A geometrical model of the electron is shown as a closed-loop standing wave. The charge path is in the form of rotating Hopf link generating a toroidal swept surface with circumference equal to the Compton wavelength. A precessing epitrochoid charge path composed of two orthogonal spinors with toroidal & poloidal current loop components of 2:1 rotary octave resulting in observed 1/2 spin. Electric charge arises a result of a slight precession due to imbalance of electrostatic & magnetostatic energy characterized by a whirl no. equal to the inverse fine structure constant. Quantum mechanical description of electric & magnetic field lines at the Planck scale is depicted.
2. Electron model
Photon -
Traveling wave of
helical geometry
Electron –
Closed-loop
standing wave
of toroidal
geometry
Spin 1 boson
Spin ½ fermion
3. Pair production and annihilation
• Collision of an electron e- and a
positron e+ results in production
of two oppositely directed photons
(gamma rays).
• A high energy photon in the vicinity
of an atomic nucleus can decay into
an electron e- and a positron e+.
4. Toroidal electron model
Parameter Symbol Relation Value Units
Electric charge e = F/E = mwC =
√(aqP
2)
1.60217E-19 C
Mass me
= E/c2 = ħc/RC =
e/wC
9.10938E-31 kg
Compton wavelength lC = h/mc = h/p 2.4263E-12 m
Compton frequency fC = mc2/h = c/lC = wC
/2p
1.2355E20 Hz
Compton radius: RC = lC/2p = aa0 =
√(E/mw2)
3.8616E-13 m
Compton angular
frequency
wC = c/RC = mc2/ħ =
e/m
7.7634E20 rad/s
Zitterbewegung angular
frequency
wzbw
= 2wC 1.5527E21 rad/s
Spin angular momentum s = ½ ħ = iw =
½meRC
2w
5.2725E-5 J∙s
Bohr magneton mB = e ħ /2m 9.274E-24 J/T
Rest Energy E = ħ c/RC 8.187E-14 J
• Toroidal electron formed
by a high energy photon
topologically confined
inside the Compton radius.
• Propagation of the rotating
spin wave describes a
current loop equal to ½
of Compton radius.
• Charge path rotation
generates toroidal swept
volume.
5. Electron model cross-section
Parameter Symbol Relation Value Units
Charge radius Re = √2(lC/4p) =
Rzbw/sing
2.7305E-13 m
Classical electron
radius
R0 = (1/4pe0) e2/mec2 =
ke(e2/mec2)
2.8179E-15 m
Compton radius RC = lC/2p = aa0 3.8616E-13 m
Helix radius Rphoton = Rhelix = lC/2p =
√(E/me∙w2)
3.8616E-13 m
Mass radius (max) Rm = ħc/E = ħ/mc 3.8613E-13 m
Poloidal radius Rp = Rm/2 1.9308E-13 m
Spindle (inversion)
radius
Rs = RC – Rm 2.6983E-17 m
Toroidal radius Rt = RC(1 + a/2p) 3.8616E-13 m
Zitterbewegung radius Rzbw
= <Rm> = lC/4p =
ħ/2mc = c/2wC =
Resing
1.9308E-13 m
Chracteristic dimensions
• Toroidal circumference equals Compton wavelength lC.
• Electron Compton radius equals lC/2p
6. Electron ring configuration
• Electron depicted as a
precessing epitrochoid
charge path composed of
two orthogonal spinors of
2:1 rotary octave
• Spin ratio of Compton
angular frequency wC and
Zitterbewegung frequency
wzbw (= 2wC) corresponds
to observed spin ½.
• Electric charge arises as a
result of a slight precession
of angular frequency we/m.
7. Electron spin precession
• Electron spin precession we/m
is due primarily to imbalance
of electrostatic and magnetostatic
energy resulting in an eccentric
whirl orbit of the charge path
about the spin axis.
• Precession follows a zoom-orbit
whirl with a periapsis advance q137
that is a function of the fine
structure constant a and the
Compton radius RC.
• Synchronization occurs every a-1
revolutions. Electric charge is a
result of a slight spin precession.
• Electron mass me = e/wC.
8. Electron toroidal electric field
• Electrostatic E-field of an
electron is shown time
averaged over one rotation
period.
• For a positron, the electric
flux is directed radially
inward.
• At distances greater than
the Compton radius RC,
the electric flux distribution
for an electron at rest is
spherically symmetric
equivalent to a point charge.
9. Electron as a rotating spin wave
• The electron continuously
generates an external dipolar
spin wave in the form of a
Archimedean spiral rotating
at the Compton frequency fC
(= 1.236E20 Hz).
