CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB FIELD OF BOUND CHARGES

International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
DOI : 10.14810/ijrap.2015.4105 67
CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF
CHARGED PARTICLES IN COULOMB FIELD OF BOUND
CHARGES
Stanislav Podosenov1
, Jaykov Foukzon2
, Alexander Potapov3
, and Elena
Men’kova4
1
All-Russian Research Institute of Optophysical Measurements, Moscow, Russia
2
Israel Institute of Technology, Haifa, Israel
3
Kotel’nikov Institute of Radioengineering and Electronics of the Russian Academy of
Sciences, Moscow, Russia
4
All-Russian Research Institute of Optophysical Measurements, Moscow, Russia
ABSTRACT
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the
Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity
theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb
centres is carried out. It is shown that the proton and electron can to create a stable connection with the
dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the
proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of
the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with
the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy
for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom
coincide with the Born orbits.
KEYWORDS
Field of the bound charges, general relativity theory (GRT), special relativity theory (SRT), equations of
motion, quasi-classics, quantization of adiabatic invariants, Bohr, Sommerfeld, Caplan, Dirac, potential
curve, level splitting, fine structure.
1. INTRODUCTION
In classic electrodynamics commonly thought that the field of rest point charge in the inertial
reference frame (IRF) is the Coulomb spherically symmetric field independently of whether the
charge is free or the sum of the forces affecting on the charge is equal to zero. However the field
of this charge with uniformly accelerated motion in accordance with the classic electrodynamics
for the noninertial reference frame (NRF) observer will be axially-symmetrical independently of
the transition method to NRF. Thus, identical physical situation in which the charges are
(identical constraint forces) results in the firelds with different symmetry!
Let us consider the charged metallic sphere. Each charge element is subjected to a force of
negative pressure from the side of the field created with the charged sphere. This force is directed
outward normally to the sphere surface. It is compensated with the crystalline force. The
crystalline bonding force is directed inside the sphere and in accordance with the [1-7] the metrics

68
or the charge element on the sphere will be axially-symmetrical with the “acceleration” directed
inward the sphere
(1),)()()())(
2
(exp= 23222120
2
1
02
dydydydy
c
ya
dS −−−
where the “acceleration” 0a is considered as positive if it is directed along the 1
y axis and this
“acceleration” is negative if it is directed opposite this axis. We point out that for each charge
element the local triad is selected so that 1
y axis coincides with the unit vector directed along the
sphere radius. Let the sphere is charged negatively. Then the scalar potential from the charge
element on the sphere has the form [3, 6, 7]
(2)},
2
)cos(1
{exp= 2
0
0
c
ra
r
dQ
dA
θ−′
−
′
where r′ is the three-dimensional (Euclidean) distance from the charge dQ up to the
observation point, θ is the angle between the radius vector r
r′ and i
r
, ||/= 0 oaai
rrr
. i
r
for each
charge element dQ is directed the the sphere centre. The “acceleration” 0a for the charges of the
negatively charged sphere (electrons) is calculated in accordance with the formula
(3),
2
=0
m
eE
a
Here e is the charge value, m is the electron mass, E is the field strength at the sphere surface.
For the positively charged sphere the “relativistic” effect will be considerably smaller as the mass
of the positive ion significantly exceeding the electron mass will be the m mass. Therefore the
field of the positively charged sphere in practice coincides with the classical field and for the
negatively charged sphere the “relativistic” corrections can be significant. Each electron on the
sphere surface belongs to tangent plane space but to the Riemann space-time. Therefore the
integration on the sphere occurs in the plane space and it is correct. Integrating we obtain
.(4)
8
=)],())/2(1([
4
)(exp
= 2
2
0
Rmc
eQ
R
r
irRi
R
iQ
A ζζζ
ζ
πζ
Φ−−Φ
−
In (4) r is the distance from the sphere centre to the observation point, )( ζiΦ is the probability
integral of imaginary argument determined as following
.(5))(exp
2
=)( 2
0
dtt
i
i ∫Φ
ζ
π
ζ
On the other hand the aggregation of the electrons at the sphere surface does not belong to the
congruence of the world lines of NRF basis and it is included to the aggregation of the world lines
belonging to the spherically symmetrical Lagrangian co-moving NRF with the metric
.(6)))((exp)sin())((exp= 22222202
drddrdydS λφθθν −+−
The expression for the metric coefficients is obtained in [2-6] and it has the form

69
.(7)1=/2)(exp 2
0
0
cm
qA
+ν
.(8)
1
=)
2
(exp
2
0
0
0
2
cm
qA
r
A
Q
r
+
∂
∂
−
λ
Thus, taking into account that the charges on the sphere are connected with the crystalline forces
we obtained the spherically symmetric field of the charged sphere (but not the Coulomb one)
specified in the Riemann space-time with the metric (4, 6-8).
2. THE PARTICLE MOTION IN THE CENTRALLY SYMMETRIC
ELECTROSTATIC FIELD OF THE BOUND CHARGES
Let us consider the motion of the charged particle with 0m mass and q charge in the field which
other Q charge creates. The charge mass creating the field is great as compared with 0m
therefore we will consider Q charge is immobile. In accordance with the idea of the general
relativity theory the test charge will move on the geodetic line in the field of the bound charges
forming the Q charge with the metric (6-8).
To describe the particle motion instead of the geodetic equation it is convenient to use the
Hamilton – Jacobi equation. It is known that when moving in the centrally symmetric field the
particle path is in plane of the immobile charge centre creating the field. /2= πθ plane is
selected.
Hamilton – Jacobi equation will have the same form as in general relativity theory when
describing the motion in the centrally symmetric gravitational field [8]
0.(9)=2
0cm
y
S
y
S
g −
∂
∂
∂
∂
νµ
µν
or we have
0.(10)=)(
1
)()( 22
0
2
2
22
0
cm
S
rr
S
e
y
S
e −
∂
∂
−
∂
∂
−
∂
∂ −−
φ
λν
In accordance with the general method of solution of Hamilton – Jacobi equation we represent S
in the form
).(11)(=
0
0
rSM
c
yE
S r++− φ
Here 0E is the constant energy, M is the permanent angular momentum. From (10) and (11) we
have
.(12))(exp)()(exp= 2
2
22
02
2
0
rd
r
M
cm
c
E
Sr λνλ +−−∫

