TUTORIAL 3. THE SELF-POTENTIAL
Here, the PTV model makes its sharpest departure from quantum mechanics. In the Bohr model, the Sommerfeld model, and the Dirac equation, and electron possesses quantised energy levels because it occupies an external Coulomb potential provided by the proton. Without the proton there are no levels. However, experiments on electron vortices in transmission electron microscopes have shown that electrons can occupy quantised orbital states with large orbital angular momentum in free space, that is, without any confining potential. The PTV model accounts for this naturally, see Paper B.
3.1 The unloaded ring (energy levels in free space)
An 'unloaded' PTV is one that has absorbed no radiation. Its Sp-3 toroidal rotation provides a self-potential, an internal energy that depends on the toroidal radius. It is simply the energy of motion of the Sp-2 guide tube around the toroidal axis.
The Self-Potential
The Sp-3 energy ε3 = m2 α²c² = hαc/R3 is added to the rest mass energy moc² to give the total unloaded ring energy:
mpotc² = moc² + hαc/R3
where R3 is the unloaded ring Sp-3 toroidal radius. The term added to the rest mass energy is the self-potential. The PTV can then possess energy levels by absorbing or emitting radiation that modifies this total energy without the presence of an external potential.
3.2 The loaded ring (absorption of radiation)
A 'loaded' PTV is one that has absorbed radiation and thereby adopted a particular energy level characterised by the Sommerfeld quantum numbers: azimuthal nΦ and radial nr. Their sum is n. In the PTV model, as n increases the Sp-3 rotational frequency decreases. This means that the highest Sp-3 frequency is at n = 1. Three radiation processes are now possible.
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An electron above its lowest state can emit a photon and drop to a lower state with higher Sp-3 frequency (natural emission).
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When the Sp-3 rotation absorbs radiation that opposes its rotation sense the Sp-3 frequency decreases and the electron moves to a higher energy state (stimulated absorption).
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For an electron above its lowest state, the incidence of a rotating radiation that is the same sense as the Sp-3 rotation increases the Sp-3 frequency and it drops to a lower n with an addition to the incoming photon (stimulated emission).
The upper case radius R3 for the unloaded ring is now replaced by the lower case r3 for the loaded case. Several crucial departures from the Sommerfeld model are now introduced.
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Sommerfeld has a radius r3 proportional to nΦ². In the PTV model, the radius r3o is invariant.
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Elliptical states are naturally accommodated arising from the same radial integral used by Sommerfeld. The major axis depends linearly on r3o.
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The increase in angular momentum with nΦ as the electron moves to higher states does not arise from an increase in Sp-3 radius but from an increase in string density N3. These are identical iterations of a string, see Figure 5.
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As well as a string density, the Sp-2 circuits take on an interlaced structure where there are n² circuits. This arises from n wavelengths and n interlacings per wavelength, see Figure 5(a) and Paper B.
Figure 5 String density and sub-strings for n = 2 and nΦ = 2. (a) One of the nΦ² sub-strings of the density function N3. The sub-string has n wavelengths each with n interlaced components to give n² circuits. (b) Iteration of the sub-string in (a) to give the density nΦ² = 4. (c) Right-end elevation of (b).
3.3 The attraction and repulsion mechanism
Paper A Paper B Paper C
One of the main foundational principle of the PTV model is that action can be redistributed between azimuthal and translational modes. This has already been exhibited in the photon structure where in a dispersive medium, translational action can be transferred into azimuthal action to reduce the speed of translation to below c. The Sp-2 rotation tends to maintain action h so if this is reduced or increased then action is taken from or deposited into Sp-3 rotation to restore it.
Paper D Paper E
Electrical Attraction — Passive Acceleration Analogy
Consider a proton and electron PTV with a common toroidal axis. When the proton approaches with opposite Sp-3 rotation sense, the electric momentum field p2f displaces electron Sp-3 rotational energy into translational energy. In seeking the next lower integer action, the electron accelerates towards the proton to increase the redistributing field strength. This is the exact analogue of passive acceleration where the rotational action is redistributed into translational action without absorbing energy .
Electrical Repulsion — Active Acceleration Analogy
When the Sp-3 rotation senses of the field and target are the same, the proton overloads the electron's Sp-3 action. The electron moves away from the proton to a weaker field to reduce this total action, redistributing the excess into energy of motion. This is the analogue of active acceleration in which energy is added to the system and the electron gains kinetic energy as a result.
Paper A: Barry R. Clarke, Reinterpretation of the Grangier experiment using a multiple-triggering single-photon model, Modern Physics Letters B , 15, 2350042 (2023).
Paper B: Barry R. Clarke, A photonic toroidal vortex model of the hydrogen atom fine structure, Quantum Studies Mathematics and Foundations, 12, 19 (2025).
Paper C: Barry R. Clarke, Geometrical interpretation of the hydrogen atom hyperfine structure, under peer review.
Paper D: Barry R. Clarke, The Lorentz force and the nature of charge from a Photonic Toroidal Vortex Model, under peer review.
Paper E: Barry R. Clarke, A heuristic model of the Bose-Einstein distribution with distinguishable photons
Tutorial 4
Tutorial 2