Journal of Modern Physics, 2012, 3, 1895-1906
http://dx.doi.org/10.4236/jmp.2012.312239 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Application of Generalized Non-Local Quantum
Hydrodynamics to the Calculation of the Charge
Inner Structures for Proton and Electro n
Boris V. Alexeev
Moscow University of Fine Chemical Technologies Prospekt Vernadskogo, Moscow, Russia
Email: Boris.Vlad.Alexeev@gmail.com
Received October 2, 2012; revised November 3, 2012; accepted November 13, 2012
ABSTRACT
The proton and electron charge inner structures are considered in the frame of the non-local quantum hydrodynamics
based on the non-local physical description. From calculations follow that proton and electron can be considered like
charged balls (shortly CB model) which charges are concentrated mainly in the shell of these balls. The proton-electron
collision in the frame of CB-model should be considered as collision of two resonators. In this case can be explained a
number of character collisional features depending on the initial and final electron energies and the scattering angles.
Keywords: Foundations of the Theory of Transport Processes; The Theory of Solitons; Generalized Hydrodynamic
Equations; Foundations of Quantum Mechanics
1. Introduction
About the basic principles of the Generalized Quan-
tum Hydrodynamics (G QH ).
I begin with the short reminding of basic principles of
GQH created in particular in [1-8]. As it is shown the
theory of transport processes (including quantum me-
chanics) can be considered in the frame of the unified
theory based on the non-local physical description. In
particular the generalized hydrodynamic equations rep-
resent an effective tool for solving problems in the very
vast area of physical problems. For simplicity in intro-
duction, we will consider fundamental methodic aspects
from the qualitative standpoint of view avoiding exces-
sively cumbersome formulas. A rigorous description is
found, for example, in the monograph [6].
Let us consider the transport processes in open dissi-
pative systems and ideas of following transformation of
generalized hydrodynamic description in quantum hy-
drodynamics which can be applied to the individual par-
ticle.
The kinetic description is inevitably related to the sys-
tem diagnostics. Such an element of diagnostics in the
case of theoretical description in physical kinetics is the
concept of the physically infinitely small volume. The
correlation between theoretical description and system
diagnostics is well-known in physics. Suffice it to recall
the part played by test charge in electrostatics or by test
circuit in the physics of magnetic phenomena. The tradi-
tional definition of PhSV contains the statement to the
effect that the PhSV contains a sufficient number of par-
ticles for introducing a statistical description; however, at
the same time, the PhSV is much smaller than the vol-
ume V of the physical system under consideration; in a
first approximation, this leads to local approach in inves-
tigating of the transport processes. It is assumed in clas-
sical hydrodynamics that local thermodynamic equilib-
rium is first established within the PhSV, and only after
that the transition occurs to global thermodynamic equi-
librium if it is at all possible for the system under study.
Let us consider the hydrodynamic description in more
detail from this point of view. Assume that we have two
neighboring physically infinitely small volumes 1
and 2 in a non-equilibrium system. The one-parti-
cle distribution function (DF) ,1 1sm
PhSV
PhSV

,,
f
trv
1
PhSV corresponds
to the volume , and the function
,,
,2 2sm
f
trv
2
PhSV
—to the volume . It is assumed in a first ap-
proximation that
,11 ,,
sm
f
trv
PhSV does not vary within
1, same as
2,,
,2sm
f
trv
PhSV
PhSV
PhSV
does not vary within the
neighboring volume 2. It is this assumption of
locality that is implicitly contained in the Boltzmann
equation (BE). However, the assumption is too crude.
Indeed, a particle near the boundary between two vol-
umes, which experienced the last collision in 1
and moves toward 2, introduces information about
the
,,
,1 1sm
f
trv PhSV
2
PhSV
into the neighboring volume 2.
Similarly, a particle near the boundary between two vol-
umes, which experienced the last collision in
C
opyright © 2012 SciRes. JMP
B. V. ALEXEEV
1896
and moves toward 1, introduces information about
the DF ,2 2sm
PhSV

