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Article
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Mathematics and Physics
The End of Klein-Gordon Equation for Coulomb Field
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
The unitary principle is used to test the mathematical procedures and conclusions of the standard
theory of the Klein-Gordon equation in a Coulomb field, and it is revealed that the exact solution of the
Klein-Gordon equation in a Coulomb field hides the inexorable wave function divergence and virtual
energy difficulty. The divergence was not found in the past because it was concealed by an unnecessary
function transformation introduced in the process of solving differential equations. Since the expected
solution of the Klein-Gordon equation for a Coulomb field does not meet the boundary conditions,
but is only a pseudo solution, and there is no other exact solution of the Klein-Gordon equation
for the Coulomb field that meets the expectation of energy quantization, which declares the end of
the Klein-Gordon equation for the Coulomb field. This conclusion is irreversible. The Klein-Gordon
equation is constructed by the evolution of the operator of relativistic momentum energy equation.
It is the foundation equation of relativistic quantum force. The end of the desired solution indicates
that relativistic quantum mechanics and even relativistic mechanics are facing severe challenges.
Keywords: Unitary principle, Klein-Gordon equation, Inevitable solution, Pseudo solution, Wave
function divergence, Energy of imaginary number.
PACS number(s): 03.30.+p—Special relativity; 03.65.Pm—Relativistic wave equations;
03.65.Ge—Solutions of wave equations: bound states; 02.30.Gp—Sp ecial functions; 02.30.Hq—
Ordinary differential equations; 32.10.Fn—Fine and hyperfine structure.
1 Introduction
Dongfang unitary principle
[1, 1]
is a basic principle that
is universally applicable to the logical test of natural and
social sciences. Dongfang’s unitary principle is as fol-
lows: there is a certain transformation relationship be-
tween different metrics, and the natural law itself does
not change because of different metrics. If the math-
ematical form of natural law under different metrics is
transformed to one metrics, the result must be the same
as the inherent form under this metrics, 1=1, and the
transformation is unitary. The theory of natural science
must conform to the unitary principle. Although the
theory conforming to the unitary principle may not be
able to correctly describe the natural law, the assump-
tion, inference and even the whole theory that do not
conform to the unitary principle must be wrong.
For a theory or experiment report, different met-
rics are selected for reasoning. If the conclusions
drawn are inconsistent, it is exposed that this theoret-
ical or experimental report is illogical or inconsisten-
t with the facts. The assumption that the speed of
light is constant
[2-5]
, the quantum number of quantum
mechanics
[6-9, 11]
, the screw double black hole gravita-
tional wave theory of LIGO signal wave
[2, 10]
, Yukawa’s
nucleon meson theory
[6]
, etc. do not conform to the uni-
tary principle. End of Yukawa’s nuclear meson theory is
the result of the unitary principle test. Yukawa meson
theory repeatedly expresses the same equation in vari-
ous patterns mathematically, thus concealing the trace
of its theory of dismembering the operator wave equa-
tion evolved by the relativistic momentum and energy
relationship. Then, it defines the radial solution of the
dismembered partial differential equation as the nucle-
ar force potential energy function and defines the mass
parameter of the radial solution as the mass of meson.
Its logic is against the spirit of science. The so-called
nuclear force meson theory actually belongs to the cate-
gory of relativistic quantum mechanics. Does relativistic
quantum mechanics also hide mathematical or physical
logic problems that can be discovered through the uni-
tary principle test?
The Klein-Gordon equations
[15-21]
and the Dirac
equations
[22-24]
of relativistic quantum mechanics
[25-27]
constitute two mo dels of relativistic quantum mechanics,
but their inferences are inconsistent, which is inconsis-
tent with the basic principle that the natural law itself
does not change because of different selectivity metrics,
which means that relativistic quantum mechanics may
hide undiscovered logic problems. The exact solution of
the Klein-Gordon equation for the hydrogen atom does
not meet the expectation, while the solution of the Dirac
equation is consistent with the expectation. Because of
this, it is generally defined that the Klein-Gordon equa-
)Citation: Dongfang, X. D. The End of Klein-Gordon Equation for Coulomb Field. Mathematics & Nature 2, 202201 (2022).
