S.S.Vilkovskii
E-mail:metglas@i.ua
20 July 2015
Abstract
Existence of zero rotations for point particles with nonzero rest mass in the field of
their quantum size is supposed. The assumption about coordinated interaction of these zero
rotations of the above particles leads to possibility of the description of their wave
properties.
Keywords: wave function, energy, mass, wavelength, frequency, spin, charge.
Now in the field of physics of elementary particles essential successes reached [1-7].
At the same time, in comparison with research of the phenomena occur in atom, internal
properties of elementary particles was not enough studied [3-5]. The equations describing
the processes occur in elementary particles, were obtained on the basis of the gauge theory
of Yang - Mills [1,5]. As the equations of quantum electrodynamics for the atom,
continuation of which they are, the equations Yang - Mills theory involves obtaining
sufficiently accurate and complete picture of the description of the phenomena. Due to the
importance of the solution of the matter the main efforts of modern physics are
concentrated on it. However, from the viewpoint of achieving a successful structure
description of atom, in which case it seems desirable to have a relatively simple theory
describing the structure of the elementary particle, which is, for example, the Bohr theory
for the atom.
Therefore, since it is impossible at present to fully solve the equations of the YangMills
for describing the internal behavior of elementary particles [1,5], parallel to modern
researches there can be useful a development of rather simple and clear model of the
description of properties of the elementary particles. For this purpose we will try to
describe first of all behavior of an electron within its quantum size simple model, like of
Bohr model for atom.
During creation of Bohr model of atom there were no such concepts, as spin of
particles, physical vacuum. Let's try to consider these phenomena.
In the theory of Bohr hydrogen atom in the description of the quantum linear
oscillator which is not considering spin: Е = hwn to the generalized coordinates and
impulses transition is made [8]. In this case for angular momentum of an electron
expression follows: M n = h [8]. Similarly, at the description of the linear oscillator
considering zero fluctuations: E n = + hw( 1/ 2). let's pass to the generalized coordinates
and impulses. Then we will pass to a special case of a rotary motion. In this case the
equation for angular momentum of an electron in atom of hydrogen takes the form:
2
M n = + h( 1/ 2) . Here the composed h / 2 corresponds a particle spin. If to believe that the
picture of a rotational motion of a particle on zero and other levels is similar to a picture of
fluctuations of the linear oscillator at appropriate levels, then, proceeding from the point
size of an electron, is advisable the following model to consider.
Let's assume that for the point particle comprising all mass and a charge of an electron
there are zero rotations on some own (internal) orbit similar to zero fluctuations of the
oscillator. Fluctuations of physical vacuum, type of the fluctuations forcing to fluctuate,
shiver according to Schrödinger an electron [9] which, besides, in the considered model
nearby electrons can exchange in coordination among themselves can be the cause holding
on average an electron in own orbit. Rotation on a circle of a point is equivalent to its
simultaneous participation in two perpendicular fluctuations displaced on a phase [8,10].
Let's note, the fluctuations of vacuum acting on the charged electron, basically should
represent photons, electromagnetic waves under the influence of which charged particle
has to make the movements close to the rotary. This model is similar to the idea of de
Broglie on the oscillators at each point in the universe tuned to the frequency of the
electron under consideration [11]. There nearby particles together tunes to frequency each
other by exchanging electromagnetic waves.
Inorder that in the considered model the spin corresponds to the experience
necessary to perform the equality / 2 mvr
r = h , where r - the radius of the orbit, r
v -
the velocity of the particle. Limiting the scope of rotation of Compton wavelength leads in
this case to the value of the particle velocity on the inner orbit comparable to the speed of
light and, accordingly, to a significant magnitude of its kinetic energy of rotation, which is
missing in the formula for the rest energy. This leads to the possibility of the assumption
that virtually the rest mass of the particle, which is on its own orbit, at rest ( 0 r
v = ), similar
to the gauge theories in the absence of the Higgs field is zero [1,5].
A particle on its own orbit may be called subparticle of elementary particle, the
parameters of which we are seeing. In this case, we can put that the subparticle mass
increases with its speed and particularly rapidly when approaching the speed of light
so, that when r
v c = -e , c>>e for an electron at rest, it becomes equal to the
experimentally observed its rest mass m0 :
00 0 ( ) /2 m v r m c r m cr r
= -» » e h . (1)
The assumption that the mechanical and magnetic moment of the electron is related
to its local motion was proposed independently by many authors [12-14]. Significant
interest is the result obtained from the Schrödinger found, confirmed by experiment, the
properties of the electron shake - Zitterbewegung [9,15]. However, theoretical
investigations in this case is not always easy due to the complexity of calculations, limits
of a class of the considered particles. Therefore, we proceed from a simple model, which
3
uses a small number of assumptions. From the equation (1) for the frequency of rotation of
a particle in an internal orbit of an electron we receive:
2
0 0 w% = 2m c / h , (2)
which coincides with the equation of equality of energy of rest of an electron and positron
with energy of the photon turning into these particles near a massive kernel:
2
0 0 hw% = 2m c , absorbing the momentum (but not energy) of the photon [8,10].
