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Journal of High Energy Physics, Gravitation and Cosmology, 2020, 6, 567578
https://www.scirp.org/journal/jhepgc
ISSN Online: 23804335
ISSN Print: 23804327
DOI:
10.4236/jhepgc.2020.64038 Aug. 13, 2020 567 Journal of High Energy Physics, G
ravitation and Cosmology
A New Theory on Electron
WaveParticle Duality
Nader Butto
Dgania, Petah Tikva, Israel
Abstract
A theory employing the vortex shape of the electron was presented to resolve
the enigma of the waveparticle duality. Conventions such as “particle”
and
“wave” were used to describe the behavior of quantum objects such as elec
trons. A superfluid vacuum formed the base to describe the basic vortex struc
ture and properties of the electron, whereas various formulations der
ived from
hydrodynamic laws described the electron vortex circumference, radius, an
gular velocity and angular frequency, angular momentum (sp
in) and magnetic
momentum. A vortex electron fully explained the associations between mo
mentum and wave, and hydrodynamic laws were essential in deriving th
e
energy and angular frequency of the electron. In general, an electron traveling
in space possesses internal and external motions. To derive the angular fre
quency of its internal motion, the Compton wavelength was used to represent
the length of one cycle of the internal motion that is equal to the
circumference
of the electron vortex. The angular frequency of the electron vortex was cal
culated to obtain the same value according to Planck’s theory. A traveling vor
tex electron has internal and external motions that create a three
dimensional
helix trajectory. The magnitude of the instantaneous velocity of the electron is
the resultant of its internal and external velocities, being equal to the internal
velocity reduced by the Lorentz factor (whose essence is presented in a de
tailed formulation). The wavelength of the helix trajectory represents the dis
tance traveled by a particle along its axis during one period of revolutio
n
around the axis, resulting in
the same de Broglie wavelength that corresponds
to the h
elix pitch of the helix. Mathematical formulations were presented to
demonstrate the relation between the energy of the vortex and its angular fre
quency and de Broglie’s wavelength; furthermore, Compton’s and de Brog
lie’s wavelengths were also differentiated.
Keywords
Superfluid Vortex, Compton Wavelength, Electron Momentum,
Broglie Wavelength, Lorentz Factor, Angular Frequency
How to cite this paper:
Butto, N. (2020)
A
New Theory on Electron Wave

Particle
Duality
.
Journal of High Energy Physics
,
G
ravitation and Cosmology
,
6
, 567578.
https://doi.org/10.4236/jhepgc.2020.64038
Received:
July 3, 2020
Accepted:
August 10, 2020
Published:
August 13, 2020
Copyright
© 2020 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
N. Butto
DOI:
10.4236/jhepgc.2020.64038 568 Journal of High Energy Physics, G
ravitation and Cosmology
1. Introduction
Waveparticle duality is one of the fundamental principles of quantum mechan
ics which is directly linked to many of its mysteries. According to this theory,
light and matter exhibit properties of both waves and particles. The wave cha
racteristic of an electron implicates many of its particle behavior; however, the
twosided nature of electron duality does not allow for it to be observable as a
particle and as a wave. Therefore, this duality addresses the inadequacy of con
ventional concepts, such as “particle” and “wave,” to meaningfully describe the
behavior of quantum objects.
Light behaves as a wave as it travels through space. However, like a tiny particle,
it gives up its energy the moment it reaches its destination. Such a duality beha
vior is not confined to light; in fact, numerous experiments have supported its
existence in photons [1], electrons [2] [3] [4] [5], neutrons [6], atoms and di
mers [7], small van der Waals clusters [8] and more recently, C60 fullerenes [9]
[10].
The idea of duality is based on a debate on the nature of light and matter dat
ing back to the 17th century, when Christiaan Huygens and Isaac Newton pro
posed competing theories of light. However, the dilemma between the wave and
particle aspects of the various components of matter and radiation arose after
Planck’s discovery of the quantum of action.
Planck’s quantum hypothesis states that a quantum of energy is related to the
frequency by the equation
E
=
hν
; at the same time, the energy is emitted in little
packets of energy called quanta, instead of a continuous emission. However, in
quantum physics, the wave is not defined—it is a sort of “clouds” of electron or
bitals around an atom which are not physical
things
but representations of prob
abilities.