,,
f
trv
r
PhSV PhSV
PhSV
into the neighboring volume
1. The relaxation over translational degrees of
freedom of particles of like masses occurs during several
collisions. As a result, “Knudsen layers” are formed on
the boundary between neighboring physically infinitely
small volumes, the characteristic dimension of which is
of the order of the path length. Therefore, a correction
must be introduced into the DF in the PhSV , which is
proportional to the mean time between collisions and to
the substantive derivative of the DF being measured
(rigorous derivation is given in [6]). Let a particle of fi-
nite radius be characterized as before by the position
at the instant of time t of its center of mass moving at
velocity . Then, the situation is possible where, at
some instant of time t, the particle is located on the in-
terface between two volumes. In so doing, the lead effect
is possible (say, for 2), when the center of mass of
particle moving to the neighboring volume 2 is
still in 1. However, the delay effect takes place as
well, when the center of mass of particle moving to the
neighboring volume (say, 2) is already located in
but a part of the particle still belongs to
.
PhSV
PhSV
PhSV
v
PhSV
2
1
Moreover, even the point-like particles (starting after
the last collision near the boundary between two men-
tioned volumes) can change the distribution functions in
the neighboring volume. The adjusting of the particles
dynamic characteristics for translational degrees of free-
dom takes several collisions. As result, we have in the
definite sense “the Knudsen layer” between these vol-
umes. This fact unavoidably leads to fluctuations in mass
and hence in other hydrodynamic quantities. Existence of
such “Knudsen layers” is not connected with the choice
of space nets and fully defined by the reduced description
for ensemble of particles of finite diameters in the con-
ceptual frame of open physically small volumes, there-
fore—with the chosen method of measurement. This en-
tire complex of effects defines non-local effects in space
and time.
The physically infinitely small volume (PhSV) is an
open thermodynamic system for any division of macro-
scopic system by a set of PhSVs. But the Boltzmann
equation (BE) [1,9-12]
B
DfDt J, (1.1)
where
B
J
is the Boltzmann collision integral and
DDt is a substantive derivative, fully ignores non-
local effects and contains only the local collision integral
B
J
. The foregoing nonlocal effects are insignificant only
in equilibrium systems, where the kinetic approach
changes to methods of statistical mechanics.
This is what the difficulties of classical Boltzmann
physical kinetics arise from. Also a weak point of the
classical Boltzmann kinetic theory is the treatment of the
dynamic properties of interacting particles. On the one
hand, as follows from the so-called “physical” derivation
of BE, Boltzmann particles are regarded as material
points; on the other hand, the collision integral in the BE
leads to the emergence of collision cross sections.
Notice that the application of the above principles also
leads to the modification of the system of Maxwell equa-
tions. While the traditional formulation of this system
does not involve the continuity equation, its derivation
explicitly employs the equation
0
a
a
t



j
r
a
, (1.2)
where
is the charge per unit volume, and is the
current density, both calculated without accounting for
the fluctuations. As a result, the system of Maxwell
equations written in the standard notation, namely
a
j
0, ,
,
a
a
tt
 

 
 
 
BD
rr
B
D
EHj
rr
,
aflaf

(1.3)
contains
l
jjj
. (1.4)
f
l
f
, The l
j

fluctuations calculated using the gen-
eralized Boltzmann equation are given, for example, in
Ref. [2,4,6]. The violation of Bell’s inequalities [13] is
found for local statistical theories, and the transition to
non-local description is inevitable.
The rigorous approach to the derivation of the kinetic
equation relative to one-particle DF f
f
K
E is based
on employing the hierarchy of Bogoliubov equations. Gen-
erally speaking, the structure of
K
f
E is as follows:
B
nl
Df
J
J
Dt 
nl
, (1.5)
where
J
is the non-local integral term. An approxi-
mation for the second collision integral is suggested by
me in generalized Boltzmann physical kinetics,
nl DDf
JDt Dt