70
For integrand we have
.(13)
)(1
)(
=)(exp
2
2
0
0
20
2
4
cm
qA
r
A
Q
r
+
∂
∂
λ
.(14)
)(1
)(
=)(exp
4
2
0
0
20
2
4
cm
qA
r
A
Q
r
+
∂
∂
−νλ
Let us find the dependence of r radial coordinate on cy /0
universal time from equation
.(15)c=
0
onst
E
S
∂
∂
Using (11-14), we find
.(16)))(1(11=
/2)(exp
/2)(exp
= 2
2
0
0
222
0
2
2
0
42
0
0
(1)
cm
qA
rcm
M
E
cm
dy
dr
c
v
++−
ν
λ
Here (1)
v is the radial “physical” component of three-dimensional velocity.
For example let us consider the fall of the negatively charged particle on the positively charged
coulomb centre.
From (16) at 0=M we find
.(17))(11= 2
2
0
0
2
0
42
0
(1)
cm
qA
E
cm
c
v
+−
As previously shown for positive charges the “relativistic” corrections stipulated for the
“acceleration” of bound charges creating the field give a small correction to the scalar potential.
Therefore
.(18))
||||
(11= 2
2
0
2
0
42
0
(1)
rcm
Qq
E
cm
c
v
−−
Let us consider the problem concerning the radial electron motion in the proton field without
quantum effects. As is known from the nuclear physics the charge distribution radius inside the
proton is cmR 13
100.8= −
⋅ . The classic electron radius is cmr 13
0 102.8= −
⋅ . The scalar
potential outside the proton can be calculated in accordance with the formula (4).

71
(19)
8
=,
8
=)],())/2(1([
4
)(exp
= 22
2
0
Rmc
eQ
Rmc
eQ
R
r
irRi
R
iQ
A δζζζ
ζ
πζ
Φ−−Φ
−
.
Outside the proton the relation is valid
.(20)0.0154=
8
=
8
= 0
2
R
r
Rm
rm
R
r
Rmc
eQ
R
r
p
e
ζ
In the last formula em , pm are the electron and proton masses respectively. Taking into account
the infinitesimality 0.0154=δ and expanding (19) into the Taylor series we obtain for the
formula (17)
.(21))(11= 20
2
0
42
0
(1)
r
r
E
cm
c
v
−−
Let the electron had zero speed at infinity that is equivalent to the relation 1=/ 2
00 cmE . In this
case from formula (21) we have
.(22))(11= 20
(1)
r
r
c
v
−−
As follows from (22) when fall on the proton firstly the electron velocity increases from zero up
to light speed c reaching the later at the diatance from the proton centre equal to the classic
electron radius 0r . Further when approaching to the centre the electron velocity begins to
decrease ant it vanishes at the distance from the proton centre equal to the half of the classic
electron radius /20r . Then when decreasing of the radius the radicand becomes negative and the
velocity becomes imaginary that is devoid of physical meaning. Thus, the obtained solution is
valid when /20rr ≥ .
The decreasing of the “physical” electron velocity is equivalent to the repulsion from the proton
side at the small distances cm13
101.4 −
⋅ which are approximately equal to the effective proton
radius.
To compare we point out that in the similar problem when moving on the particle radius in the
gravitational field specified with the Schwarzshil'd metric the “physical” velocity is determined
with the formula [9]
.(23)=)(11=
(1)
r
r
r
r
c
v gg
−−
In (23) gr is the gravitational radius.
When formal substitution epmkme →2
or /20 grr → the solutions (22) and (23) in the distance
from the gravitational radius coincide. Near the gravitational radius these solutions are
qualitatively differed from one another. In (23) the “physical” particle velocity increases with the

72
radius decrease, achieves the light velocity c at the gravitational radius and it goes to infinity at
0→r .
In our case (22) the value of the particle velocity like in SRT does not exceed the velocity of light
in free space achieveing the later at 0= rr .
The solution of the similar problem in SRT [8] results in the relation
.(24)
)(1
1
1=
20
(1)
r
rc
v
+
−
As follows from (24) the electron velocity when approaching to the centre increases with the
radius decrease and it tends to the velocity of light c at 0→r . At 0= rr , cv 0.866=(1)
. It is
clear that in our consideration at 0> rr the maximum deviation of the velocity from the SRT
solution does not exceed 14%. At 1/0 =rr the solutions (22) and (24) coincide. At 1>/0 rr the
velocity behaviour is changed: in our case it decreases and in SRT it increases.
Let us investigate the radial motion of ultrarelativistic electrons which even at infinity have
infinitely large energy. Supposing in (17) ∞→0E we obtain for any r , cv =(1)
. For similar
case the same result will be obtained in SRT and GRT.
Let us investigate more detailed the relation (21) introducing the dimensionless electron energy
E and dimensionless velocity v in accordance with the formula
.(25)=,=
(1)
2
0
c
v
v
cm
E
E
e
According to agreed notations we have
.(26))(1
1
1= 20
2
r
r
E
v −−
As follows from (21) for any 1≥E value v has a maximum equal to unity at the point 0= rr .
For greater energy values the velocity growth curve from infinity to 0r point will be more smooth
than for smaller energy values. For ultrarelativistic particles the growth curve coincides with the
straight line 1=v . After maximum the velocity curve will fall and at the some radius value v
goes to zero. The distance minr corresponding to zero valuation of the velocity will be the closely
to the Coulomb centre, the more the particle energy is. It follows from the formula
.(27)
1
= 0
+E
r
rmin
It follows from (27) that only electrons with infinitely large energy E reach the Coulomb centre.
However it is not the case. Formulas (26), (27) are used for the proton with the finite radius. For