In 1923, Louis de Broglie proposed a hypothesis stating that electrons and other
discrete bits of matter, then conceived only as material particles, hold properties
of waves. Within a few years, de Broglie’s hypothesis was tested by a doubleslit
experiment which demonstrated that the electron stream acts like a light, prov
ing de Broglie correct. In 1928, Neils Bohr announced an understanding of the
complementary relation between the wave aspects and the particle aspects of the
same phenomenon in what is known as the complementarity principle, reflect
ing his argument of the nonnecessity to use the words “wave” and “particle”
at the same time and the prohibition of questions such as “What is light?” and
“What is an electron?” According to Bohr, we must confine ourselves into ask
ing how things are observed to behave under a specified set of circumstances, a
solution that denies the possibility of saying anything meaningful about a world
that is not being observed, and at the same time, a limitation that was completely
unacceptable to many physicists, including Einstein.
On the basis of experimental evidence, Einstein made an independent propos
al of a sort of revival of the corpuscular theory of light, including the concept of
the quantum of action in the form of energy or light quanta. In 1905 he first
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showed that light, then considered a form of electromagnetic waves, must also be
thought of as particlelike, localized in packets of discrete energy. In his theory
of photoelectric effect, he posited that when light is shown on certain objects,
electrons will be released; if a photon of an energy greater than that of the elec
tron hits a solid, that electron will be emitted. Einstein’s theory of photoelec
tric effect contributed largely to de Broglie’s theory and was a proof that waves
and particles could overlap. Moreover, Compton’s observations of his Comp
ton effect (1922) could be explained only if light has a waveparticle duality. Lat
er, in 1927, the wave nature of electrons was experimentally established by Clin
ton Davisson and Lester Germer and independently by George Paget Thomson
[11].
The problem of waveparticle duality remains unsolved until this day. In quan
tum mechanics, an electron may appear to us as a particle or as a wave depend
ing on how we “look” at it, and mainly because it is considered as a cloud of
structureless point. There is no theory at present that describes the shape or
predicts the size of an electron, its mass, or its charge, or that quantifies the par
ticle in a meaningful calculation or the relationship between the electron as a
particle and electron as a wave.
To provide a suitable interpretation and resolve this ostensible paradox, a new
theory is presented to demonstrate that the waveparticle duality can be ex
plained by considering the wave as a trajectory of a rotating electron that travels
in space. Here, the vacuum is considered a superfluid and the electron an irrota
tional vortex made up of this fluid. Applying the hydrodynamic laws, a formula
tion to measure the circumference of the vortex gives the same value of Comp
ton wavelength, whereas an equation to measure the angular frequency of the
vortex demonstrates that it has the same wave frequency as the electron accord
ing to Planck’s theory. Thus, the local physical reality of the electron as a vortex
determines the results of local measurements.
2. The Electron Structure
According to current experiments and theories, the electron is a structureless,
pointlike object. It is not made of anything else and its entire mass is concen
trated in its extensionless center. Nonetheless, the electron displays properties
that normally result from an extended structure, namely, angular momentum
(spin), magnetic moment, and some sort of an internal oscillation. In 1928,
when Paul Dirac presented the wave function of the electron in the Dirac equa
tion, it became clearer that there must not only be an internal oscillation but also
some internal motion at the speed of light.
Another article proposed that the vortex shape of the electron is a condensa
tion of the vacuum which provides the correct relationship between the parame
ters of the electron—mass, density volume, time, constant angular momentum
(spin), electric charge, and magnetic moment. A brief description of the electron
structure is shown in Figure 1.
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Figure 1. Artistic representation of the magnetic field around the electron vortex.
In the electron vortex, virtual photons flow in a spiraling downward motion in
the funnel due to the pressure gradient normal to the vortex center and that acts
along the central axis of the vortex spin. It follows down moving up around the
vortex and returns from the upper side of the central axis in the mouth of the
vortex generating the magnetic momentum. The vertical magnetic pressure gra
dient is normal to the horizontal electrostatic pressure gradient created by the
vortex and acts along the central axis of the spin. The threedimensional mag
netic field has a negative pole that sucks the energy from the vacuum and a posi
tive pole that pulls the energy from the positive magnetic pole. Furthermore, the
rotating vortex exercises a rotation effect on the magnetic field, resulting in two
movements 90˚ on from the other.
The acting electrostatic and magnetic forces become interlinked in selfbalancing
feedback loops that provide great stability to the vortex structure shape of the
particle as a whole. These are the two forces that generate movement and create
currents. The currents are the avenues of movement made up of Higgs particles
and surrounding these are spaces or crevices through which currents of lesser den
sity comprising Higgs particles flow.
Hydrodynamic laws are applied to describe the behavior of a single electron in
space, and to describe its dynamics as a vortex particle in connection with de
Broglie’s particle–wave hypothesis and Planck’s theory, its energy and wave fre
quency, vortex angular frequency and finally, the electromagnetic wave charac
teristics and free electron travel in space.