202201-2 X. D. Dongfang The End of Klein-Gordon Equation for Coulomb Field
tion is suitable for zero spin particles and the Dirac equa-
tion is suitable for 1/2 spin particles. However, the Dirac
equation does not come from the relativistic mechanical
law of spin particles, and this definition of the scope of
application is farfetched. It does not conform to the u-
nitary principle to define the scope of application of the
equation according to different particles after the equa-
tion is constructed rather than at the beginning of the e-
quation construction. According to the basic principle of
quantum mechanics in which mechanical quantities are
replaced by operators to construct wave equations, spins
should have corresponding mechanical laws and wave e-
quations constructed with operator principles. What are
the classical mechanical laws and relativistic mechanical
laws of 1/2 spin particles? The Dirac theory, as the
mainstream of relativistic quantum mechanics, has no
answer to this question, while relativistic quantum me-
chanics only defines the scope of application of the Dirac
equation and the Klein-Gordon equation. The Klein-
Gordon equation is the operator evolution equation of
relativistic momentum equation, which should be the
foundation equation of relativistic quantum mechanics.
Some theories of modern physics and their develop-
ment theories often hide distorted mathematical calcu-
lations and biased explanations. The unitary principle
has strong logic insight, which enables us to constant-
ly find new problems and always find effective methods
to deal with problems and draw reliable conclusions. S-
tarting with this article, I will gradually introduce some
irreversible conclusions of relativistic quantum mechan-
ics tested by unitary principle.
2 Klein-Gordon equation of Coulomb field
After reading the Schr¨odinger equation of the hydro-
gen atom for the first time, we can easily associate the
relativistic wave equation of the hydrogen atom, which
is exactly the Klein-Gordon equation that will be read
later. In the era of paper-based literature, it is not like
now that every time you encounter a problem, you can
find a pleasing answer in Internet search. It always takes
too much time to consult paper documents, so many
problems must be reasoned independently. Adhering to
this habit can integrate power and enhance confidence
to reveal the truth. When I independently solved Klein-
Gordon equation in those years, I encountered difficulties
in exact solution divergence and imaginary number ener-
gy, and believed that this was a new challenge to relativ-
ity from the development of theory of relativity. Later,
I read the exact solution of the Klein-Gordon equation
of zero spin meson in the Coulomb field from Problems
in Quantum Mechanics, and learned that standard theo-
ry sometimes evades key problems and describes formal
calculation results as scientific conclusions. This is al-
so the motivation for further studying the truth of the
Dirac equation.
The exact solution of the Schr¨odinger equation in the
π
−
meson Coulomb field is highly similar to the exac-
t solution of the Schr¨odinger equation for the hydro-
gen atom. The only difference is their masses. The key
to accurately solving the Schr¨odinger equation is to ig-
nore the boundary condition of the atomic nucleus size,
which requires that the series solution be interrupted in-
to a polynomial, so that the energy quantized eigenvalue
can be obtained naturally. The formal exact solution of
the Klein Gordon equation of π
−
meson in the Coulom-
b field belongs to the standard theoretical solution, and
unnecessary function transformation is introduced in the
process of solving the equation, which is a key point. In
a Coulomb field with a nuclear or central charge number
of Z, the potential energy of the π
−
meson is
V = −
A~c
r
(1)
Where A = Zα, Z is the number of nuclear charges, α
is the so-called fine structure constant, and r is the dis-
tance from the electron to the nucleus. The relativistic
momentum energy relationship of particles in the field is
c
2
p
2
+ m
2
c
4
= (E − V )
2
(2)
Substitute equation (1), replace the momentum with the
momentum operator, and act on the wave function to ob-
tain the stationary Klein-Gordon equation of π
−
meson
−~
2
c
2
∇
2
+ m
2
c
4
ψ =
E −
A~c
r
2
ψ (3)
The form of this equation in spherical coordinates is
1
r
2
∂
∂r
r
2
∂ψ
∂r
+
1
r
2
sin θ
∂
∂θ
sin θ
∂ψ
∂θ
+
1
r
2
sin
2
θ
∂
2
ψ
∂ϕ
2
+
A
2
r
2
+
2AE
~cr
−
m
2
c
4
− E
2
~
2
c
2
ψ = 0 (4)
Separating variables, it supposes that
ψ = R (r) Y (θ, ϕ) (5)
Substituting it into equation (4) yields
1
R
d
dr
r
2
dR
dr
+
A
2
+
2AE
~c
r −
m
2
c
4
− E
2
~
2
c
2
r
2
= −
1
Y
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
+
1
sin
2
θ
∂
2
Y
∂ϕ
2
= λ (6)
Mathematics & Nature (2022) Vol. 2 No. 1 202201-3
The angular equation is Legendre equation
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
+
1
sin
2
θ
∂
2
Y
∂ϕ
2
+ λY = 0 (7)
The exact solution of the equation satisfying the boundary conditions is Y
lm
(θ, ϕ), and the eigenvalue is
λ = l (l + 1) , l = 0, 1, 2, ··· , m = 0, ±1, ±2, ··· , ±l
(8)
According to equation (6), the radial equation of the Klein-Gordon equation of Coulomb field is,
d
dr
r
2
dR
dr
+
A
2
− l (l + 1) +
2AE
~c
r −
m
2
c
4
− E
2
~
2
c
2
r
2
R = 0 (9)
This is a second-order ordinary differential equation.