73
the model point proton one should use the general fomulas (17) and (4). In accordance with (4)
for the scalar potential near the coulomb centre the formula holds
(28))],([
2
)(exp
=)(
2
0 δ
δ
πδ
i
Ri
Q
RA Φ
−
whence we find at 1= >>δz the expression
.(29)
4
=)(
2
0
e
mc
RA
It follows from the last formula and (17) that the electrons with energy (30) can have zero
velocity at the Coulomb centre
1.(30)4= −
e
p
m
m
E
Thus, electrons with high but final energy can reach the point Coulomb centre.
Let us elucidate the physical meaning of the origin of the electron stop in the proton field.
Equating in (26) the radicand to zero we find the function )(rE acting as the potential curve in
the nonrelativistic theory.
.(31)|1=|)( 0
r
r
rE −
It follows from the analysis (31) that as a result of the presence of the modulus the )(rE function
at 0< rr does not damp but it increases and becomes equal to unit at /2= 0rr , and this function
goes to infinity at 0→r . Thus, when the distances are smaller then 0= rr the “effective
repulsion” takes place. This repulsion results in the electron stop at minrr = in accordance with
(27). As 1=)(∞E the potential well with single depth is formed. At 1<= 1EE the radial
motion of the electron in the proton field is finite. The electron radially oscillates within the
limits from minr up to )/(1= 101 Err − . At 1>= 1EE the motion is infinite. The electron reaching
when fall the minr value shifts at infinity after the “reflection”.
In SRT the solution of such problem at 0=M results in the fall of the charge on the coulomb
centre. Similar situation takes place with zero moment in the Newtonian mechanics and in GRT
.
Let us find the electron motion on the circular orbits in the proton field. Assuming in (16)
0=(1)
v that corresponds to the circular orbits at 0≠M we obtain the expression for the
potential curve ))/((=)( 2
0 cmrErU e
.(32)|1|1= 0
222
2
r
r
rcm
M
U
e
−+
The considered problem is similar to known S. A. Caplan problem in GRT [8], [9]. To compare
with the Caplan solution we make a formal substitution /20 grr → . The meaning of this
substitution is as follows. After the substitution the approximate metric (7,8) after the Taylor

74
expansion coincides with the exact GRT Schwarzshil'd metric. The circular orbits in the Caplan
problem are determined in the Schwarzshil'd field.
For simplicity we introduce the dimensionless E energy values (25), a moment and x
dimensionless radius in accordance with the formulas
.(33)
2
,=,= 0
g
gge
r
r
r
r
x
crm
M
a →
After substitution the relation (32) has the form
.(34)|
2
1
1|1= 2
2
xx
a
U −+
The radiuses of the circular orbits and corresponding energy and moment values are determined
by the extremums of the effective potential energy )(xU . Minimum of the function corresponds
to the stable circular orbits and maximum corresponds to the unstable orbits. Solving jointly the
equations set ExU =)( and 0=)(xU′ we obtain
.(35)|
2
1
1|
2
=),
2
1(1= 3/2
1,21,2
2
2
1,2
xx
a
E
a
ax −−±
Plus sign in the formula for the orbit radiuses corresponds to the stable orbit and minus sign
corresponds to the unstable one. In order to compare we present the solution of the Caplan
problem in our designations
(36)|
1
1|
2
=),
3
1(1=
1,21,2
2
2
1,2
xx
a
E
a
ax −−±
and the expression of the effective potential )(0 xU in GRT.
.(37))
1
)(1(1=)( 2
2
0
xx
a
xU −+
Iit is clear if we introduce in (34) radicand the expression with the modulus and expand into
Taylor series limiting with the first expansion term we will obtain the potential curve (37).
However this is correct only for 1>>x . It follows from the (37) analysis this expression is
determined only for the values 1≥x . If not 0U becomes an imaginary one.
For (34) such imitations do not exist. Potential curve charts (34) for different values of a
moment are presented in [8], [9] and for our case potential curve charts are presented in [3].
Curve charts (34) at different a in 1≥x are similar with the charts (37). They are differed only
with maximum and minimum. You could see that with the comparison the expressions (35) and
(36).

75
For example, at 2=2
a maximum and minimum coordinates for our case coincide. In GRT it
takes place at 3=2
a . For our case the nearest to the centre stable circular orbit corresponds to
the value 01 42= rrr g → . Corresponding energy is 0.919=27/32=1E , and the velocity of
the circular motion is determined from the formula
.(38)
1
=
2
2
00
2
0
0
c
v
gcm
E
−
of the energy conservation law which is similar to the energy conservation law for the particle in
the constant gravitational field in GRT [8]. In (38) the velocity v is calculated in accordance with
the formula
(39),= 0
00 dyg
cdl
v
ccv 0.577=/33= .
In GRT we have respectively grr 3=1 , 0.943=8/9=1E , /2= cv [9].
At 2<2
a for our case the curve does not have the extremums (in GRT for 3<2
a ). With the
growth of the a moment from 2 to ∞ the coordinates of maximums are decreased from
02 4= rr to 02 2= rr (in GRT from grr 3=2 to grr 1.5=2 ). maxE energy is increased from
0.919=maxE to ∞=maxE (in GRT from 0.943=maxE to ∞=maxE ).
The nearest to the centre unstable circular orbit corresponds to the value 01 2= rrr g → ,
∞=maxE , cv = (in GRT grr 1.5=1 , ∞=maxE , cv = ).
In fig. 1 four dimensionless potential curves ),( axU for different values of the dimensionless a
moments in the Caplan problem in the Coulomb field of the bound charges are presented. The
lowest curve corresponds to 0=a and it does not have extremums.
The second curve from the bottom corresponds 2=a when the coordinates of maximum and
minimum of the potential curve coincide. The third curve from the bottom has maximum equal to
unit at 1.665=a , and the fourth curve at 5=a has maximum equal to 2.551.