• The electron acts as a spinning
dipole antenna with virtual
radiation of a pair of entangled
photon wavetrains emitted
along the spin axis.
10. Electric field of an oscillating electron
• Oscillation of the electron
at frequencies less than the
Compton frequency fC in
response to excitation by an
external EM field results in
generation of observed EM
waves in resonance with
the imposed frequency.
• Entangled states represent
different points on the same
wavefront.
• Acceleration over time Dt
creates a local flux field
distortion with the farfield
flux pointing in direction of
the retarded initial starting
position.
11. Electron toroidal magnetic field
• Magnetostatic B-field of an
electron is shown time
averaged over one rotation
period.
• External magnetic field is
toroidal while the internal
field is poloidal.
• Magnetic flux is concentrated
in the central region with
increased potential magnetic
energy.
13. Electron energy storage
• During acceleration of an electron, kinetic energy is stored in the magnetic field.
• The radiation field dissipates during and subsequent to electron deceleration as
the electromagnetic field regains symmetry.
14. Electrostatic & Magnetostatic energy
vs. Velocity ratio b
Variation of electron energy as a function of velocity ratio b (= v/c)
15. Electromagnetic energy E vs. Lorentz factor g
After Bergman
The Lorentz factor g is inversely proportional to the Lorentz contraction g.
g = 1/√(1 – v2/c2) = 1/√(1 – b2) = 1/g
Electromagnetic energy of an electron as a function of Lorentz factor g.
16. Electron Compton radius RC vs. Lorentz factor g
• Absorption of energy causes electrons to contract in size increasing the
wave function curvature, kinetic energy and volumetric energy density.
Variation in electron radius as a function of Lorentz factor g (= 1/√(1 – b2)
17. Electron Compton radius RC vs. Velocity ratio b
Variation in electron radius as a function of velocity ratio b (= v/c)
• The Compton radius reflects an equilibrium between torsion and the
gravitomagnetic field. Contraction in radius occurs as spin remains
constant and torsion decreases accordingly.
18. Electron mass energy MeV/c2 vs. Velocity ratio b
Relativistic increase in electron mass energy as a function of velocity ratio b (= v/c)
19. Electron Inductance L & Capacitance C
vs. Velocity ratio b
Relativistic variation in electron inductance L and capacitance C
as a function of velocity ratio b (= v/c = Df/p = r/g)
20. Electron wave-function eigenstates in a deep
harmonic oscillator 2D potential well
• Electron represented as a
resonant spin density wave
confined in an oscillating
deep potential well in a
quantum vacuum.
• Zitterbewegung corresponds
to the motion of the center of
charge around the center of
mass with a frequency twice
the Compton frequency.
22. Electron/positron pair production
Electron Compton wavelength, zitterbewegung wavelength, and de Broglie
wavelength compared to the wavelength of an energetic photon required
for pair production of an electron e- and positron e+.
25. Electron Coulombic repulsion
• Maximum Coulomb repulsive
force between electrons occurs
at closest proximity equal to a
separation distance ar Compton’s
radius RC.
• The radiated EM wavefronts are in
the form of of Archimedean spiral
forms. Two electrons on approach
are repelled by constructive EM
wavefront interference.
• Positron spin waves rotate in a
direction opposite to electrons.
Electron and positron interaction
give rise to destructive interference.
26. Spin wave precession
• Electric charge is intrinsic to matter and exhibits similarities to topological
and vortical charge.
• Electric charge of the electron has dimensions of spin angular momentum
and appears to be the result of a slight precession equal to the inverse
fine structure constant a-1.
27. Elementary Particles of Matter
Mass, charge and spin characteristics of fundamental particles and anti-particles
28. Electric and color charge symmetry
• When there is symmetry,
there is conserved charge.
• Symmetry alone does not
provide a dynamical origin
of charge or define underlying
fundamental dimensionality
29. Electric charge vs. topological charge
• Geometrical relation of electric charge (e-, e+) to topological charge F
is illustrated in the form of a Tusci couple (2-cusp hypocycloid) which
corresponds to Special Unitary Group SU(2)
30. Spinor representation of ½ spin characteristic
Spinor examples of 720 degree rotation to return to initial orientation
involving two rotational frequencies differing by a factor of 2.
31. Quark color charge
Twist ribbon represention of quark color charge interactions
and equivalent 3-phase circuit
32. Neutral Pi-meson (Pion)
• Neutral Pion p0 consists of a
spin 0 neutral pi-meson
composed of positron-electron
pair (Sternglass model)
• The positron and electron pair
each have spin angular
momentum oriented either
parallel or anti-parallel to each
other.