3 The primitive solution of the radial equa-
tion and its termination
Quantum mechanics uses natural boundary condition-
s to determine the special solutions of Schr¨odinger equa-
tion, Klein-Gordon equation and other wave equations.
The boundary conditions for the wave equation of elec-
trons and π
−
mesons in the Coulomb field usually do not
consider the size of the atomic nucleus or central charge,
specifically,
lim
r→0
ψ = 0, ψ (0 < r < ∞) ̸= ±∞, lim
r→∞
ψ = 0 (10)
This boundary condition is actually a rough boundary
condition. The Klein Gordon equation of π
−
meson
treated by the standard theory usually imitates the solu-
tion of Schr¨odinger equation of the hydrogen atom, and
the following function transformation is introduced first,
R (r) =
H (r)
r
(11)
Substitute the function R and its corresponding deriva-
tive
dR
dr
=
1
r
dH
dr
−
H
r
2
,
d
dr
r
2
dR
dr
= r
d
2
H
dr
2
(12)
into equation (9), turning the radial equation into a sec-
ond order differential equation about H,
d
2
H
dr
2
−
m
2
c
4
− E
2
~
2
c
2
−
2AE
~c
1
r
+
l (l + 1) − A
2
r
2
H = 0
(13)
Then introduce the dimensionless variable
ξ = βr, β =
2
~c
m
2
c
4
− E
2
, ε =
AE
√
m
2
c
4
− E
2
(14)
Thus, equation (13) is reduced to a dimensionless equa-
tion
d
2
H
dξ
2
−
1
4
−
ε
ξ
+
l (l + 1) − A
2
ξ
2
H = 0 (15)
Finding the solution of equation (15) in the following
form
H = e
−ξ/2
u (ξ) (16)
Then the second order optimal differential equation for
u (ξ) is obtained
d
2
u
dξ
2
−
du
dξ
+
ε
ξ
−
l (l + 1) − A
2
ξ
2
u = 0 (17)
The boundary condition (10) of the wave function re-
quires that the series solution of equation (17) be inter-
rupted into polynomials, which is generally understood
as that function (16) satisfies the boundary (10). Let
the interrupted series solution of equation (17) to be
u (ξ) =
n
k=0
b
k
ξ
s+k
, s > 0, b
0
̸= 0 (18)
Its first and second derivatives are
du
dξ
=
n
k=0
(s + k) b
k
ξ
s+k−1
d
2
u
dξ
2
=
n
k=0
(s + k) (s + k − 1) b
k
ξ
s+k−2
Substitute these into the optimal differential equation
(17) to get
n
k=0
(s + k + 1) (s + k) − l (l + 1) + A
2
b
k+1
− (s + k − ε) b
k
n
k=0
b
k
ξ
s+k−1
= 0 (19)
202201-4 X. D. Dongfang The End of Klein-Gordon Equation for Coulomb Field
This relation determines the following recursive relations satisfied by the coefficients of the series
s (s − 1) − l (l + 1) + A
2
b
0
= 0
(s + 1) s − l (l + 1) + A
2
b
1
− (s − ε) b
0
= 0
.
.
.
(s + k + 1) (s + k) − l (l + 1) + A
2
b
k+1
− (s + k − ε) b
k
= 0
.
.
.
(s + n) (s + n − 1) − l (l + 1) + A
2
b
n
− (s + n − 1 − ε) b
n−1
= 0
(s + n − ε) b
n
= 0
(20)
The first equation is the index equation, which has
two roots
s =
1
2
−
l +
1
2
2
− A
2
1
2
+
l +
1
2
2
− A
2
(21)
The negative root obviously does not satisfy the bound-
ary, so it is abandoned. It is generally believed that
taking the positive root
s =
1
2
+
l +
1
2
2
− A
2
(22)
satisfies the boundary conditions. Then the coefficient
recursive relationship of the p olynomial is determined as
b
k+1
=
s + k − ε
(k + 1) (2 s + k)
b
k
(23)
Because the series breaks to the b
n
ξ
s+n
term, that is,
b
n+1
= 0, and b
n
̸= 0. The final form of equation (20)
yields s + n − ε = 0, that is,
1
2
+
l +
1
2
2
− A
2
+ n −
AE
√
m
2
c
4
− E
2
= 0 (24)
The quantized energy formula is thus obtained
E =
mc
2
1 +
A
2
(
n+
1
2
+
√
(
l+
1
2
)
2
−A
2
)
2
(25)
The above mathematical calculation process and conclu-
sion seem to be perfect.