76
Figure 1. Dimensionless potential curves ),( axU for different values of the dimensionless a moments
in the Caplan problem in the Coulomb field of the bound charges
In fig. 2 the curve showing the arms of the stable and unstable orbits in the Caplan problem for
the field of the bound charges is presented. The lower reach determines the radiuses of the
unstable orbits, and upper reach determines the radii of the stable orbits. The curve calculation is
carried out in accordance with the formula (35).
Figure 2. Curve of the dependence of dimensionless distance x on a moment in the Caplan problem in
the Coulomb field of the bound charges
It should be noted that in the Newton mechanics at the centrally symmetric gravitational field the
stable circular orbits exist at any distance from the centre. In the Coulomb field of the bound
charges the minimum radius 2=a exists. At smaller radius circular orbits do not exist. Similar
restriction takes place in GRT where minimum radius is 3=a .
Although outwardly the Caplan solution and our one do not greatly differ, however the important
basic difference takes place. The presence of the expression with the modulus in the effective
potential energy )(xU in (34) results in the additional minimum value )(xU in the 1/2=x
point. In is clear that x derivative from the )(xU function in this point has a discontinuity as at
the radial fall of the particle that is at 0=a . In 1/2=x point the attraction changes on the
repulsion and )(xU function unrestrictedly increases when the radius vanishes. The presence of
the additional centrifugal energy makes the function increase )(xU faster when vanishing than at
Effectivepotentialenergy
[0,55] Dimensionless distance

77
the zero moment. Aforesaid is shown in fig. 3 where the plot of function (34) is continued to
0.5<õ .
Dimensionless distance
Figure 3. Potential curve in the Coulomb field of the bound charges
The presence of the maximum at the potential curve in the Schwarzshil'd field results in the
gravitational capture if the particle energy is maxEE > . For our case at 1.5=> maxEE the
capture does not occur as any E value always “meet” the potential curve at 0>>0.5 x (see fig.
3). We remind that the dimensionless value x on the X-axis is the ratio )/(2/= 0rrrrx g → .
Thus, similarly to the Newton mechanics at any high energy the particle rounds the attractive
centre and goes into infinity. Even as opposed to the Newton mechanics it takes place at the radial
fall on the Coulomb centre too. It is significant to note that the stable electron equilibrium in the
proton field at 0=1/2= rrx → exists. In the classic consideration it is absent.
The presence of the additional “potential well” with minimum value 0=)( 0rU (see fig. 3)
permits the existence of the finite electron motion in this well. Thus, the proton and the electron
can form the stable connection “neutron” with the dimensions of the order of 0r . Of course the
considered approach has only a methodological interest as the quantum effects must already
become apparent at the distances considerably greater 0r .
Let us consider the electron trajectory at the proton field. As is known [8] the trajectory is
determined with the equation onstMS c=/∂∂ . Whence we have
.(40)
)(exp)()(exp
=
2
2
22
02
2
02
λνλ
φ
r
M
cm
c
E
r
rMd
+−−
∫
It is of the methodological interest to investigate the electron trajectories in the atom limits
similarly to the planetary motion in the sun gravitational field for the electron motion in the
proton field at the distances compared with the atom dimensions 0rr >> . Therefore the proposed
theory must only result in insignificant correction data as compared with the general Coulomb
interaction.
To calculate the correction data to the trajectory we will assume as in [8] from the radial part of
the action before the moment of its M differentiation (12-14).

78
.(41)
)()()(
= 2
0
2
2
0
22
0
2
4
0
42
0 rd
rr
r
rrcm
M
rr
rE
cmSr
−
−
−
−
−∫
After the transformation of integration variable
,=0 rrr ′−
using easily verifiable formula
,2=1)( 2
2
0
222
0
c
E
EmEcm
′
+′−
where E′ is the nonrelativistic particle energy (without the rest energy) we obtain with the
required precision the expression
.(42)
582
2= 2
22
0
2
0
2
00
22
00
2
2
0 rd
r
cmrM
r
Emrcmr
c
E
EmSr
−
−
′+
+
′
+′∫
When formal substitution epmkme →2
or /20 grr → the structure of the radial part of the action
rS is close to the similar value used to describe the planetary motion in the centrally symmetric
gravitational field in GRT. The difference is revealed only in the last factor at 2
1/r . In GRT
22
0
2
06 cmr value is subtracted from 2
M (in our designations). The rest of the terms coincide.
As is known [8] the correction factors in the first two terms of the radicand are only influenced by
the changing of the connection between the energy, particle moment and parameters of its orbit.
The factor subtracted from 2
M results in the systematic drift of the orbit perihelion. Considering
that 222
0
2
0 /5 rcmr term is the small small correction relatively 22
/rM after the expansion of the
intergrand into a series we obtain the relation
(43),= 0
rrr SSS δ+
where 0
rS corresponds (42) at zero correction, and rSδ is determined with the equality
.(44)
82
22
5
=
2
2
00
22
00
2
2
0
2
2
0
22
0
r
M
r
Emrcmr
c
E
Emr
rdrcm
Sr
−
′+
+
′
+′
∫δ
Increment rSδ∆ during the orbital period of the electron in accordance with the
nonrelativistic mechanics at the Coulomb field after the integration in (44) will have the form
.(45)
5
=
2
0
22
0
M
rcm
Sr
π
δ∆
As the trajectory is determined with the equation