. (1.6)
Here,
is non-local relaxation parameter, in the sim-
plest case—the mean time between collisions of particles,
which is related in a hydrodynamic approximation with
dynamical viscosity
and pressure p,
p
, (1.7)
where the factor
is defined by the model of collision
of particles: for neutral hard-sphere gas,
= 0.8
[11,12]. All of the known methods of the kinetic equation
derivation relative to one-particle DF lead to approxima-
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV 1897
tion (1.6), including the method of many scales, the
method of correlation functions, and the iteration me-
thod.
In the general case, the parameter
is the non-lo-
cality parameter; in quantum hydrodynamics, its magni-
tude is correlated with the “time-energy” uncertainty rela-
tion [7,8].
Now we can turn our attention to the quantum hydro-
dynamic description of individual particles. The abstract
of the classical Madelung’s paper [14] contains only one
phrase: “It is shown that the Schrödinger equation for
one-electron problems can be transformed into the form
of hydrodynamic equations”. The following conclusion
of principal significance can be done from the previous
consideration [7,8]:
1) Madelung’s quantum hydrodynamics is equivalent
to the Schrödinger equation (SE) and leads to description
of the quantum particle evolution in the form of Euler
equation and continuity equation. Quantum Euler equa-
tion contains additional potential of non-local origin
which can be written for example in the Bohm form.
2) SE is consequence of the Liouville equation as re-
sult of the local approximation of non-local equations.
3) Generalized Boltzmann physical kinetics leads to
the strict approximation of non-local effects in space and
time and after the transmission to the local approxima-
tion leads to parameter
, which on the quantum level
corresponds to the uncertainty principle “time-energy”.
4) Generalized hydrodynamic equations (GHE) lead to
SE as a deep particular case of the generalized Boltz-
mann physical kinetics and therefore of non-local hy-
drodynamics.
In principal GHE needn’t in using of the “time-en-
ergy” uncertainty relation for estimation of the value of
the non-locality parameter
. Moreover the “time-en-
ergy” uncertainty relation does not lead to the exact rela-
tions and from position of non-local physics is only the
simplest estimation of the non-local effects. Really, let us
consider two neighboring physically infinitely small
volumes 1 and 2 in a non-equilibrium sys-
tem. Obviously the time
PhSV PhSV
should tends to diminish
with increasing of the velocities of particles invading
in the nearest neighboring physically infinitely small
volume ( or ):
u
2
PhSV
1
PhSV
n
H
u
. (1.8)
But the value
cannot depend on the velocity direc-
tion and naturally to tie
with the particle kinetic en-
ergy, then
2
H
mu
, (1.9)
where
H
is a coefficient of proportionality, which re-
flects the state of physical system. In the simplest case
H
is equal to Plank constant and relation (1.8) be-
comes compatible with the Heisenberg relation. Possible
approximations of
-parameter in details in the mono-
graph [6] are considered. But some remarks of the prin-
cipal significance should be done.
It is known that Ehrenfest adiabatic theorem is one of
the most important and widely studied theorems in
Schrödinger quantum mechanics. It states that if we have
a slowly changing Hamiltonian that depends on time, and
the system is prepared in one of the instantaneous eigen-
states of the Hamiltonian then the state of the system at
any time is given by an the instantaneous eigenfunction
of the Hamiltonian up to multiplicative phase factors
[15-19]. Since the establishment of this theorem many
fundamental results have been obtained, such as Landau-
Zener transition [15,16], the Gell-Mann-Low theorem
[17], Berry phase [18] and holonomy [19].
The adiabatic theory can be naturally incorporated in
generalized quantum hydrodynamics based on local ap-
proximations of non-local terms. In the simplest case if
Q
is the elementary heat quantity delivered for a sys-
tem executing the transfer from one state (the corre-
sponding time moment is ) to the next one (the time
moment ) then
in
t
e
t

12QT

tt
,
(1.10)
where ein
and T is the average kinetic energy.
For adiabatic case Ehrenfest supposes that
12
2,,T
 
,,
(1.11)
where 12
are adiabatic invariants. Obviously for
Plank’s oscillator (compare with (1.9))
2Tnh
. (1.12)
Conclusion: adiabatic theorem and consequences of
this theory deliver the general quantization conditions for
non-local quantum hydrodynamics.
2. Generalized Quantum Hydrodynamic
Equations
Strict consideration leads to the following system of the
generalized hydrodynamic equations (GHE) [6] written
in the generalized Euler form:
: Continuity equation for species

 

0
0000
1
0
I,
tt
t
pq R
m
 
 

 
 
 

 








 
 

 
v
r
vvvv
rr
FvB
r
(2.1)
a
nd continuity equation for mixture
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV
Copyright © 2012 SciRes. JMP
1898

 

0
1
0000
v
tt
pq
tm


 
 
 










 

r
vvvvF
rrr
0
I0.