Problems in Quantum Mechanics reviewed a period of
history about Klein Gordon. Shortly after Klein-Gordon
equation was proposed in 1926, in order to try to explain
the fine structure observed in the spectrum of hydrogen
like atoms, calculations similar to those given above were
made. The calculated results do not agree with the ex-
perimental observations; for the fine structure splitting
of the energy level with the principal quantum number,
the predicted energy broadening value is
E
n,n−1
− E
n,0
=
mZ
4
e
8
~
4
c
2
n
3
n − 1
n − 1/2
(26)
This is much larger than the experimental value. The re-
sults did not meet expectations. Later, it was explained
that the Klein-Gordon equation ignored the spin of the
electron and could only describe the spin free particles.
The formula for the fine structure should be very simi-
lar to equation (25) obtained by Dirac equation. Is that
really the case?
The solution of the equation can often inspire people
to find the deficiency of the equation. However, accord-
ing to the unitary principle, the scope of application of
an equation cannot be defined by the solution of the
equation. The quantum mechanical wave equation de-
scribing the particle spin must have a corresponding me-
chanical equation, so as to ensure that the consistency
of the operator principle of quantum mechanics is not
broken. From a mathematical point of view, the ba-
sis for the above standard theory to choose between the
two roots (21) of the index equation is to understand
the boundary condition (10) on wave function ψ as the
boundary condition on H (ξ),
lim
ξ→0
H ̸= ±∞, H (0 < ξ < ∞) ̸= ±∞, lim
ξ→∞
H = 0 (27)
This is unreasonable. The function substitution (11)
conceals that the radial wave function cannot obtain
the value satisfying the boundary strip (10). This can
be found by writing the complete formal wave function.
Note that A = Zα, substitute (16), (18) and (22) into
(11) to obtain the complete formal wave function
ψ = e
−
1
2
βr
n
k=0
b
k
β
1
2
+
√
(
l+
1
2
)
2
−Z
2
α
2
+k
r
√
(
l+
1
2
)
2
−Z
2
α
2
−
1
2
+k
Y
m
l
(θ, ϕ) (28)
Mathematics & Nature (2022) Vol. 2 No. 1 202201-5
Obviously, when l = 0
ψ
l=0
= e
−
1
2
βr
b
0
β
1
2
+
√
1
2
2
−Z
2
α
2
r
√
1
2
2
−Z
2
α
2
−
1
2
+
n
k=1
b
k
β
1
2
+
√
1
2
2
−Z
2
α
2
+k
r
√
1
2
2
−Z
2
α
2
−
1
2
+k
Y
m
l
(θ, ϕ) (29)
The solution of the equation diverges at the coordi-
nate origin:
lim
r→0
|ψ| = ∞ (30)
The standard theoretical solution of the Klein-Gordon
equation of Coulomb field do es not satisfy the boundary
condition (10).
Another key problem is that Klein-Gordon equation
of Coulomb field hides imaginary energy of anti-natural
law. The quantized energy formula (25) is investigated.
Reviewing the quantized energy formula (25), it is clear
that to ensure that the radical
(l + 1/2)
2
− Z
2
α
2
is a
real number, the following conditions must be met
l +
1
2
2
− Z
2
α
2
> 0 (31)
namely
Z 6
1
α
l +
1
2
(32)
In the case of l = 0, the specific result of this inequality
is Z 6 1/2α = 68.5, which means that when Z > 69,
there will be imaginary numbers of energy, which is not
in line with the fact.
In conclusion, the complete wave function of the
Klein-Gordon equation of the Coulomb field is only a
formal wave function, which does not meet the bound-
ary conditions used to determine the special solution of
the equation. The unnecessary function transformation
(14) is introduced in the process of obtaining the wave
function of this form, thus covering the divergence of
the wave function at the origin of coordinates. In fac-
t, without introducing the function replacement in the
form of (14), the second order differential equation (13)
about H (r) can be directly solved according to the above
method, and then substituted into (11) to obtain the
complete form of formal radial wave function. This for-
mal solution is exactly the expected solution, which is
merged as following:
R (r) =
1
r
e
−
Amc
~
√
(n+s)
2
+A
2
r
n
ν=0
b
ν
r
s+ν
b
ν
= −
2 (n − ν + 1) Amc
~ [(s + ν) (s + ν − 1) − l (l + 1) + A
2
]
(n + s)
2
+ A
2
b
ν−1
E =
mc
2
1 +
A
2
(s+n)
2
, s =
1
2
+
l +
1
2
2
− A
2
(l = 0, 1, 2, 3, ··· ; n = 0, 1, 2, 3, ···)
(33)
When l = 0, this formal radial wave function diverges
at the coordinate origin, so the exact solution cannot
meet the boundary conditions, which means that π
−
meson must fall into the atomic nucleus or the oppo-
site sign charge moving in the Coulomb field must fall
into the charge center of the Coulomb field. This is ob-
viously contrary to the fact of cosmic structure. On the
other hand, the appearance of virtual energy of the K-
lein Gordon equation of Coulomb field requires limiting
the number of nuclear charge or central charge, which
is not in accordance with scientific logic. Theoretically,
there can be a Coulomb field with a large number of cen-
tral charges, but it is impossible for an odd sign charge
to move in such a Coulomb field to generate imaginary
energy.