79
,== const
M
S
M
S r
∂
∂
+
∂
∂
φ
then during the period we have
.==
00
M
S
M
S
M
S rrr
∂
∆∂
−
∂
∆∂
−
∂
∆∂
−∆
δ
φ
Using (45) and taking into account that
,2=
0
π
M
Sr
∂
∆∂
−
we obtain
.(46)
5
2= 2
2
0
22
0
M
rcmπ
πφ +∆
Second term in (46) determines the perihelion shift of the electron orbit around the proton. The
solution of this problem in SRT results in the perihelion shift five times smaller than our shift and
six times smaller than in GRT at the formal substitution epmkme →2
or /20 grr → . Thus, the
developed apparatus is much closer to GRT than SRT.
Let us consider the nonradial motion of the ultrarelativistic electrons in the proton field (analog of
the light ray propagation in the centrally symmetric gravitational field). We consider by definition
that even at infinity cv →∞ .
The radial part of the action (41) after the transformation of integration variable
,=0 rrr ′−
provided 1>>E , EcmE 2
00 = and the impact parameter ρ is connected with the M moment
in accordance with the formula [1]
(47)=
0cEm
M
ρ
is reduced to the form
.(48)
64
1= 2
2
0
2
2
00
rd
r
r
rr
r
c
E
Sr
′
+
′
−
′
+∫
ρ
Firstly let us conside the case when 1/0 =ρr . For this case (48) has the form
.(49)
4
1= 2
2
00
rd
rr
r
c
E
Sr
′
−
′
+∫
ρ
The last expression accurate within the multiplier and substitution grr →02 coincides with the
radial part of the eikonal for the light ray propagation at the gravitational field in GRT [8]. Using
the derivation [8] we obtain that under the influence of the Coulomb attraction field from the

80
proton the trajectory of the ultrarelativistic electron is curved forming the curve with the
concavity towards the centre. The angle between two asymptotes of this curve is differed from
π by δφ determined from the equality
.(50)
4
= 0
ρ
δφ
r
Let us analyzed the motion of the ultrarelativistic electrons when the impact parameter ρ is of
the same order of 0r . To analyze the motion we use the formula (16) which at 1>>E is reduced
to the form
.(51))(11= 20
2
2(1)
r
r
rc
v
−−
ρ
Obviously the approaching of the electron to the proton will stop when the radial velocity
component 0=(1)
v . The turning point will be determined from zero radicand in (51).
This results in the relation between the impact parameter and the turning point coordinate
represented in the form
.(52),,
|1|
=
0
1
0
1
1
2
1
1
r
r
r
rr
r
≡≡
−
ρ
ρρ
The curve (52) has minimum in point
4(53)=2,= 11 minr ρ
and two asymptotes: the vertical one 1=1r and the slanting line 1= 1 +rY . The presence of the
modulus in the denominator expands the field of the variable existence up to the values 01 →r .
The second branch of a curve appears which when variating of the argument from 1=1r to
0=1r changes respectively from ∞ to zero. Using near the Coulomb centre the formula (28) for
zero component of 4 – potential in (16) influences only on zero approximation behavior at
01 →r . The dependence of 11 ρ≡R on 11 rr ≡ is presented in fig. 4.

81
Positionofatuningpoint Impact parameter
Figure 4. Curve of dependence of 1r ultrarelativistic particle on 1ρ impact parameter in the Coulomb
field of the bound charges.
The particles with 4<1ρ suffer the “pseudotrapping”
To analyze the solution obtained we write the similar equation for the curve in GRT for the
ultrarelativistic particle in the Schwarzshil'd field. In accordance with [8] and [9] for the desired
curve we have
.(54),,
1
= 22
2
3/2
2
2
gg r
r
r
rr
r
≡≡
−
ρ
ρρ
The curve (53) has minimum in point
(55)
2
33
=,
2
3
= 22 minr ρ
and two asymptotes: the vertical one 1=2r and the slanting line 2= rY .
Both considered curves at 1>1r and 1>2r are similar. They are differed from only with
minimums and their minimal values. However at 1<2r the curve (53) is not determined as the
radicand becomes negative. Thus, as opposed to our case in GRT the second branch of a curve is
absent. In GRT the ultrarelativistic particle flying from infinity with the impact parameter
2.6=< 22 minρρ does not meet the turn curve and consequently its gravitational capture occurs.
The capture cross-section is determined with the formula
.(56)6.75=
4
27
== 2
2
22
2 g
g
gmin r
r
r π
π
πρσ
In our case at 4=< 11 minρρ the electron “passes” under the first turn curve but “meets” the
second branch of this curve (see fig. 4). The peak value of the point of contact coordinate with the
second branch is determined from the apparent equation