3. The Vortex Model and De Broglie’s Hypothesis
The essence and physical relationship between a particle and a wave remains an
unresolved problem in physics. Louis de Broglie developed a hypothesis [12] re
lating the dual wave and particle behavior that can be applied to electrons.
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In his hypothesis, he first used Einstein’s equation of matter and energy
2
,E mc=
(1)
where
E
represents the energy of matter,
m
its mass, and
c
the speed of light.
He then proceeded with Planck’s theory, which states that every quantum of a
wave has a discrete amount of energy given by Planck’s equation
,E hf=
(2)
where
E
represents energy,
h
Plank’s constant (6.62607 × 10−34 J∙s) and ƒ
fre
quency.
According to him, the energies produced in both equations should be the
same; thus,
2
,mc hf=
(3)
Substituting
c
/
λ
for ƒ, de Broglie arrived at a final expression relating the wa
velength
λ
and particle momentum
mc
with speed of light:
2
mc hc
λ
=
, (4)
where two relationships can be derived:
h mc
λ
=
(5)
and
h mc
λ
=
. (6)
In this de Broglie wavelength formula relating
λ
and
mc
of a wave/particle
[13], supposed that the mass of the electron is
me
= 9.109 × 10−31 kg,
h
= 6.63 ×
10−34 and
c
= 2.998 × 108, then the derived wavelength will be 2.42779 × 10−12 m,
which corresponds to the Compton wavelength.
However, this relation does not explain the mechanism that connects the wave
with the particle. By contrast, the electron as a vortex explains fully the associa
tion between momentum and wave. The rate of rotation of the fluid in the irro
tational vortex is greatest at the center and decreases progressively with distance
from the center until no gradient pressure remains on the boundaries of the
vortex where the flow is laminar and the friction null. However, the speed of ro
tation in every point of the vortex is the speed of light
c
.
In hydrodynamics, the velocity of the fluid element instantaneously passing
through a given point in space in a vortex with radius
r
is constant in time, thus,
the circulation or the vorticity Γ is defined by
2,rcΓ= π
(7)
Because Γ
m
is a conserved momentum, 2π
rcm
is constant, which corresponds
to the Planck constant.
Accordingly,
2,
ee
rcΓ=π
(8)
ee
hm= Γ
(9)
and
constant
ee
hmΓ= =
. (10)
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Supposed the electron mass is 9.109 × 10−28 g and Planck’s constant is 6.6262 ×
10−27 erg∙s, then
7.274
e
hm =
,
From these values, the radius of the electron can be calculated as
13 20
3.863509856 10 m
2
e
e
rc
−
Γ
π×= =
.
If
2,
rcm h mc
λ
π==
(11)
then
2r
λ
π=
(12)
Knowing that the radius of the vortex is the Compton radius, 3.86 × 10−13 m,
the circumference is 2π
r
= 2.42408 × 10−12 m, which agrees well with the
CODATA 2014 [14] value for the Compton wavelength of the electron
2.4263102367(11) × 10−12 m [15]. Therefore, we can conclude that the Compton
wavelength is produced by one rotation cycle of the electron vortex.
4. Vortex Angular Frequency and Electron Frequency
According to Planck’s theory, photons of frequency ƒ produces an energy
.
e
E hf=
(13)
The frequency of the electron is equivalent to the frequency of the photon
having the same energy thus,
e
f Eh=
, (14)
where ƒ
e
=
E/h
= 0.511 MeV/
h
= 8.1866 × 10
−
7 ergs/
h
. Therefore, ƒ
e
= 1.2355 ×
1020 cycles/s.
Another way to describe the vortex is by its angular velocity. Being material
particles, vortices share similar mechanical properties with waves. All stream
lines of the vortex tubes rotate around their axis and the vortices have a mea
surable rotating angular frequency that can be described in units of time (rota
tions per second).
Because the rotational velocity
ω
of the vortex is
2,
cf
r
ω
= = π
(15)
the frequency ƒ can be expressed as
.
2c
fr
=π
(16)
Assuming
v
=
c
= 2.889 × 108 and 2π
r
= 2.42408 × 10−12 m, the frequency is
2.998 × 108
/
2.42408 × 10−12 = 1.2367 × 1020 cycles/s, which coincides with the
electron frequency derived from Planck’s equation.
5. Relation between the Energy of the Vortex and Its
Frequency
In hydrodynamics, the force
F
that moves the vortex is directly related to the
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pressure that creates the vortex (or the socalled dynamic pressure
Pd
) and the
area
A
:
d
F PA=
. (17)
The
Pd
representing the fluid kinetic energy is given by
2
1
2
d
Pv
ρ
=
, (18)
where
ρ
and
v
=
c
represents the density and velocity of the fluid.
