natures of particles in a model of a particle, the particle"s wave function must satisfy both the Classical wave equation (which ensures that the wave function can represent a vibration of the space-time continuum) and the Schrödinger wave equation (which ensures that the wave function can represent a quantum of energythus a particle) (Simpson. G.D. 2012). location in space and time. The position of a test point in space as it is affected by the wave motion can be represented as a displacement vector drawn from the starting location of the point to its current location. The wave function defines the magnitude and direction of the displacement vector at any location in space and at any timetherefore it completely and precisely defines the pattern of vectors that form the structure of the particle that it describes.
In the case of Electromagnetism there is a single vector field that describes the motion of an Electromagnetic wave in this way, it is known as the Hertzian vector field (Ornigotti. A. et al, 2014), (J. J. Sein, 1989). The Electric and Magnetic fields can both be derived from the Hertzian vector field by differentiation with respect to space and time. (Ornigotti. A. et al, 2014) "in a vacuum a single Hertz vector written as the product of a scalar potential and a constant vector, naturally arises as consequence of the transversality of the electromagnetic fields" Therefore, a wave function that describes a field of vectors representing Hertzian vectors can also represent a wave function describing Electric and Magnetic field vectors. If the wave function satisfies both the Classical wave equation and the Schrödinger wave equation then it can also represent a vibration of space-time and a potential solution for a Quantum particle.
In the case of a Classical wave function describing an Electromagnetic wave, the displacement vectors described by the wave function are of a physical charge displacement from the otherwise neutral vacuum state. Thus, the vacuum is seen to be polarised by the displacement by an amount with units of Volt-Meters. The presence of the displacement from the neutral vacuum state requires a certain energy, and the total (rest) energy of the particle"s wave function is the sum of the resulting Electric and Magnetic fields (Traill. D. A, 2020); This paper presents two such solutions, one representing an electron and one representing a positron. In addition, it is shown that the correct Classical fields are produced by them, matching those of the real particles, and that the Quantum Mechanical requirements of the De Broglie equations are also met by them.

Lorentz Transformation of the Wave Function Solutions
The Schrödinger wave equation is not Lorentz Invariant, whereas the Klein-Gordon or Dirac equations (which are Lorentz Invariant) are usually used to model electrons/positrons. In my model, however, as the mass of the electron/positron does not appear as a point particle but is present as the mass equivalent of the total energy of in the Electromagnetic field (Electric and Magnetic fields combined) in the fields derived from the wave-function equation (Traill. D. A , 2020), the mass term in the Klein-Gordon and Dirac equations would be zero as the energy of the field is already accounted for in the other terms in these equations. The Klein-Gordon equation reduces to the Classical wave equation when the mass term is zero, and my wave function solutions are solutions to the Classical wave equation.
The Klein-Gordon equation (OxfordReference, 2021): which is the Classical wave equation.
The Classical wave equation is Lorentz Invariant, so my wave function solutions are also Lorentz Invariant. Therefore, my wave-function solutions are compatible with Relativity even though the Schrödinger wave equation (to which they are also solutions) is not a Relativistic wave equation. Another point to make here is that the wave functions are solutions to the Schrödinger wave equation (which applies in a frame that is at rest) and from Relativity we know that all inertial reference frames are equivalentwhich means that Physics is the same for any moving observer. Therefore, we can conclude that the wave function in the moving reference frame is still a solution to the Schrödinger wave equation in that reference frame, even though that frame is moving with respect to us, and the Schrödinger wave equation itself is not Lorentz invariant.
in the waves all having zero amplitude at the same time during their oscillation cycle, which would cause problems with the conservation of energy. This model has been taken further and it has been found that in order to explain charge, the required structure is that of a spinning spiral wave, with inward or outward flowing phase. These equations are formulated based on my initial analysis of the form of the structure that must be requiredthat of a spinning spiral with a spherical wave distribution. The equations were further refined with the use of the 3D modelling software developed by the author to simulate the wave functions. Also, the properties of these modelled wave functions were analysed to refine the amplitude term of the wave functions such that the electric potential of the modelled particles matches that of the real particles. Having done this gives good confidence that the resulting equations are a true description of the real fundamental particles.
The images located in  show graphical representations of the fields derived from these wave functions using a 3D vector modelling program developed by the author to aid in the visualization and testing of proposed wave function solutions.
For the Electron: For the Positron: Where:

The Solutions Satisfy the Wave Equations
The wave nature of the particles is being modelled here as a vibration of the space-time continuum (a physical charge displacement) and the particle nature is modelled as localized quanta of this wave energy (a temporally stable 3D wave structure). In order to satisfy both the wave and particle natures of particles in the model, the wave function must satisfy both the Classical wave equation (which ensures that the wave function can represent a real, physical vibration of the space-time continuum) and the Schrödinger wave equation (which ensures that the wave function can represent a quantum of energythus a stable particle) (Simpson. G.D. 2012).
Classical wave equation: The wave function describes a field of rotating displacement vectors which can each be thought of as comprising two orthogonal Quantum Harmonic Oscillators (OpenStax, 2020); one along each axis of the complex plane. The vectors trace out a circle, such that at any given time half of the total energy is present as Kinetic energy and half as Potential energy. The amount of energy in each these forms depends on the phase of each of the component Quantum Harmonic Oscillators. In their simple harmonic motion oscillation, each oscillates between full KE and full PE, but when one has full KE the other has full PE and vice-versa.
See Appendix A for the proof that a Quantum Mechanical wave-function can be modelled as a field of Real 3D vectors and still satisfy both the Schrödinger and Classical wave equations.
Due to Equipartition of energy in a Classical wave (Duffin. R.J, 1969):