82
(57),
1
=
1
2
1
1
r
r
min
−
ρ
The solution of this equation has the form
0.83.(58)=
2
4
= 11
2
1
1
minminmin
maxr
ρρρ −+
As opposed to GRT the electron “flying” under the turn curve is not captured with the proton
because of the proton repulsion prevents. This repulsion appears at the distances 0< rr .
From 02= rr up to 0= rr distances the electron trends to the capture. Therefore we denote the
pseudotrapping cross- section by pσ
.(59)416== 22
0
2
0
2
1 gminp rrr πππρσ →
Let us consider one more case when the electron at infinity has a negligible velocity ∞v as
compared with the velocity of light c that corresponds to 1=E .
In GRT the section of the gravitational capture is determined from the requirement of the
maximal value of the potential curve (37) 1=)( maxxU . 2=maxx , 2=a correspond to this
energy value. The impact parameter ∞vcrg /2=ρ and the capture cross-section
.(60))(4== 2222
2
∞v
c
rr ggmin ππρσ
All particles with ∞vcrg2<ρ are captured gravitationally.
In our case the potential curve )(xU is specified with the relation (34). Solving jointly the
system of equations 1=)(xU and 0=)(xU′ we obtain for the maximum (see fig. 1, the second
curve from the top)
.(61)|
2
1
1|
2
=1),
2
1(1= 3/2
11
2
2
1
xx
a
a
ax −−−
From (61) we find
1.309.(62)=1.665,=2.773,=
16
162222
= 1
2
2
xaa
++
The impacr parameter in our designations ∞vcr /3.33= 0ρ and the pseudotrapping cross- section
.(63))(2.77)(11.1== 2222
0
2
∞∞
→
v
c
r
v
c
r gp πππρσ

83
In our case the proton can not capture the electron so far as at 0< rr the potential curve
begins to grow and the electron energy 1=E will “intersect” with this curve (see fig. 5).
Dimensionless distance
Figure5. Potential curve for the nonrelativistic particle having at infinity negligible velocity.
As opposed to GRT such particle will not be captured with the proton as it “meets” with an
increasing branch of the curve after the passing of the maximum
It is known from the classical relativistic electrodynamics when moving in the Coulomb field [8]
that at |<| QeMc and 0<Qe the particle trajectories represent the spirals with the fanishing
radius r at the angle ∞→φ . The fall time of the particle at origin is the finite quantity.
Particularly for the nonrelativistic particle at infinity (the electron in the proton field) the
condition of the fall on the centre is equivalent to the equality
(64)< 0
∞v
cr
ρ
In our case the impact parameter when pseudotrapping is 3.33 times as much than in SRT. A
significant feature of our solution as opposed to SRT and GRT is the possibility of the stable
static balance of the electron in the proton field at the distance 0r from the proton centre. The
radial oscillations relatively 0r are allowed. This permits the (qualitative) possibiliy of the
existence of the neutral stable particle with the dimensions of the order of 0r .
3. QUANTIZATION OF ADIABATIC INVARIANTS IN CENTRAL FIELD OF
BOUND CHARGES
It is of interest to consider the elementary possibilities of the accounting of quantum effects. As is
known Bohr and Sommerfeld explained the spectrum of the hydrogen atom by means of the
quantization of adiabatic invariants. Then Sommerfeld had attempted to take into account
relativistic corrections in the network of the mechanical analog. He permitted that relativistic
corrections can explaine the splitting of terms degenerated in the nonrelativistic theory. Thus,
Sommerfeld wanted to create a theory of fine structure.

84
It should be noted that in a way he pulled it off and he obtained formulas for the fine structure of
hydrogen atomic levels in the old Bohr theory before the creation of the quantum mechanics
without the solution of the Dirac equation.
The structure of the proposed approach is close to the Bohr and Sommerfeld one but it has the
distinctions of kind.
1. As opposed to the Sommerfeld approach using the motion of the electron in the proton field in
the SRT flat space-time we are working in the Riemann geometry stipulated for the field of
elements of proton bound charges.
2. The world line of the electron in the proton field in the Sommerfeld theory corresponds in our
case to the electron geodesic line in the Riemann space-time. In our approach the proton field is
absent in explicit form. It becomes apparent in the form of the cuved space-time geometry.
As the atom dimensions are of the order of 8
10−
cm and the nuclear sizes are of the order of
13
10−
cm then factors of the metric (6) can be represented by means of (7,8) in the form
(65)),(exp=)(1=)(1=)(exp 202
2
0
2
λν −−−
r
r
crm
e
where e is the electron charge, 0m is the electron mass.
As is known [8], the 4-vector determined with the equality is called the generalized momentum
µP
.(66)= µµ
y
S
P
∂
∂
−
Let us calculate the radial component of the 4-pulse of the electron in the proton field. From (11)
and (12) we find
.(67))(exp)()(exp= 2
2
22
02
2
0
λνλ
r
M
cm
c
E
Pr +−−−
As opposed to SRT we apply the quantum condition to the “physical” radial component of 4-
pulse determined with tetrads. The expression for these tetrads has the form [2-4]. Tetrad indices
are parenthesized.
(68),=,=,||=,
||
= (0)
(0))(
)(
µµ
µµαα
α
µ
α
µ
αα
µ
αµ
α δ
δ
VeVege
g
e
.(69)=)()(exp=
1
= )(
2
2
22
02
2
0
11
)(
r
rr P
r
M
cm
c
E
P
g
P −+−−−
−
− ν
Following to Bohr and Sommerfeld we state quantization conditions in the form
(70),=,=)(
hnMdhndrP r
r
φφ∫∫
where rn and φn are integral numbers. As a result we have the equality