Therefore, the internal force of the vortex can be expressed by
2
1
2
F cA
ρ
=
. (19)
The area of the vortex is approximately a circle, and its radius when stretched
will cause the vortex radius to double in size; therefore,
A
= 2π
r
2. Correspon
dingly,
22
F cr
ρ
= π
. (20)
If the righthand side of the equation is multiplied and divided by a time
t
,
then
2
ct r c
Ft
ρ
π
=
. (21)
The quantity (
ct)
is equivalent to the distance
L
,
L
π
r
2 is equivalent to the vo
lume
Q
,
ρQ
is equivalent to the mass
m
, and 1/
t
is equivalent to the frequency
ƒ
.
Therefore,
F mcf=
. (22)
The energy of the rotating electron around its axis is
E
= force × distance. As
the electron rotates a distance of 2π
r
in one cycle,
2E rcmf= π
, (23)
which gives the rest energy of the vortex electron. Furthermore, the frequency
can be derived as
.
2E
frcm
=π
(24)
Since
2,rcm hπ=
(25)
then
.
E
fh
=
(26)
In this case, the frequency indicates the number of passages of a single elec
tromagnetic wave within one second of time. Planck’s constant is the energy
found within one cycle.
6. Flow of Free Electron in Space
De Broglie’s hypothesis also states that each portion of energy with a rest mass
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m
0 may associate a periodic phenomenon of the frequency
ƒ
0, such that
hfƒ
0 =
m
0
c
2, where
ƒ
0 is the rest mass frequency [15].
He used the special theory of relativity to find that, in the frame of the observ
er of the electron energy packet moving with a velocity
v
, its frequency was ap
parently reduced to
1
22
02
1,
v
ff c
= −
(27)
where
1
22
2
1v
c
−
is the Lorentz factor, a factor by which time, length and
relativistic mass change for an object while it moves. The expression appears in
several equations of special relativity and arises in derivations of the Lorentz
transformations. However, the mechanism from which this factor was derived is
not known.
However, if
( )
12
2 22
0
1mc hf v c= −
,
0
fc
λ
=
and
( )
( )
12
2 22
1
mc hc v c
λ
= −
, (28)
then
( )
( )
12
22
1h mc v c h p
λ
= −=
, (29)
also known as the “de Broglie wavelength” for a particle moving at
v
to a fixed
observer [16].
However, the mechanism that associates the mass with frequency is not ex
plained and the origin of Lorentz factor is not clear.
The electron as a vortex in free space will never be still; it will move conti
nuously around its axis. In order to complete one cycle, the rotating electron
vortex needs to complete a distance of 2π
r
which is equivalent to
ct
, with
t
being
the time needed to complete one cycle and
c
the speed of light.
Consequently,
2r ctπ =
if
1f
t=
.
The frequency of the rest electron is
0
2c
fr
=π
, (30)
with the electron vortex circumference 2π
r
corresponding to the Compton wa
velength
λ
.
Moreover, because
2rcm h mc
λ
= =π
, (31)
.
h
mc
λ
=
(32)
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Every microparticle is endowed with two types of motion, an internal motion
and an external motion. A vortex electron travelling in space has internal and
external velocities. The internal velocity
v
⊥ is in a plane normal to the external
parallel velocity
v
//. The magnitude of the instantaneous velocity
v
0 of the elec
tron is the resultant of its internal and external velocities, which is always equal
to “
c
,” the velocity of light in free space.
As shown in Figure 2, due to Magnus effect, the vortex deviates an angle α
and a vortex drift on a threedimensional helix trajectory.
The helix can be considered as a threedimensional wave whose pitch is
equivalent to the “wavelength”. The pitch of the helix is the distance travelled by
the particle around the axis in one period of revolution.
The angular velocity is a vector whose magnitude measures the rate at which
the radius sweeps out an angle and whose direction shows the principal axis of
rotation.
Thus, the velocity
v
0 of the electron of mass
m
with respect to the origin O is
separated with trigonometric functions and can be resolved into components
parallel to (
v
//) and perpendicular to (
v
⊥) the radius vector
r
. Here,
v
// is the lon
gitudinal velocity while
v
⊥ is the rotational velocity of the electron around its
axis as presented in Figure 3.
Figure 2. Threedimensional helix electron trajectory connecting the particle with the
wave.