• The orbital angular momentum
is equal to twice the spin
angular momentum (L = ħ). A
spin 0 neutral pion has a life-
time of ~0.83E-16 sec and can
decay into gamma rays.
• Binding energy is decreased
when magnetic moments are
parallel due to mutual repulsion.
33. Positively charged Mu-meson (Muon)
• Positive Muon m+ consists of a
positron-electron pair and orbital
positron (Sternglass model)
• Positron e+ is at rest relative to
the precessing frame KP of the
central pair system consisting
of a spin 1 neutral pi-meson p0
1.
34. Positively charged Pi-meson (Pion)
• Positive pion p+ consisting of
a positron-electron pair and
orbital positron (Sternglass
model)
• A positron e+ is in orbital
motion relative to the
precessing frame Kp of a
central pair system consisting
of a spin 1 neutral pi-meson
p0
1.
35. Negatively charged Tau-meson (Tauon)
• Negative Tauon t- consisting of
a positron-electron pair and
orbital electron (Sternglass
model).
• Electron e- is in orbital motion
relative to the precessing
frame Kp of the central pair
system consisting of a spin 1
neutral pi-meson p0
1.
36. Gluon field Fresnel zone pattern
• In-phase and out-phase gluon
field Fresnel zone patterns
with Airy diffraction cross-
section.
• Quark separation in a proton
with Compton radius RCp of
2.1031E-16 m suggest a gluon
field with a minimum distance
of ~0.33E-15 m or ~1.56 RCp.
37. Electron-positron coaxial wave interference
• Coaxial coupling of an e-e+ pair with
aligned spins as a hypothetical quark
model with a massless spin 1 field.
• A charge screen formed by depolar-
ized Planck mass dipoles prevents
Coulombic attraction.
• The electron and positron oscillate
along the spin axis alternating
between a state where charge
screening is effective at close
distance and further out where
charge screening is ineffective
thus exhibiting asymptotic freedom.
39. Proton model
• A proton composed of
electrons and positrons
equivalent to QCD quark
representation.
• Proton absorption of an
antineutrino results in
transformation into a
neutron with emission of
a positron and neutrino
in inverse beta decay.
• The bulk of the proton’s mass
is due to kinetic energy of
component quarks.
40. Gravitational force on electron
In free fall, the acceleration of gravity and associated frequency
difference is equal to zero
41. Electric flux lines
• Faraday electric flux lines fE are
postulated to be an effect of
synchronous alignment of
positive and negative Planck mass
dipoles of the quantum vacuum.
• Alignment results in tension along
the flux lines due to attraction of
positive and negative masses and
compression perpendicular to the
flux lines.
• Electric field intensity E is a function
of the product of the Planck electric
dipole moment m0P and the Planck
mass mP.
42. Magnetic flux lines
• Magnetic flux lines fB are interpreted
as vortex filaments of rotating Planck
mass dipoles. Magnetic field lines
are the result of spin axis alignment
of multiple dipoles.
• Magnetic field intensity H is a measure
of the vortex strength of entrained
dipoles
• Photons and electrons are spin waves
in the Planck quantum vacuum
composed of positive and negative
Planck masses. Photons are traveling
waves. Electrons are standing waves.
45. Abstract
A comprehensive description of the nature of light, electricity and gravity is provided in
terms of quantum wave mechanics. Detailed models include the photon as a travelling
electromagnetic wave and the electron as a closed loop standing wave formed by a
confined photon. An electron is modeled as a torus generated by a spinning Hopf link
as a result of an imbalance of electrostatic and magnetostatic energy. Electric charge is a
manifestation of a slight precession characterized by the fine structure constant. The
physical vacuum as a polarizable medium enables wave propagation and appears
ultimately to be quantized at the Planck scale. Standing wave transformations for objects
in motion are reviewed and Lorentz Doppler effects compared. The mechanism for
generation De Broglie matter waves for objects in motion is depicted including the inverse
effect of induced motion of an object by synthesis of contracted moving standing waves.
Gravity is viewed as a frequency synchronization interaction between coupled mass
oscillators. The acceleration of gravity is described by a spectral energy density gradient.
Antigravity corresponds an inversion of the naturally occuring energy density gradient.
Gravitons are shown to be phase conjugate photons. The metric of curved spacetime
corresponds to the electromagnetic wave front interference node metric. Hence, the
gravitational field becomes quantized.
Quantum Wave Mechanics
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