 





vB
(2.2)
Momentum equation for species

 



11
0000 0
1
0000 0
00 00
I
pq
tt m
qpq
mt m
pp
t

 

 
 

 
 


 


0
t







v
r

 

 

 


vvvvFvBF
rr
vvvvFvBB
rr
vv vv
r
 

 
 
00 00
11
000 0000
,,
I2I
I
dd.
st elst inel
p
qq
pmm
m
q
m
JmJ


 

 
 


 






vv vv
rr
vF vvFvBvvvB
r
vv vv
(2.3)
Generalized moment equation for mixture

 




11
0000 0
01
0000 0
00 I
pq
tt m
qpq
mt m
pt

 
 

 
 

 









0
t







v
r






 


 




 


vvvvFvBF
rr
vvvvFvBB
rr
vv
r
 

 
 
0000 00
11
000 0000
I2I
I0
q
m
pp
qq
pmm
 

 

 


 



  

vvvv vv
rr
vF vvFvBvvvB
r
(2.4)
Energy equation for component

22
2
00
00 0
122
0000000
2
000
331
22222 2
15 15
22 22
17
2
vv
pnpnv p
tt
vpn vp
t
v

 


 
 


 








 
vv
r
Fvvvvv v
r
vv
r
0
00
5
n
n




v
v
 
 
  
 



2
2
00 0000
2
11
20
000
11 1
0000
15
222
13 5
222 2
pp
ppvn m
vq q q
vpp n
mmm
p
t

 

 





 
 
  
 
 
 
vvvvF v
F FvBvBvB
Fv FvvvF
rr
 

11
0
1
0
p
n



v F
F
0
22
,,
dd.
22
st elst inel
qn
mv mv
JJ

 
 





 
 
 
 

vB
vv



(2.5)
B. V. ALEXEEV 1899
and after summation the generalized energy equation for mixture

22
2
00
00
122
00000 00
2
000
33
+
222 222
151 5
222 2
1
2
vv
pnpn v
tt
vp nv
t
v


 


 
 


 










r
Fv v vvv
r
vv
r
0 0
00
15
p n
pn
 







vvv
v v
  

 
 


2
2
00 000
2
11 11
20
00 00
11
00 0
1
000
715
222
13
222
5
2
pp
ppvn m
vq
pvp
m
qq
pnn
mm
p
t

 


 

 

 


 







 
 
 
vv vv
Fvv FFFvB
vBvFF vF
Fvvv
rr
 


1
00.qn
 

 


FvB
(2.6)

1
Here F
BIq
are the forces of the non-magnetic origin,
magnetic induction,
unit tensor,
charge of
the
component particle, p
static pressure for
component,
internal energy for the particles of
component, 0hydrodynamic velocity for mixture.
For calculations in the self-consistent electro-magnetic
field the system of non-local Maxwell equations should
be added (see (1.3)).
v
It is well known that basic Schrödinger equation (SE)
of quantum mechanics firstly was introduced as a quan-
tum mechanical postulate. The obvious next step should
be done and was realized by E. Madelung in 1927—the
derivation of special hydrodynamic form of SE after in-
troduction wave function
as

,,, ,

i,,,
,,
x
yzt
zte
xyzt xy

. (2.7)
Using (2.7) and separating the real and imagine parts
of SE one obtains
22
0
tm




rr
2


 , (2.8)
and Equation (2.8) immediately transforms in continuity
equation if the identifications for density and velocity

 , (2.9)

m
r
v (2.10)
introduce in Equation (2.8). Identification for velocity
(2.10) is obvious because for 1D flow



1
vm
xm
px v
mx




1
k
Etpx
x




v
, (2.11)
where
is phase velocity. The existence of the condi-
tion (2.10) means that the corresponding flow has poten-
tial
m. (2.12)
As result two effective hydrodynamic equations take
place:

0
t


 v
r, (2.13)
2
2
11
22
vU
tmm




 

v
rr
. (2.14)
But
2
2
22
1
2


 




r, (2.15)
and the relation (2.15) transforms (2.14) in particular
case of the Euler motion equation
1U
tm
 

 

 

vvv
rr
, (2.16)
where introduced the efficient potential
2
21
42
UU m


 


r
. (2.17)
Additive quantum part of potential can be written in
the so called Bohm form
2
22
1
42
2m
m