The inexorable divergence of the exact solution of the
Klein-Gordon equation and the imaginary quantized en-
ergy declares the end of the Klein-Gordon equation of
the Coulomb field.
4 The original solution of radial equation
and the end of solution
The function transformation (11) introduced by the
standard theory of the Klein-Gordon equation in solv-
ing the radial equation (9) diverts the reader’s attention
from logic, thus making the reader prone to an illusion
that the boundary conditions of the wave function can
be satisfied only by taking the positive root from the t-
wo roots of the index equation, so the complete form of
the equation solution at the origin of the coordinates is
covered up. If we directly solve the radial wave equa-
tion (9) without introducing function substitution (11),
we will easily find that the root of the index equation
202201-6 X. D. Dongfang The End of Klein-Gordon Equation for Coulomb Field
cannot make the exact solution meet the boundary con-
ditions. Many equations in modern physics, even if their
expressions are slightly different, it is necessary to calcu-
late and solve them in person or try to prove some basic
equations independently. From this, we will eventually
understand some of the logic and operation of modern
physics.
The general theory for solving the radial equation (9)
is to first optimize the equation and obtain the optimal
differential equation with respect to one factor of the
radial wave function, while the other factor is the ap-
proximate solution of the equation when r → ∞. The
simplified procedure is to set the radial wave function
R = e
−ar
Ω (34)
where
a =
√
m
2
c
4
− E
2
~c
(35)
Calculate the derivative of R and related terms
dR
dr
= e
−ar
dΩ
dr
− ae
−ar
Ω
d
dr
r
2
dR
dr
=
d
dr
r
2
e
−ar
dΩ
dr
− ae
−ar
Ω
= e
−ar
r
r
d
2
Ω
dr
2
− (2ar − 2)
d
Ω
dr
+ a (ar − 2) Ω
(36)
Then (34) and (35) are substituted into the radial equation (9) to obtain the optimal differential equation
r
2
d
2
Ω
dr
2
+
2r − 2ar
2
dΩ
dr
+
A
2
− l (1 + l) + 2
AE
~c
− a
r +
a
2
−
m
2
c
4
− E
2
~
2
c
2
r
2
Ω = 0
Using (35), the above equation is simplified as
r
2
d
2
Ω
dr
2
+
2r − 2ar
2
dΩ
dr
+
A
2
− l (1 + l) + 2
AE
~c
− a
r
Ω = 0 (37)
Seek the interrupted series solution of this equation
Ω =
n
ν=0
b
ν
r
s+ν
,
dΩ
dr
=
n
ν=0
(s + ν) b
ν
r
s+ν−1
,
d
2
Ω
dr
2
=
n
ν=0
(s + ν) (s + ν − 1) b
ν
r
s+ν−2
(38)
Substitute it into equation (37) to obtain
∞
ν=0
(s + ν) (s + ν + 1) + A
2
− l (1 + l)
b
ν
− 2
(s + ν) a −
AE
~c
b
ν−1
r
s+ν
= 0 (39)
Therefore, the coefficients of the series satisfy the recursive relation
s (s + 1) + A
2
− l (1 + l)
b
0
= 0
(s + 1) (s + 2) + A
2
− l (1 + l)
b
1
− 2
(s + 1) a −
AE
~c
b
0
= 0
.
.
.
(s + ν) (s + ν + 1) + A
2
− l (1 + l)
b
ν
− 2
(s + ν) a −
AE
~c
b
ν−1
= 0
.
.
.
(s + n) ( s + n + 1) + A
2
− l (1 + l)
b
n
− 2
(s + n) a −
AE
~c
b
n−1
= 0
2
(s + n + 1) a −
AE
~c
b
n
= 0
(40)
The above recursive relationship includes index equa-
tion and energy eigenvalue. The radial wave function
can be determined only after the index s and energy
eigenvalue are determined. The first equation is the in-
dex equation. The first equation is the index equation.