Figure 3. In threedimensional space, the vector
r
of a moving particle is the radius vec
tor from the origin. The angular velocity is a vector whose magnitude measures the rate at
which the radius sweeps out an angle and whose direction shows the principal axis of ro
tation and is given by the righthand rule.
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The internal rotation velocity
v
⊥ of the vortex is determined as
( )
12
22
0
v vv
⊥= −
. (33)
The angle
α
can be determined by the trigonometric function
( ) ( )
( )
( )
12
12 12
22 22 2 22
00 0 0 0 0
sin 1vv vv v vvv vv
α
⊥
==−=− =−
. (34)
Assuming
v
0 =
c
and
v
// =
v
,
( )
12
22
sin 1 vc
α
= −
, (35)
which is the origin of the Lorentz factor.
Although the electron’s resultant velocity with respect to an observer is given
by
c
, its internal and external velocities would be different according to this ob
server since there is an additional velocity
v
between the frames.
The changes in the internal velocity
c
due to the external motion of the elec
tron is very small for low external velocities. However, external velocities very
close to “
c
” bring about significant changes in the internal velocities. For exam
ple, the electron vortex “at rest” with respect to an observer
S
has an internal ve
locity “
c
,” but the same electron when observed from a frame
S'
moving with a
relative velocity
v
with respect to
S
will have an internal velocity
( )
12
22
1c vc−
which is less than “
c
”.
Moreover, the angular momentum can be considered as a rotational analogy
of the linear momentum
p
which is directly proportional to mass
m
and linear
speed
v
0:
0
p mv=
. (36)
The electron’s angular momentum depends on its instantaneous velocity
v
0
which in turn depends on the internal motion reduced by Lorentz factor as de
scribed in
( )
12
22
0sin sin 1v v c c vc
αα
⊥
= = = −
. (39)
From the relation
λ
=
h
/
p
= (
h
/
mv
0) with the angular momentum, the wave
length
λ
of the travelling electron can be expressed in terms of
( )
( )
( )
12
22
0
1h mv h mc v c
λ
= = −
. (37)
Considering the mass of the electron travelling at 1 × 105 meters per second,
λ
~7.3 × 10−9 m, or approximately the radial size of an atom. This proves that the
de Broglie wavelength is completely different from the Compton wavelength, as
illustrated in Figure 4.
7. Conclusions
The electron waveparticle properties can be accurately described using classical
laws of the Newtonian mechanics that can exhibit particle and wave properties
simultaneously. The electron is treated as a superfluid irrotational vortex, thus,
its local physical reality as a vortex determines the results of local measurements.
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Figure 4. Compton wavelength is generated by one cycle or rotation around the electron
vortex axis while de Broglie wavelength is the pitch of the helix generated by an electron
travelling in space.
Hydrodynamic laws are applied to study the behavior of such a vortex which has
internal rotation, angular momentum, angular speed, and angular frequency.
The electron is endowed an internal motion and an external motion, the former
being a circular motion in a plane with a radius characteristic of the mass of the
electron and produces a Compton wavelength equal to the circumference of the
electron vortex. With the combination of the internal and external motions caused
by the Magnus effect, the electron drifts on a threedimensional wave attaining a
pitch equivalent to the de Broglie wavelength. The internal rotational motion of
the electron travelling in space at a relatively low speed attains the speed of light
speed, but is reduced by the Lorentz factor with the electron travelling at high
speeds. The magnitude of the instantaneous velocity of the particle is the resul
tant of its internal and external velocities, and is equal to the internal velocity
reduced by the Lorentz factor.
Therefore, the de Broglie wavelength represents the wavelength of a three
dimensional helix derived from the electron momentum equation whereas the
Compton wavelength represents the rotation of one cycle of the internal motion
derived from the circumference vortex equation. In this regard, the electron fre
quency derived from Planck’s equation represents the angular frequency of the
rotating electron vortex and has a fixed value while de Broglie’s wavelength
represents the helix wave frequency which is variable and dependent on the
electron’s external velocity.
By postulating an electron vortex structure with internal rotation motion, we
developed a simple “physical” theory combining elegantly the mechanics with
the mathematics that might improve our understanding of the mysteries of na
ture,
i
.
e
., “super conductivity,” “quantum mechanical tunneling,” and that can
explain doubleslit experiments. Future experimental studies are needed to con
firm the vortex structure of the electron. The electron structure as well as the
origin of its mass, energy, and electric charge will be discussed separately in dif
ferent papers.
Acknowledgements
The author would like to thank Enago (http://www.enago.com) for the English
language review.
This research did not receive any specific grant from funding agencies in the
public, commercial, or notforprofit sectors.
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Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this pa
per.
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