 


r
 . (2.18)
Then
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV
1900
2
2
2
1.
42
m
2
qu
UUU U
Um








r
2
p
 
 
(2.19)
Some remarks:
a) SE transforms in hydrodynamic form without addi-
tional assumptions. But numerical methods of hydrody-
namics are very good developed. As result at the end of
seventieth of the last century we realized the systematic
calculation of quantum problems using quantum hydro-
dynamics (see for example [1,20]).
b) SE reduces to the system of continuity equation and
particular case of the Euler equation with the additional
potential proportional to . The physical sense and the
origin of the Bohm potential are established later in [7,8].
c) SE (obtained in the frame of the theory of classical
complex variables) cannot contain the energy equation in
principle. As result in many cases the palliative approach
is used when for solution of dissipative quantum prob-
lems the classical hydrodynamics is used with insertion
of additional Bohm potential in the system of hydrody-
namic equations.
d) The system of the generalized quantum hydrody-
namic equations contains energy equation written for
unknown dependent value which can be specified as
quantum pressure
of non-local origin.
In the following I intend to apply generalized non-lo-
cal quantum hydrodynamic Equations (2.1)-(2.6) to in-
vestigation of the proton and electron internal charge
structures.
3. The Charge Internal Structures of Proton
and Electron
Let us consider a positive charged physical system
placed in a bounded region of a space. Internal energy
of this one species object and a possible influence of
the magnetic field are not taken into account. The char-
acter linear scale of this region will be defined as result
of the self-consistent solution of the generalized non-
local quantum hydrodynamic Equations (2.1)-(2.6). Sup-
pose also that the mentioned physical object for simplic-
ity has the spherical form and the system (2.1)-(2.6) takes
the form [21,22]:
Continuity equation:
 

222
0 0
2 2
00
222 2
111 1
0
r r
rr
rvrv p
rv vqr
ttrrtrrrr
rrr r


 


 

 
 

 
 
 

 
 

 
 


, (3.1)
Momentum equation:


 
22 2
0 0
00
2 2
23
0
222
00 0
22
11
11
2
2
r r
rr
r
rr r
rv rv
pq
vvq
tt rrrrtr
rr
rv pp
rv vqv
rt rrrrt
rr

 
 
 



 



 


 







 
 


 
 


 



(3.2)


2
00
2
22
10.
rr
rpv pv
r
rr rr
rr








 


Energy equation:
2222
00 000
2
222222
00 0000
2 2
13 13115
22 2222
115 15117
222222
rr rrr
rr rrrr
vpvprv vpqv
tt rr
r
r vpvvpvrvpv
rtr
rr

 


 

 
 



 
 
 






 

 
 

 



22 22
00000
2
2
22 2
0
13 1
22
1151 0.
rrrrr
r
qq
qvvpqvvrv
rrrrt r
r
pq
rpvrp
 





 
 
 

 

 
 





 





22
22
p
q
rr
rrr r
rr

 



(3.3)
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV 1901
Moreover let us admit that stationary physical system
is at the rest, namely and
00
r
v0
t
.
Is it possible to obtain the soliton type solution for this
object under these stiff conditions? Let us show that the
system (2.1)-(2.6) admits such kind of solutions. For
mentioned case system (2.1)-(2.6) can be written as (see
also (3.1)-(3.3):
Poisson equation:
2
2
1
r4πrq
rr





 , (3.4)
where
—scalar electric potential and q is the positive
charge (per the unit of volume) of the one species quan-
tum object.
Continuity equation:
2p
rr
 
 0rq
r
 





. (3.5)
Momentum equation:
0
pq
rr


 . (3.6)
Equation (3.5) is satisfied for all parameter of non-lo-
cality if Equation (3.6) is fulfilled.
Energy equation takes the form:
2
2
2
2
2
15
2
15 0
2
qq
rp
rr
r
p
r
rrr
p
q
rrr
























(3.7)
or using(3.6)
2
2
0
p
r








2
q
rp r
rrr





 . (3.8)
From (3.8) follows
2
p
C
rr







22
q
prr

, (3.9)
where C is constant of integration. If the non-locality
parameter
does not depend explicitly on and the
left side of Equation (3.9) turns into zero by
r
0r
, then
. Equation (3.9) is written as
0C
2
0
p
rr







q
p
(3.10)
or using (3.6) and the relation mq e
0
p
rq



С
pCq
, (3.11)
which leads to the solution with new constant of integra-
tion
. (3.12)
From (3.6), (3.12) one obtains
1
ln qC C