85
.(71)
2
=,=)()(1 2
22
22
02
2
020
π
φ h
hndr
r
n
cm
c
E
r
r
r h
h
+−+∫
In this equality after the integration we identify 0E values depending on quantum numbers rn
and φn . The integral can be represented in the form
),(=,//2 2
2
022
0
2
c
E
cmAdrrCrBA −−+−∫
.(72)=,= 2
2
0
2
022
2
2
00
c
Er
nC
c
Er
B −hφ
As we consider the finite motion of the electron in the proton field the A value is obviously
negative. The integrand has two roots for positive r values that obviously corresponds to the
perihelion and the aphelion of the electron orbit. The integration is expected to be carried out
from one root up to the other and vice versa with the changing of the sign before the root in the
integrand. The similar problem with other constants has been solved in [11]. On the basis of
which we have
).(73)(2=//2 2
C
A
B
drrCrBA −−+−∫ π
As a result we obtain the following equation for the determination of energy levels
.(74)=1
/
1
22
2
0
2
0
22
0
22
0
2
00
hrn
Mc
Er
cEcmc
Er
−−
−
Introducing the dimensionless energy E in accordance with (25) after cumbersome algebraic
transformations the last equation is reduced to the form
(75),=,=,]
)(
[1=
2
1/2
222
2
c
e
E
nn
E
r
h
ααα
α
α
φ
′
+′−
′
+ −
where α is the fine structure constant. The last relation with the renormalized fine structure
constant α′ exactly coincides with the relation of the Sommerfeld fine structure [12] obtained by
means of the solution of the Dirac equation when the electron moves in the Coulomb field. As
when motion of the electron in the atom the relation is fulfilled with pinpoint order of accuracy
(76),1= ε−E
where ε is the small positive value and the fine structure constant 1/137=α , then it is
obviously that in our case the first approximation corresponding to zero ε in the right side of
equation (75) results in the exact Sommerfeld formula.

86
The physical meaning of ε value is the total dimensionless energy of the electron in the proton
field with the reversed sign after deduction of the dimensionless rest energy. Taking into account
that 1=α′ we obtain from (75) and (76) the expression
).(77)
2
(1=,=)],
4
31
([1
2
= 2
2
22
2
2
2
n
nnn
nnnn
r
α
αα
αα
ε φ
φ
−′+−
′
+
′
Transferring to the dimension energy 2
0= cmW ε− and introducing 1/2= +jnφ we obtain in
the prescribed approximation the following formula for the level energy in hydrogen atom
).(78)
4
3
1/2
1
(
22
= 3
2
0
4
2
2
0
2
njn
cm
n
cm
W −
+
−
′
−
αα
Let us analyze the last formula. As is known from the quantum mechanics the first term
determines energy levels of hydrogen atom calculated by means of the nonrelativistic
Schrödinger equation. The second term represents the additive which is stipulated for the fine
level splitting. This additive is calculated by means of the Dirac equation. The second term in the
formula (78) in the prescribed approximation exactly coincides with the Dirac theory and the first
term in (78) slightly differes from the conventional one. In our case energy levels are slightly
underrated as compared with standard sW values. In order to compare we represent specific
values
9)0.00266%(7
1
=100%
2
=100% 22
2
nnW
WW
s
s
⋅⋅
− α
The obtained estimation is within the error estimate of theoretical and experimental energy values
of the Balmer series. The total width of the fine structure [13] determined as the distance
between 1/2=1 −nj and 1/2=2j levels at prescribed n coincides with the similar value from
the Dirac theory. Let us consider the quantization of circular electron orbits in the proton field
with the effective potential energy (32). Our aim is to ascertain how above mentioned circular
orbits in the proton field (35) similar to the Caplan orbits in GRT with the Bohr orbits in
hydrogen atom are connected. Obviously in the case of circular orbits we must nullify the radial
quantum number rn in (75). Then the orbital quantum number φn will coincide with the main n.
This results in the following value of the dimensionless energy E .
.(80)=,=,
1
1
=
2
2
2 c
e
nn
n
E
h
α
α
φ
+
It is clear that owing to smallness of the fine structure constant α we write the last relation in the
form
.(81)
8
3
2
1= 4
4
2
2
nn
E
αα
+−

87
Let us consider stable electron orbits in the proton field in accordance with the relation (35)
selecting the plus sign before the radii. We consider the case when 1>>a that is equivalent to
1/ 0 >>rr . Just that very case is realized in the hydrogen atom when radii of Bohr orbits are
considerably greater than the electron classical radius. The relation (35) is transformed into the
form
.(82)
8
1
1=),1/(2= 21
22
1
a
Eaax −−
Equating the energy of stable Caplan orbits from (82) to the energy of stable Bohr orbits (81),
restricting by terms with 2
α and using (33) we obtain the expression
.(83)== 0
hn
crnm
M e
α
It follows from the last relation that Caplan stable orbits in hydrogen atom exactly councide with
Bohr orbits.
Let us consider the nearest to the cenrte stable Caplan orbit which in accordance with the above
mentioned has the value 2=2
a or 01 4= rr . For this case we find from formulas (35)
.(84)
32
27
=1E
Equating the energy from the last energy formula from the relation (80) we obtain
0.017.(85)=
5
27
= αn
It is clear that last formula does not satisfy the quantization conditions as n should be integral
and positive number. It tells that the developed Sommerfeld quantization method does not work
at short distances from the proton centre.
Concerning unstable Caplan orbits for the proton, none of them is not compatible with
Sommerfeld quantization conditions. Thus, the applied Bohr - Sommerfeld quantization method
in the Riemannian space results in the findings close to the Sommerfeld ones. It tells that the
proposed version of the new metric theory at least is not absurd.
In conclusion we point out that the proposed work is the further development of a new original
theory of bound charges which first was published in [2, 3, 14, 15]. Electrostatic field of bound
charges determines the Riemann space-time geometry non-connected with the Einstein’s general
relativity theory but restricting the domain of its applicability in some particular cases. This
proves in [15]. Similarly to general relativity theory the elementary problems are solved at the
field of bound charges and they are compared with the GRT results by means of the simplest
substitution of the coupling constants. Quantization of adiabatic invariants is realized. One of the
most valuable results of the theory of bound charges is the cancellation of divergence of the field
of point charges [16].
REFERENCES
[1] Podosenov S. A., (1982) in col. Discussion problems of the theory of relativity and gravitation,
Nauka, Moscow, pp. 95.
[2] Podosenov S. A., (2000) “The space, time and the classic fields of the bound structures,” Izd.