Equality (multiplicity) of wavelength of a subparticle 0
l
%
and length of its orbit r
l ,
which is typical for nuclear orbits, also from (2) follows:
0 00 2 / 2 /2 2 r
l pw p p == = = = сT с mc r l % % h . (3)
Ifthe subjects surrounding an electron are in rest, which have to make the main
contribution to his behavior, the average speed of their electrons is equal to zero. Let's
consider for simplicity that the centers of own orbits of these electrons are in a motionless
state. If to assume that trembling (in our model - rotation) electrons, thanks to interaction,
becomes coordinated, their movement has to create a standing wave of frequency w0
% which
will hold a subparticle of the motionless electron considered by us in own orbit. In this
case, the existence of (3) can be explained as follows. In order that interaction of an
electron (at rest) with the electromagnetic wave having the frequency determined by a
formula (2) had resonant character, this wave must to pass through a certain point of its
internal orbit of distance equal to length of its wave over the period of one complete
revolution of a particle on an orbit. It does not contradict that charged particles exchange
photons many of which merges in a continuous electromagnetic wave.
Thanks to longitudinal and transverse Doppler effect the picture of interaction of
electromagnetic waves and a moving electron significantly becomes complicated. There is
a question: what frequency has to correspond to a moving electron? The observer, with
respect to which the electron moves as a result of delay of time must accept its frequency
equal 1 0
2 2 w w % % = -1 / v c [8]. But this is the frequency with which the moving electron
impact on the still surroundings.
On the other hand, by virtue of the invariance of moving systems, according to the
equation (2) the moving electron must comply with frequency:
2 2 22 22
0 0 w w % % = = - =- 2 / (2 / 1 / ) / / 1 / . mc m c v c v c h h (4)
The frequency determined by the equation (4) has to correspond also to the frequency
of the photon which is given rise an electron - in positron couple each of which particles
moves with a speed v . To maintain the momentum it is necessary that it occurred near the
massive nucleus having the same speed. Thus, the wave of this frequency can really
interact with an wave of environment of electron. Defined by the equation (4) the
frequency of the wave w% , we will assume the corresponding to a moving electron.
Proceeding from it, we can present behavior of an electron in the form of superposition of
two flat waves with frequencies w 0
% and w% identical, because of equal interaction,
4
amplitude. Note, at values v c << the results obtained below for the wavelength of an
electron does not depend on which of the frequencies w% or w1
% has been taken, since
2 2
0 10 0 ww ww w %% %% % - »- » v c / 2 .
In the case under consideration discrepancy of frequencies w 0
% and w% has to
influence behavior of a moving electron creating areas in a bigger and smaller measure
favoring to location of an electron. Rotation point in a circular orbit is a superposition of
two perpendicular linear waves. The linear combination of two harmonic oscillations will
be the decision for the oscillator with the harmonious compelling force. As we already
noted, under the influence of a flat electromagnetic wave charged particle makes the
movements close to the rotary. Based on this we can represent the behavior of the electron
by the superposition of two plane waves of frequencies w 0
% and w% of the same, due to
the equal interaction, amplitude. The exact value of the received sum of fluctuations of two
waves, and their approximate value for the case w w» 0
% % , (v c << ) can be represented as [8]:
,(5)
where 0 w ww w = - =D ( )/2 /2 %% % ,
0
k kk k = - =D ( )/2 /2 %% % . For the waves with close frequencies
corresponding to the last equality, the first oscillating multiplier acts as amplitude to
process of high frequency w 0
% .