Because b
0
̸= 0, s (s + 1) + A
2
− l (1 + l) = 0. The two
Mathematics & Nature (2022) Vol. 2 No. 1 202201-7
roots of this equation are
s =
−
1
2
−
l +
1
2
2
− A
2
−
1
2
+
l +
1
2
2
− A
2
These are two roots of the original index equation, which
is obviously different from the two roots (21) of the
special-shaped index equation obtained by introducing
function transformation to solve the radial equation (9).
The boundary condition (10) of the wave function re-
quires that the negative root be discarded. The second
root is usually taken without thinking,
s = −
1
2
+
l +
1
2
2
− A
2
(41)
However, when l = 0, this root is actually a negative
number, so the radial wave function given by (34) take
the form
R = e
−
√
m
2
c
4
−E
2
~c
r
n
ν=0
b
ν
r
−
1
2
+
√
1
4
−A
2
+ν
namely
R =
e
−
√
m
2
c
4
−E
2
~c
r
r
1
2
−
√
1
4
−A
2
b
0
+
n
ν=1
b
ν
r
ν
(42)
The wave function boundary condition (10) cannot be
satisfied. This is because
lim
r→0
|R| = lim
r→0
e
−
√
m
2
c
4
−E
2
~c
r
r
1
2
−
√
1
4
−A
2
b
0
+
n
ν=1
b
ν
r
ν
= ∞
(43)
The exact solution of the Klein-Gordon equation of
Coulomb field diverges at the coordinate origin. From
the mathematical point of view, the radial equation (9)
does not meet the exact solution of the definite solution
condition; From a physical point of view, allowing the
radial wave function to diverge at the origin of coordi-
nates means that the universe collapses, which does not
conform to the fact. It has been explained previously
that when l = 0, if A > 1/2, that is, Zα > 1/2, and
the number of nuclear charges Z > 1/2α = 68 .5, the in-
dex s is an imaginary number, resulting in an imaginary
energy, which does not conform to the fact.
The last equation of the recursive relation (40) is
the energy eigenvalue equation. Since 2b
n
̸= 0,
(s + n + 1) a −AE/~c = 0, the formal energy eigenvalue
equation is obtained by substituting (35),
(s + n + 1)
m
2
c
4
− E
2
− AE = 0 (44)
Then substitute (41) into the above equation to obtain
the positive formal positive definite eigenvalue
E =
mc
2
1 +
A
2
(
n+
1
2
+
√
(
l+
1
2
)
2
−A
2
)
2
(45)
The result of substituting (41) into equation (45) is e-
quation (25). The non relativistic approximation of this
result is consistent with the Bohr energy level formu-
la. But its mathematical process is untenable, and the
hidden imaginary number difficulty cannot be solved.
The above formal solution of primitive state directly
exposes that the exact solution of the Klein-Gordon e-
quation of Coulomb field hides fatal problems that do
not meet the b oundary conditions of the wave equation
and exist imaginary energy, which seriously violates the
mathematical and physical meanings. Such dual mathe-
matical and physical logic difficulties cannot be avoided
by new definitions. Therefore, Klein-Gordon equation of
the Coulomb field has only formal solution, which cannot
be used to describe the quantum state of the Coulom-
b field. From (34), (35), (a2), (39), (42) and (45), the
formal solution of primitive state is merged as follows:
R = e
−
Amc
~
√
(n+s+1)
2
+A
2
r
n
ν=0
b
ν
r
s+ν
b
ν
= −
2Amc (n − ν + 1)
~ [(s + ν) (s + ν + 1) − l (1 + l) + A
2
]
(n + s + 1)
2
+ A
2
b
ν−1
E =
mc
2
1 +
A
2
(s+n+1)
2
, s = −
1
2
+
l +
1
2
2
− A
2
(l = 0, 1, 2, 3, ··· ; n = 0, 1, 2, 3, ···)
(46)
202201-8 X. D. Dongfang The End of Klein-Gordon Equation for Coulomb Field
Compared with (33), the indexes s in the two solutions
are different, but the final results are the same.
(33) and (46) are only the formal solutions of the
Klein-Gordon equation of the coulomb field. In addi-
tion, Klein-Gordon equation of the coulomb field has no
other reasonable solutions. Therefore, Klein-Gordon e-
quation of the coulomb field has no practical significance,
and all relevant deformation equations obtained by dis-
torting mathematics are just props for mass production
papers, which are meaningless. So, as the main equation
of relativistic quantum mechanics, what about the Dirac
equation?