Testing the electron solution with the Schrödinger wave equation
Referring to Eqs. 1 and 4: In Spherical coordinates (Gradient, Divergence, Curl, and Laplacian, LibreTexts, 2020) the Laplacian of is: As the wave function is spherically symmetrical, all the vectors at the same distance from the origin are identical, so the terms involving and are zero.
So ∇ reduces to: Thus, the Schrödinger wave Eq. 4 becomes: Thus: As from by differentiation of Eq. 1, we can also say that: Also: Thus:

Testing the electron solution with the Classical wave equation
Referring to Eqs. 1 and 3: So: Thus: And, substituting Eqs. 6a, 10a and 15a into Eq. 3: 3) too, so the electron wave function (Eq. 1) is a solution to it also.

Testing the positron solution with the Schrödinger wave equation
Referring to Eqs. 1 and 4: In Spherical coordinates the Laplacian of is: As the wave function is spherically symmetrical, all the vectors at the same distance from the origin are identical, so the terms involving and are zero.
So ∇ reduces to: Thus, the Schrödinger wave Eq. 4 becomes: Thus: As from by differentiation of Eq. 2, we can also say that: Also: Thus: So, from Eqs. 8b, 9b and 11b, we can see that LHS = RHS of the Schrödinger wave equation (Eq. 4), so the wave function (Eq. 2) is a solution to it: ∇ Eq. 9b equals Eq. 11b:

Testing the positron solution with the Classical wave equation
Referring to Eqs. 2 and 3: So: Thus: And, substituting Eqs. 6b, 10b and 15b into Eq. 3: So LHS = RHS of Classical Wave equation (Eq. 3) too, so the positron wave function (Eq. 2) is a solution to it also.

The Wave Function and Electromagnetism
Each of the measurable fields in Electromagnetic Theory (J. J. Sein, 1989), (Traill, D.A, 2002), (Kouzaev, G.A, 2010), and their connection back to the wave function, can be expressed quite simply by the following set of equations and illustrated by Figure 1.
Note: Eq. 20 is derived from Eq. 3 as the Classical wave equation states that ∇ , so H is equivalently

Analysis of the Wave Functions
Both wave functions represent a field of rotating charge displacement vectors. The pattern described by the phases of the field of rotating vectors is that of a spinning spiral wave. The phase wave phase flows either away from or towards the centre of the particle (Figures [3, 4a, 4b]). The Electron spins with the phase wave flowing outward and the Positron with the phase wave flowing inwards (Traill, D.A, 2002).
The angular frequency in the wave function is derived from the following three known equations.
(24) Substituting Eq. 23 and Eq. 24 into Eq. 22 and solving for we have: Then to convert to angular frequency: (26) Substituting Eq. 25 into Eq. 26 and using the electron mass gives: Radians per Second This describes the rate of rotation of each of the vectors in the vector field that describes the Electron/Positron wave function.
The centre of the Electron comprises a vector that rotates around a fixed position at the particle"s centre over time. As time progresses this vector propagates radially outwards, at the speed of light, away from the centre thus forming the phase wave spiral. Therefore, the phase of each vector depends not only on time, but also on distance from the centre of the wave function.
(1) The vector rotation rate, determined by: (2) The propagation delay for a wave travelling radially outwards at the speed of light, given the vector rotation rate above: Thus, the total phase change (for an electron) at a distance from the centre is the sum of these two factors: Every point on the wave function spiral comprises a vector that rotates around a fixed location in space as the phase waves pass through each point.
In a classical wave, each point in the medium supporting that wave (such as the water molecules in a water wave) moves in a circular motion as the wave passes. The frequency of this circular motion is the same as that of the wave. However, when two waves of equal frequency (but travelling in opposite directions) combine to form a standing wave, each point in the medium rotates at twice the angular frequency of each of the two component waves.
The Classical interpretation of the De Broglie wave is that of a beat frequency of the upstream and downstream components (with respect to the particle"s direction of motion) of the Electron"s IN/OUT wave, so: up electron Which is the Energy/Mass relationship as it should be.
This can also be proved using the displacement amount of the Quantum Harmonic Oscillators (OpenStax, 2020) in the wave function vector field. The units of the vectors in the wave function are Volt-Meters, so they need to be converted to just Meters, then the energy of the Oscillators calculated (See Appendix C).
The electron rest mass has also been calculated from my 3D model by integrating the Electric and Magnetic field vector energies over a small volume (cube) of space with the electron wave function at the center of the volume. The resulting rest-mass value is 100.46% of the actual electron rest-mass. (Traill. D. A, 2020)