C
C
. (3.13)
with new constants of integration С and 1. Let us
use these constants as scales, namely 0
10
Cq,
denoting of dimensionless values by wave 0,qqq
0

. Equation (3.4) transforms into dimensionless
equation

22
expAr r
rr







 , (3.14)
where the form-factor is introduced
2
000
4π
A
rq
. (3.15)
For other equations one obtains
22
1q
rrq
rq
r







A
, (3.16)
22
1p
rrp
rp
r







pCC q
A
, (3.17)
where the scale for pressure is 0100
. Equa-
tions (3.16), (3.17) have the same dimensionless solu-
tions. Definition (3.15) for dimensionless factor A can be
written as
2
000
4πqrA

1
2
4πCrA
. (3.18)
It means that 0,cap 0 can be considered as
the scale of proton capacity per unit of volume and for
scale of volume 3
00
4π
3
Vr
r
the scale of proton capacity
is equal to if
013A.
Figures 1 and 2 reflect the solutions of (3.14), (3.16)
correspondingly for 13A, Maple notations are used
(v,

vDt r
q
,q).
Cauchy conditions for these calculations:
v00 1,


v00 0Dr
;
1
0eq

,

000
q
Dq r
.
From Figures 1 and 2 follow that solutions exist for
this case in the domain less than 0
From (3.16) follows that the proton charge qpr is equal
to
3rr.
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV
1902
Figure 1.
,r

v=
 
D
tr
r
v=
for proton, red line
r

.v=
Figure 2.
qq
r

= for proton.

23
00
00
32
00
0
22
00 00
0
4πd4π
1
4πd
d
pr pr
pr
pr
or
2
d
1ln
p
r
pr
rr
rqr
q
rr
r



rr
pr
r
r
qrqrrrq
q
rqA rr
rqr
q
rrr
rqr



















(3.18)
2ln
p
r
pr pr
rr
q
qr r




qr
(3.19)
if ,00 0pr .
Relation (3.19) delivers the natural boundary condition
for the external area of proton. For example if the restric-
tions 0
p
r
rr
, 03
0.75 π
r
p
r
q
qr
2
00
pr
rq
and (there-
fore 0
3
4π
p
r
r
q
r
) are introduced then
0
ln 4π
3
pr
pr
rr pr
q
q
rr




~
(3.20)
and in the vicinity of
p
r
rr we have
4π
exp 3
pr
p
r
rr
rr
qC r







3
. (3.21)
Obviously (3.21) can be written using the dimensional
(cm
) form-factor q
F
p
rqpr
qr rFq . (3.22)
For the chosen scales
0
2
0
1
4π
4πpr
Arq

. (3.23)
As we see the choice of scales is the question of con-
venience by the interpretation of the experimental data
and the corresponding choice of the Cauchy conditions.
Now we can apply the previous theory to the calcula-
tion of the internal electron structure. As before we in-
tend to consider the electron at the rest placed in the
self-consistent intrinsic electric field without an intrinsic
magnetic field.
In our electron model, we no longer regard the elec-
tron as a point-like particle. Similar to the proton’s elec-
tric charge, which has continuum distribution inside of
the proton, we make the same basic assumptions based
on the application of the non-local theory. On this step of
investigation no reason to introduce the simplest model
of electron spin like a spinning electrically charged ball
or much more complicated theory which leads to the
magnetic charge continuum distribution inside of the
electron using the Dirac monopole speculations [23,24].
Obviously Poisson Equation (3.4) transforms into
2
2
14πrq
rr
r




 , (3.24)
where
—scalar electric potential and
is the
negative charge (per the unit of volume) of the one spe-
cies quantum object. In other words is absolute charge
q
q
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV 1903
density for electron. Two other Equations (3.16), (3.17)
do not change the forms
22
1q
rrq
rq
r






 , (3.25)
A
22
1p
rrp
rp
r






 , (3.26)
A
A
is the same dimensionless factor (3.15) but where
eq
and

2
expr
2
Ar
rr






 . (3.27)
Figures 3 and 4 reflect the solutions of (3.27), (3.25)
correspondingly for 13A, Maple notations are used

v
(v
Dt r
qq
, ). Cauchy conditions for
this calculations:
 
0 1,


v0

0 0
r

v0D;