5 Conclusions and comments
Based on the idea of unitary principle, the above con-
tent tests the relativistic Klein-Gordon equation of the
Coulomb field and proves that its exact solution does not
meet the boundary conditions, and the energy eigenval-
ue implies imaginary energy, which is a pseudo quanti-
zation energy. Specifically, the wave function follows the
statistical significance of Born’s interpretation, but the
Klein-Gordon ground state wave function of the Coulom-
b field is divergent, which means that the meson will fall
into the atomic nucleus or central charge when enter-
ing the Coulomb field, resulting in an increase in the
mass of the atomic nucleus or central charge. Applying
it to the atomic system means that the universe seems
to have collapsed long ago, obviously violate the laws of
nature. In the Klein Gordon equation of the Coulom-
b field, when the number of central charges or nuclear
charges exceeds 68, imaginary energy will appear. This
requires that the number of central charges or nucle-
ar charges in the Coulomb field model does not exceed
68, which obviously does not conform to the fact of the
atomic structure in nature. According to the unitary
principle, the Coulomb field is not necessarily limited to
the hydrogen like atom model. It can have a model with
a large enough central charge number. A particle with
a different sign charges move in a Coulomb field with a
large central charge number. This model can solve the
additional problems caused by the existence of spin or
not. It can be seen that there is no scientific basis for
specifying the Klein-Gordon equation to describe parti-
cles with zero spin. In a word, Klein-Gordon equation
of the Coulomb field has no eigen solution conforming to
physical meaning. The eigenwave functions and energy
eigenvalues of the Klein-Gordon equation of mesons giv-
en by the standard theory are pseudo solutions, which
determine the end of the Klein-Gordon equation of the
Coulomb fields. This irreversible conclusion urges us
to further test other relativistic quantization theories of
Coulomb field.
From the point of view of mathematical steps, the
function transformation introduced in (11) conceals the
problem that the exact solution of the Klein Gordon
equation of Coulomb field does not meet the wave func-
tion boundary conditions. If the wave function is allowed
to diverge, the index equation cannot choose between the
two roots. The so-called quantized energy has anoth-
er set of different eigenvalues, which does not conform
to the unitary principle, and the imaginary energy can-
not be eliminated. Without introducing the radial wave
function transformation, the Klein-Gordon equation of
the coulomb field is solved directly to obtain the primi-
tive solution. The problem that the boundary conditions
cannot be satisfied is obvious. From the perspective of
physical logic, if the relativistic momentum energy rela-
tionship is regarded as an accurate mechanical law, and
the basic principle of quantum mechanics to construct
wave equations by replacing mechanics with operators
is correct, then according to the unitary principle, the
Klein-Gordon equation derived from the combination of
special relativity and quantum mechanics becomes the
first-choice equation to describe the quantization law of
Coulomb field. Furthermore, according to the unitary
principle, if relativistic mechanics is regarded as a the-
ory that accurately describes high-speed motion, then
the relativistic description of physical laws is only differ-
ent in accuracy from that of Newtonian mechanics, and
there is no essential difference between them. However,
from the Schr¨odinger equation of quantum mechanics to
the Klein Gordon equation of relativistic quantum me-
chanics, the basic physical model of the Coulomb field
has changed substantially. The Schr¨odinger equation
has a solution that conforms to the physical meaning,
while the Klein-Gordon equation has no solution that
conforms to the physical meaning. This shows that the
relativistic Klein-Gordon equation does not conform to
the unitary principle.
The unitary principle can be used to test enough
mathematical paradoxes hidden in modern physics. The
reason for the existence of these mathematical paradoxes
may be that the basic mathematical equations of mod-
ern physics take the speed of light as the singularity.
That is, because of the theory of relativity. Time and
space are two basic physical concepts. However, modern
physics believes that relativistic mechanics is an accu-
rate theory because it reversely modifies the definition
of time and space with the relative speed and the speed
of light, while Newtonian mechanics follows the natu-
ral order that speed can only be defined based on the
concept of time and space first, so it can only be re-
garded as a low-speed similarity theory. In this way, the
high-speed quantum theory described by the relativis-
tic Klein-Gordon equation is the advanced theory of the
low-speed quantum theory described by the Schr¨odinger
equation, and the Schr¨odinger equation is regarded as
the low-speed approximation of the Klein-Gordon equa-
tion. However, the exact solution of the Klein-Gordon
equation is subversive compared with the exact solution
of Schr¨odinger equation and subverts the natural world.