Derivation of the Classical Electric Potential for the Electron and Positron
For the Electron wave function, the Electric Potential (V ) is ( ), which in spherical coordinates is [10].
When viewed close-up, the spinning spiral and charge layers that comprise the Electron/Positron are clearly visible. Due to the fast spinning of the spiral (and outward or inward phase flow), or at large distance scales where the undulations of the spinning charge layers are small by comparison, the fields appear to become smooth and be of a continuous nature [ Fig. 6-11]. So, for example, the Electric Potential for the Electron/Positron in this case appears to be the RMS (Root-Mean-Squared (The RMS Voltage of an AC Waveform, electronics-tutorials, 2015)) of Eq. 41a or 41b, which is equal to the classical equation: or For the Proof that this is the case, see Appendix B.

The Quantum Mechanical Wave-Function
In Classical Physics, the wave function describes a field of displacement vectors; that is the complex number of the wave-function describes X-axis and Y-axis displacements from the origin on an Argand diagram. In Quantum Mechanics, however, the wave function is a complex scalar value rather than a vector, although http://apr.ccsenet.org Applied Physics Research Vol. 14, No. 1; 2022 the complex number of its wave-function is in the same mathematical form. However, the complex "i" term in the wave equation simply represents a 90-degree rotation (orthogonal vector quantities), and so the equation can be reformulated purely Classically, with no complex numbers by introducing a rotation matrix in place of the "i" term.
In Quantum Mechanics the square of the wave function gives the probability density of finding a particle at a point in space: (43) In order to get vector quantities in Quantum Mechanics one must apply a vector operator to the probability, such as applying the position or momentum operators to the wave-function.
The Classical wave function, on the other hand, remains a vector and its magnitude is equivalent to the Quantum Mechanical scalar wave function: See Appendix A for the proof that a Quantum Mechanical wave-function can be modelled as a field of Real 3D vectors (with a rotation matrix in place of the complex "i" term) and still satisfy both the Schrödinger and Classical wave equations.
So, given that my Classical wave function model is compatible with the Quantum Mechanical wave function, this work sheds new light on the wave/particle duality issue in Quantum Mechanics. The Classical wave functions presented here for the particles is fundamentally a wave structure, yet the energy of this wave structure remains localized (quantized) and its properties appear to be associated with a single pointat the center of the wave function structure. As this wave structure is not a point particle but is spread through space with varying degrees of energy density in different locations, we can see how the probability amplitude referred to in Quantum Mechanics may be associated with the energy density and other properties of the dynamic waves of the Classical wave function.

Normalization of the Wave-Function
In Quantum Mechanics is it important that a wave-function be normalizable; that is the area under the curve of the wave function be finite, such that the square of the wave-function can represent a probability density of finding the particle. The probability integrated over the whole wave-function must be 1, so that the particle exists somewhere in space. An infinite probability doesn"t make physical sense, so in order that the wave-function represents a real particle, it must be Normalizable.
As my wave function solutions (Eqs. 1 & 2) contain a ⁄ term they would not, on face value, appear to be normalizable due to an infinity at . However, if in reality, a function is known to be finite, as is the case for an electron or positron, then it may be considered Normalizable (Wave Mechanics, 2018), (Physics -Stack Exchange discussion, 2018). As my wave functions are comprised of Quantum Harmonic Oscillators with real displacements in space, at size scales less that this displacement distance, the nature of the function will no longer be ⁄ , thus restricting its amplitude and preventing an infinite quantity (Gopalakrishnan, S, 2006). For these wave-functions, the order of magnitude of this displacement is meters (see (OpenStax, 2020) on the physical displacement for a Quantum Harmonic Oscillator; in this case assuming all of the energy of an electron was a single Quantum Harmonic Oscillator at the particle"s center). Thus, there is a natural limit on the ⁄ factor in the wave-function amplitude. Thus, these wave-functions are compatible with Quantum Mechanics, and their probability density may be calculated via Eq. 43, with the use of Eq. 44.

Conclusion
The wave functions presented here describe particles with all the correct properties for an Electron and a http://apr.ccsenet.org Applied Physics Research Vol. 14, No. 1;2022 Positron and satisfy the requirements of both the Classical and Quantum Mechanical interpretations.
The wave function represents a field of rotating charge displacement vectors with units of Volt-Meters. The spinning vectors form a phase wave that describes a spinning spiral. The resulting phase wave flows either away from or towards the centre of the particle. Interactions between the phase waves of two or more particles can be shown to be the cause of the Electrical and Magnetic attraction/repulsion between charged particles due to momentum exchanges between the wave structures (Traill, D.A, 2020).
In general, the concepts use to build these two wave equations could be applied to all particles in Physics. The key principles are: 1. The frequency of the waves that comprise the three-dimensional wave structure of the particle is based on the particle's mass (via the calculation shown above).
2. A particle's charge is defined by either an outward or inward flowing phase wave. A neutral particle would have no net phase flow inward or outward, but may contain regions of either inward or outward flow, which cancel out in the region surrounding the particle.
3. The completed wave function must satisfy both the Classical and Schrödinger wave equation.

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