In particular, for photons the square of amplitude defines intensity, and the last - the
probability of finding a particle in space. Quite logically in this model to assume that amplitude
of fluctuations of high frequency plays a similar role for an electron. Using (4), (5) write the
approximate value for frequency w at v c << . As a result we receive the equation de Broglie
connecting the frequency and energy for this case [10]:
2
00 k w w ww =D = - » = % %% /2 ( )/2 /2 E / m v h h . (6)
For the module of a wave vector k of an electromagnetic wave of length lw it is had:
2 2 23 2
00 0 0 2 / / 2 ( ) / 2 (1 / 1 / 1) / 2 / 4 / 2 w
k k k k v c c v c mv c = =D = - = - - » = pl w w % %% % % h . (7)
The module of the amplitude, which determines the probability of finding the electron in
space, is undergoing similar changes at half-wave length: / 2 w
l . Considering it and using
(7), believing that the relation of speeds of an electromagnetic wave and an electron
equally to c v/ , we come to de Broil's equation for wavelength [10,11]:
0
( / 2) / ( / ) [(2 / ) / 2] / ( / ) / w
ll p == » c v k c v h mv . (8)
In this case the resonance of the second order - the repeating sequence of not resonant
interaction of a wave and an electron, in particular, will exist in atom orbits which length is
multiple to electron wavelength.
0
0 0 00
cos( ) cos( ) ( , ) ( , ) 2 cos( / 2
/ 2) cos[( ) / 2 ( ) / 2] 2 cos( ) cos( )
o
u a t k x a t kx t x t x a t
k x t k k x a t kx t k x
w w yj w
ww w w
=× - +× - = = × D× -
-D × + - + » - -
% % %% %
% %% % %% %
5
Asamplitude of high-frequency fluctuations as we see from the equation (5),
represents a wave, the decision for it, as usual for waves, has to be from the differential
equation connecting, eventually, values of frequency and wavelength. Knowing the
relationship between the kinetic energy Ek
and an impuls p, defining w and l values,
we can receive this equation. Let's enter the operator g
)
the translating sine into a cosine,
and vice versa, which role imaginary unit can play, mutually converts the real and
imaginary (sine and cosine) of the complex value. Doing replacement E E k k ® + E U =
leading to an additive in the equation of Uy where Uxyz (, ,) - potential energy, we come
to Schrödinger's equation [9] allowing to find the solution in a complex look:
2
i t mU h h ( / ) ( /2 ) ¶ ¶ =- D + y yy .
When obtaining wavelength of an electron (8) we, in particular, for the above
reasons, reduced length of the corresponding electromagnetic wave entering in (7), twice.
Similarly, we had to increase the frequency (6) twice: 2 / w = × Ek
h . In Schrödinger's
equation at the operator for energy in this case that equality remained, there has to be then
a multiplier 1/2 which is reduced with a similar multiplier at the operator of impulses. The
phase speed determined by a formula v k =w/ in this case is equal to particle speed.
However according to (5) it will be group speed [8]: vk k = =D D w w / / % % . The phase speed
determined from (5) in the case under consideration is equal to velocity of light [8]:
ф 0 0 v k = = w / с
% % that is the expected result since in this case the original image of the
resulting waves are electromagnetic waves.
At the same time, as on experience de Broil's frequency, in difference from
wavelength, does not give in to measurement [10], practically this change of frequency in
the considered model nothing will change. In this regard it is possible to believe that in a
formula for frequency in this approach nothing is to be changed. Only in this case at
substitution of energy and an impulse (mass and the speed of a particle) in expression of
w = kv (according to the aforesaid equivalent D =D w kv% % ) equality will not be carried
out.
In this model the subparticle in an orbit moves with almost light speed and in lack of
the movement possesses zero mass. In this regard, the issue of the gyromagnetic ratio of
the electron in this case requires a special study.
The wave function, acting as a low-frequency multiplier of the equation (5), a good
description of the area above the Compton wavelength, the external behavior of microparticles
in a large range of its properties, regardless of the second multiplier. It is possible
to assume that high-frequency function has to describe behavior of an electron in scales of
its Compton wavelength, namely, electron subparticles, and also in considerable area
regardless of the first multiplier.
6
The differential equation for high-frequency function j of equality (5) in the
assumption that particle speed in own orbit is equal r
v c » , and the speed of the center of
an orbit v c << , due to the fact that its solution is primarily a function must satisfy
0 0 j w ( , ) cos( ) tx t kx = - % % it is as follows:
2 2 2 (1/ ( , ) / ( ) ( , ) 0 c tt t )[¶ ¶ -D = j j r ] ρ
r r
.
The muon is a particle similar to an electron, the size of mass differing from it [1,4].
Therefore for it the conclusions similar the aforesaid are apparently fair. In this model
wave properties of neutral particles can be explained to that they consist of charged
particles.
All elementary particle possess of wave properties. It can be assumed that the
particles that make up the elementary particles with not the zero rest mass, make on the
internal orbits the local movement similar to the movement of an electron in the field of the
quantum size. Existence of these periodic processes in elementary particles does not
contradict Standard model in which the elementary particle consists of dot particles [1,4].
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