88
"Sputnik+" Company, Moscow.
[3] Podosenov S. A., Potapov A. A., Sokolov A. A., (2003) “The pulse electrodynamics of the wideband
radio systems and the fields of the bound structures,” "Radiotekhnika", Moscow.
[4] Podosenov S. A., (2011) “The new method of the field calculation in the space-time of the bound
structures,” Monograph, LAP LAMBERT Academic Publishing.
[5] Podosenov S. A., Foukzon J., Potapov A. A., (2010) “A Study of the Motion of a Relativistic
Continuous Medium, Gravitation and Cosmology”, Gravitation and Cosmology, Vol. 16, No 4, pp.
307-312.
[6] Podosenov S. A., Foukzon J., and Potapov A. A., (2012) “Electrodynamics of a Continuous Medium
in a System with Specified Structure”, Physics of Wave Phenomena, Vol. 20, No. 2, pp. 143-157.
[7] Podosenov S. A., (1997) Russian Physics Journal, Vol. 40, No. 10, pp. 985. Springer, New York,
ISSN 1064-8887 (Print), 1573-9228 (Online).
[8] Landau L. D., Lifshits E. M., (1973) “Field theory,” Nauka, Moscow.
[9] Zel’dovich Ia. B., Novickov I. D., (1967) “Relativistic astrophysics,” Nauka, Moscow.
[10] Zommerfel’d A., (1958) “Electrodynamics,” IL, Moscow.
[11] Bergman P. G., (1947) “Introduction in relativity theory,” IL, Moscow.- 380 p.
[12] Berestetskiy V. B., Lifshits E. M., Pitaevsckiy L. P., (1968) “Relativistic quantum theory,” Part 1,
Nauka, Moscow.- 480 p.
[13] Davidov A. S., (1963) “Quantum mechanics,” Fiz-matgiz, Moscow.- 748 p.
[14] Podosenov S. A., Potapov A. A., Foukzon J. and Men’kova E. R., (2014) “Geometry of Noninertial
Bases in Relativistic Mechanics Continua and Bell’s Problem Solution,” International Journal of
Recent Advances in Physics IJRAP. Wireilla Scientific Publications, Vol. 3, No. 1, pp. 23–37.
[15] Podosenov S. A., Foukzon J., Potapov A. A., and Men’kova E. R., (2014) “About Nonlinear Classic
Field Theory of Connected Charges,” International Journal of Recent Advances in Physics IJRAP.
Wireilla Scientific Publications, Vol. 3, No. 2, pp. 1–20.
[16] Podosenov S. A., (1997) “Space-time structure and fields of bound charges,” Izvestyia Vuzov, Vol.40,
No. 10, pp. 63-74.
Authors
Stanislav A. Podosenov (M’96) was born in Arkhangelsk, Russia in 1937. He received the degree in
theoretical physics in 1963 and the Ph. D. degree in relativistic mechanics of deformable media in 1972,
both from Moscow State University physical Faculty, Russia. From 1963 to 1970, he lectured physics at
Moscow State Technical University, where he completed his postgraduate study. Since 1970, he has been
working at the All-Russian Research Institute of Optophysical Measurements on problems of relativistic
mechanics and electrodynamics, noninertial frames of reference in special and general relativity, and
nonstationary neutron transfer theory. Presently, he is working in the field of EM pulses radiation theory
and their interaction with conducting bodies. Dr. Podosenov is the laureate of prize of the Russian
Federation government 2002 in the field of science and engineering.
Jaykov Foukzon was born in 1954. He is the Doctor of physical and mathematical Sciences, Professor of
Israel Institute of Technology, Haifa, Israel. The main subjects of scientific works: mathematical physics,
differential geometry of manifolds, transcendental number theory, non-classical set theory, deterministic
chaos and fractals, differential game theory, modern theory of homing and computer-aided design of
missilery.
Alexander A. Potapov was born in 1951. He is the Doctor of physical and mathematical Sciences. Dr.
Potapov has been working as a chief researcher at the Institute of radio engineering and electronics of
Russian Academy of Sciences named after V. A. Kotelnikov. He is working in the field of radio physics,
dispersion and diffraction of electromagnetic waves, radiolocation, recognition and image and signal
processing, determinate chaos, modern topology, fractal analysis, fractional operators, scaling effects. Dr.
Potapov is the academician of Russian Academy of Natural Sciences RANS and the academician of
Academy of Engineering Sciences named after A. M. Prokhorov. He is the professor of Kazan State
Technical University named after A. N. Tupolev. Dr. Potapov is the Chief Editor of the magazine
“Nonlinear world”, Member of editorial board of the international magazine “Electromagnetic
Phenomena”. He is Honorable radio operator of the Russian Federation. Dr. Potapov is the author of the
basic researches in the field of fractal approaches to radio physics, radio engineering and radar problems

89
and broad spectrum of adjacent scientific and technology paths. He is working in this field from 1979. Dr.
Potapov is the initiator of the pioneer researches and developments on fractal theory application in radio
physics, scaling effects and fractal operators in radio systems in Russia. He is the author of more than 400
scientific publications, includes 10 monographs.
Elena R. Men’kova was born in Moscow, Russia in 1955. She received the Ph.D. degree in 1984 from
Russian Chemical-technological University, Russia. Since 1984 she has been working as a Junior
Researcher, and then Senior Researcher at the All-Russian Research Institute of Optophysical
Measurements. Her research interest focuses on the works connected with radiation and measurement of
pulse electromagnetic fields.

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CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB FIELD OF BOUND CHARGES