The divergence of the standard solution of the Klein-
Mathematics & Nature (2022) Vol. 2 No. 1 202201-9
Gordon equation of the π
−
meson, which imitates the
Schr¨odinger equation in solving method, means the col-
lapse of the universe. The existence of its imaginary
energy requires the existence of an upper limit on the
number of nuclear charges. These deductions that vi-
olate the laws of nature prove that the Klein-Gordon
equation of the Coulomb field violates the unitary prin-
ciple and can only be terminated. Physical logic cannot
be cause and effect inverted because of personal wor-
ship, and time and space cannot be different because
of people or theories. The so-called Minkowski space-
time and Schwarzschild space-time are both artificially
defined. When Klein-Gordon equation is written into
the so-called Schwarzschild space-time form, it seems
infinitely profound and daunting. The college system
forces generations of students to recite random concepts,
equations and conclusions of modern physics, and even-
tually become one of the masters in this field of physic-
s. End of Yukawa’s nuclear meson theory and Klein-
Gordon equation today reflects the serious defects of the
college system. Those wrong physical theories are dif-
ficult to be discovered or even refused to be discovered
because of the product of the college system.
There is a simple truth that the development theory
of a theory whose basic principles hide logical contradic-
tions must contain more logical contradictions. Accord-
ing to the definition of relativistic quantum mechanics,
the simple harmonic oscillator and the zero-spin par-
ticle in the Coulomb field should be described by the
Klein-Gordon equation, while the 1/2 spin particle in
the Coulomb field should be described by the Dirac equa-
tion. This leads to a unitary problem. Does the 1/2 spin
simple harmonic oscillator model exist? The focus of sys-
tematic examination of relativistic quantum mechanics
is to examine the mathematical processing of the Klein-
Gordon equation and the Dirac equation. First of all,
we will discuss the Klein-Gordon equation of the zero-
spin π
−
meson in the Coulomb field and prove that the
inevitable solution of the Klein-Gordon equation of the
π
−
meson does not meet the boundary conditions and
implies imaginary energy, so it has no physical signifi-
cance, thus declaring the end of the Klein-Gordon equa-
tion of the Coulomb potential. However, a large number
of documents
[28-43]
claiming to have obtained the exact
solution of Klein-Gordon equation distort mathematics
and cover up the truth, so we will not comment on them
one by one.
Physical problems need to have clear conclusions, and
they can have clear conclusions. Historically, Klein-
Gordon proposed Klein-Gordon equation based on rel-
ativistic momentum and energy equations. Klein and
Gordon did not benefit from this, and they failed to find
the irrationality of the equation at that time only be-
cause of the limitations of mathematical treatment of
the equation. However, mass producers of later papers
benefited a lot from the Klein-Gordon equation. The
relativistic momentum energy relationship will prove to
be actually incorrect, although it is often claimed to be
confirmed by experiments that cannot be repeated. It
is not our interest and responsibility to criticize a large
number of reputable SCI journals that vigorously pro-
mote modern physics papers created by distorting math-
ematics, and such specific criticism is also likely to cause
general hostility. However, there will always be people
in the world who can and will be sure to carry out logical
tests to expose the distorted mathematics, sophistry and
lies of modern physics. Perhaps readers have to wonder
whether modern physics is under the control of a dif-
ferent religion. The distortion of mathematics to mass
produce papers may be caused by the pressure of work,
but these works are defined as excellent scientific theories
and breakthrough scientific achievements because they
are published by prestigious academic journals, which
puts the truth that is constantly discovered today and
tomorrow into a hopeless situation.
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Latest findings
The End of Teratogenic Simplified Dirac Hydrogen Equations
The two-component radial wave function of the Dirac equation of hydrogen is decomposed by linear combination function, which leads to the difference in the range of energy eigenvalues of the new first-order differential equations and the corresponding two simplified second-order differential equations constrained by the same energy parameter, belonging to the teratogenic simplified Dirac equation. The teratogenic simplification theory of the Dirac equation of hydrogen atom introduces the term ``decoupling", which is far from scientific logic, thus deleting a recursive relationship or corresponding second-order differential equation whose eigenvalue set does not meet the expectation, and only retaining the other one whose eigenvalue set value meets the expectation. Such an obvious logic problem has not been discovered or intentionally covered up, reflecting the real background of modern physics. Here, we first use the machine proof method to prove that the coefficient of the series solution of the teratogenic simplified first-order Dirac equation system should satisfy the linear recurrence relationship system without solution, which also proves that the teratogenic simplified first-order Dirac equation system has no eigensolution. Then the mathematical proof of the absence of solution of the teratogenicity simplified first-order Dirac equation system is given from different aspects. The simple truth is that the inconsistency of the eigenvalues of the two teratogenic simplified second-order differential equations destroys the existence and uniqueness theorem of the solutions of the differential equations. The essence of decoupling is to intentionally delete one of the two parallel second-order differential equations. The methods and conclusions do not conform to the unitary principle. It is concluded that the decoupled eigensolution of the teratogenic simplified first-order Dirac equation is pseudo-solution. The teratogenic simplified Dirac equation for the hydrogen-like atoms is therefore ended, and this conclusion is irreversible.