0qe

,

000
r

q
Dq .
From Figures 3 and 4 follow that solutions exist for
this case in the domain less than 0
From calculations follow that proton and electron can
be considered like charged balls (shortly CB model)
which charges are concentrated mainly in the shell of
these balls. Relativistic consideration (see also [22]) can-
not change this conclusion based on principal of non-
local physics.
1.1rr .
Figure 3.
r

v=

, (solid line);

D
qq
r

= for electron. Figure 4.
In the developed theory spin and magnetic moments of
proton and electron can be introduced without changing
the main conclusions. Really the mentioned characteris-
tics correspond to
—internal energy for the particles
of
—species. For example for electron
,,elelspel m
tr
r
v=
(dashed
line) for electron.

,
where ,el sp2
,el
, and m
 pBp
mint; m—elec-
tron magnetic moment;
B
int —intrinsic electron mag-
netic induction. But
2
m
e
e
pm
 , then 2
el eff
const
.
As it follows from (2.6) written in the spherical coor-
dinate system by the condition eff
all previous
calculations take place with the additional relation
5cons t
2eel
p
m

ep
.
4. To the Theory of Proton and Electron as
Ball-Like Charged Objects
The affirmation that proton and electron can be consid-
ered as the ball-like charged objects radically changes the
theoretical results of scattering. The wave length
is correlated with the particle impulse
p
p as
p
hp
, this relation leads to condition of the particle
localization. At very low electron energies when the
wave length is much more than the proton radius
p
r
~
the
scattering is equivalent to that from “point-like” spin-less
object. The localization begins when
p
r
and im-
pulse is about 1 GeV/c. At high electron energies
p
r
,
the wavelengths is sufficiently short to resolve substruc-
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV
1904
ture.
It is reasonable to write down the Rosenbluth formula
for elastic scattering expressed as
ep
22
Mott
dd
dd 1
EpM
p
GG
22
2t
an
2
pM
G


 

, (4.1)
where
Mott
d
d



is the Mott differential cross-section
which includes the proton recoil. It corresponds to scat-
tering from a spin-0 proton. Formula (4.1) formally is
valid for extended charged object. With this aim Rosen-
bluth introduced the dimensionless electric
E
G and
magnetic
M
G form factors. Formula (4.1) contains also
Lorentz invariant quantity
2
22 0
4
p
q
Mc
p
. The form
factors are a function of 2
p
q
rather than scalar produc-
tion
2
pp
pp
E E
in the three dimensional space and
generally speaking cannot be considered in terms of the
Fourier transformation of the charge and magnetic mo-
ment distribution. If and are initial and final
electron energies than

2
2
2
1
p
qE
c


2
E


pp
pp, (4.2)
or

2
12
p
q
Mc







22
p
q

pp
pp ; (4.3)
only if
2
22 1
4
p
q
Mc

2
 
pp
pp

2
, (4.4)
one obtains with typical approxima-
tion
2
p
q

2

pp
pp MM
GG
EE
GG ,

pp
pp
 

xpi
.
In the limit (4.4) the Fourier transforms are used like


23
de
Ep
Gq q
ppr
 
 
xpi

rr , (4.5)


23
de
Mp
Gq
ppr

rr . (4.6)
Rosenbluth formula is derived for a spin-half Dirac
particle, then for magnetic moment
e
M
μ
S
.
The typical experimental correction leads to additional
coefficient
. (4.7)
All calculations depends significantly on the choice
the approximation for the charge
of
density
qr. Until
now the following
qr approximations are considered:
point-like, exponential dependence which gros smaller
with the distance from he proton center, Gaussian, uni-
form sphere, Fermi function. In general the calculations
are sensitive to the choice of
w
t
2
22 0
4
p
p
q
M
c
.
20
pp
q
we have For low
2
dd
d
E
Mott
dG


. (4.8)
For high


21
pp
q
2.79 e
M
μ
S
22
Mott
dd 12 tan
d 2
Mp
dG

 

 
 
. (4.9)
From the first glance it seems that the theo
scattering needs only in recalculation (4.5), (4
ne
ry of elastic
.6) using
w models for the internal proton and electron charge
distribution. But situation is more complicated.
The proton-electron collision in the frame of CB-
model should be considered as collision of two resona-
tors.
In this case can be explained a number of character
collisional features depending on the initial and final
electron energies and the scattering angle. At low
2
2
p
q
 
pp
pp
c
one observes not only the elastic peak
but also the resonance curves typical for the excitations
in resonators in luding discrete and continuum spectra.
Resonance curves disappear when the cavity cannot
serves as resonator. It is the situation which is well-
known from radio physics. For example the usual reso-
nance band corresponds to approximately 8 GeV and the
wave length ~14
1.5 10cm
, [25]. It leads to the relation
~0.1
pr
r
. This relation is typical for axially symmetric
resonators, (see for example [26]). This paper contains
calculations for the complex shaped cavity with axial
symmetry; the resonance frequency is about 70 GHz with
the character length ~35 mm. In the definite sense it is
the similar situation; the cavity contains very compli-
cated topology of the electro-magnetic field. Curves of
the equal amplitudes of the intensity of electric field cre-
ate domains in the form of many “islands”—caustic sur-
faces of electromagnetic field. These “islands” could be
the origin of specific features of the electron scattering
usually related with partons or quarks. In support of this
conjecture can be indicated the results [27]. In [27] M.
Popescu realizes the analyses of the cross-sections by
collisions πp
and πp
. It was noticed that the ex-
perimental data converge to show that the curves giving
the total cross-section versus energy are smoothed curves,
with large s (resoce) for medium energies and a peaknan
Copyright © 2012 SciRes. JMP
B. V. ALEXEEV 1905
slow variation with some waving on the high energy side.
Popescu points out that the curves exhibit striking re-
semblance to scattering curves obtained by electron, neu-
tron or X-ray scattering in liquid and amorphous materi-
als. Starting from this observation Popescu developed a
method to extract structural information on the proton
internal structure, in the hypothesis that ions are scattered
by unknown internal centers when they knock the proton.
The result of this analysis was very interesting because
the total number of scattering centers was estimated as 20
- 30 with the severe variance with the number of quarks
now supposed to be integrated in the proton. It should be
added that mentioned number of scattering centers is
typical for the quantity of “islands” in resonators for the
mentioned conditions.
In this connection another interesting problem is aris-
ing. Can be experimentally confirmed the resonator
model for the electron? In this case it is reasonable to
remind one old Blokhintsev paper published in Phys-
ics-Uspekhi as the letter to Editor [28]. He considered the
process of the interaction neutrino
and electron e
with transformation of electron in
-meson:
e

 . In this case the energy density W can
be estimated as
e
Wg


,
where
(4.10)
g
is Fermi constant, ,,
e

arewave
functions for el
ectron,
-m
ng I.S. Sha
eson a
spondingly ollowipir
nd neutrino corre-
o, Blokhintsev esti-. F
mated
g
as

2
0
gc
, (4.11)
with 16
10
0~ cm. His conclusion consists in affirm
strong interaction
place when the ve length
ss tha
) c be considered as estim
for revealing of the reson
hintzev supposes that fulfi
ni
of the inner charge distribution of the pro-
the frame of the non-local quan
s to following main results:
shell of
th
ty of
el
[1] B. V. Alekseev, “Matematicheskaya Kinetika Reagiru-
yushchikh Gaz of Reacting Gases,
Nauka, Mosco
a-
tion that the
takes
of electron and neutrino
of the neutrino
wa
wave pacn 0
.
0
. (4.12)
The inequality (4.12an
ket le
ation
ance electron properties. Blo-
lling of (4.12) leads to the sig-
ficant changes in the Compton effect and to other
changes in electro-magnetic interaction of electrons. It is
possible also to wait for the influence of the resonance
electron effects on investigation of hypothetical neutrino
oscillations.
5. Conclusions
Investigation
ton and electron in
hydrodynamics lead
tum
1) From calculations follow that proton and electron
can be considered like charged balls (shortly CB model)
which charges are concentrated mainly in the
ese balls. In the first approximation this result does not
depend on the choice of the non-locality parameter.
The proton-electron collision in the frame of CB-
model should be considered as collision of two resona-
tors. Curves of the equal amplitudes of the intensi
ectric field create domains in proton in the form of
many “islands”—caustic surfaces of electromagnetic field
which can serve as additional scattering centers. It can
open new way for explanation a number of character
collisional features depending on the initial and final
electron energies without consideration partons or quarks
as scattering centers.
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