MATHEMATICS & NATURE
Mathematics,Physics, Mechanics & Astronomy
Random Vol. 2 Sn: 202202
https://mathnature.github.io
c
⃝ Mathematics & Nature–Free Media of Eternal Truth, China, 2022 https://orcid.org/0000-0002-3644-5170
.
Article
.
Mathematics and Physics
The End of Teratogenic Simplified Dirac Hydrogen Equations
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
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
corresp onding 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
mo dern 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
asp ects. 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.
Keywords: Dirac equation; linear combination function transformation; teratogenic simplified e-
quation; existence and uniqueness theorem of solution; unitary principle.
PACS number(s): 03.65.Pm—Relativistic wave equations; 03.65.Ge—Solutions of wave equations:
b ound states; 02.30.Gp—Special functions; 02.30.Hq—Ordinary differential equations; 32.10.Fn—
Fine and hyperfine structure
1 Introduction
As the enlightenment equation of relativistic quan-
tum mechanics
[1-7]
, the Klein-Gordon equation of the
Coulomb field has been ended because of the irreparable
contradiction hidden in logic
[8]
. Is there any logic con-
tradiction hidden in the Dirac equation or its evolution
equation, which is the mainstream equation of relativis-
tic quantum mechanics?
Dongfang unitary principle
[9-11]
is an effective logical
standard for testing the self-consistency of various theo-
retical logic, which is widely applicable to reveal whether
there are logical contradictions hidden in the theories of
natural science and social science. The reliability of this
principle lies in stating a simple fact: there is a cer-
tain transformation relationship between different met-
rics, and the natural law itself does not change because
of different metrics. If the mathematical form of natural
law under different metrics is transformed to one met-
rics, the result must be the same as the inherent form
under this metrics, 1=1, and the transformation is uni-
tary. The unitary principle has made many important
breakthroughs. Its application to quantum mechanics
has led to the discovery of the morbid equation of quan-
tum numbers
[10]
and the discovery of the incompatibility
of multiple operator evolution equations of the angular
motion law
[12]
, and led to the end of the Yukawa nu-
clear force meson theory and the end of Klein-Gordon
equation for the Coulomb field
[12]
.
As we all know, the relativistic Dirac equation is con-
sidered to be an accurate wave equation describing the
motion of microscopic particles. Its great influence and
the mystery of popular science description have led to
various named Dirac equations, some of which are not
)Citation: Dongfang, X. D. The End of Teratogenic Simplified Dirac Hydrogen Equations. Mathematics & Nature 2, 202202 (2022).
*The author published this article on the Internet with the title “MECHANICAL NON-SOLUTION PROVING OF FAKE DIRAC
EQUATION” and the signature Rui Chen in the early years, with many equations typing errors. The rewritten article has changed the
title and abstract and greatly revised the main body, indicating a clear scientific position.
202202-2 X. D. Dongfang The End of Teratogenic Simplified Dirac Hydrogen Equations
Dirac equations in essence, and the conclusions and even
methods are specious. In short, by comparing the solu-
tions of various Dirac equations, it is easy to find whether
the Dirac wave function and energy level formula of the
Coulomb field satisfy the existence and uniqueness the-
orem. Various Dirac equation theories do not conform
to the unitary principle, that is, they destroy the ex-
istence and uniqueness theorem for the solution of the
same physical model, and more than one of them must
contain significant logical defects.
New equations are obtained from the original Dirac
matrix equation
[1-5]
through function transformation,
some of which are reasonable, some are unreasonable
and even deformed. The solution of the deformed Dirac
equation is formal or pseudo-solution. However, in mod-
ern physics, when the formal solution of the equation
that does not conform to the mathematical op eration
law is consistent with the expected solution, the real
solution that conforms to the mathematical operation
law but does not conform to the expectation is often re-
moved by new definitions or interpretations. This devel-
opment pattern prevented later physicists from finding
wave equations that are more accurate. In fact, the es-
tablishment and solution of some equations in modern
physics imply enough irreconcilable mathematical con-
tradictions, even though the equations themselves may
not be correct, but they are rendered as major scientific
research achievements.
Here I will focus on the simplified equation of the dis-
tortion of the Dirac equation for the hydrogen atom.
First, the non-existence of the solution of the terato-
genic simplified first-order Dirac hydrogen equation is
proved by using the method of machine solving the re-
cursive relationship group and the method of mathemat-
ical logic proof. Then the non-existence of the solution
of the teratogenic simplified second-order Dirac hydro-
gen equation and the deceptive nature of the so-called
decoupling technique is analyzed. The problem origi-
nates from quantum mechanics, but its demonstration
process is purely mathematical.
2 First order teratogenic simplified Dirac
equation system
The treatment of a wave equation of quantum bound
state system often focuses on obtaining the expect-
ed quantized energy formula. To solve the bound s-
tate Schr¨odinger equation or Klein-Gordon equation, the
boundary condition requires that the power series solu-
tion must be interrupted to any polynomial, so the recur-
rence relationship
[13]
of the coefficients of the series needs
to be interrupted to any general term, and the eigenval-
ue set and the quantized energy formula are naturally
derived. Its strict mathematical basis is the differential
equation optimization theorem
[14, 15]
. To solve the Dirac
equation with at least two wave function components,
the original equation is usually transformed into a set of
differential equations with respect to the wave function
components. According to the boundary conditions, two
power series must also be interrupted into arbitrary poly-
nomials. This means that the two recurrence relation-
ship groups for determining the series coefficients must
be interrupted to any general term, thus the Dirac wave
function and Dirac energy level formula as the solution of
the Dirac equation are obtained. According to the the-
orem of optimal differential equation, the existence and
uniqueness of the solution of Schr¨odinger equation can
be proved. However, some non-orthodox second-order
Dirac equations and new first-order Dirac equations on-
ly focus on finding the Dirac formula of energy level-
s, and the given solutions actually violate the existence
and uniqueness theorem of the solutions of Dirac equa-
tions. These second-order and first-order equations are
substantially different from the original Dirac equation,
which is called the teratogenic simplified Dirac equation.
It is generally believed that the Dirac equation suc-
ceeds in describing the fine structure of hydrogen and
hydrogen-like atom
[16-18]
. From then on, multifarious
first-order and second-order Dirac equations
[19-27]
were
introduced more and more by much literature. It seem-
s very ingenious to introduce linear combination func-
tion transformation to transform the original first-order
Dirac equation system into a simplified first-order Dirac
equation system and then into a simplified second-order
differential equation. Its first-order and second-order e-
quations are precisely the teratogenic simplified Dirac
equation. A modern quantum mechanics book
[28]
in-
troduces its “decoupling” solution in detail. This sec-
tion discusses the nonexistence of the eigensolution of
the quasilinear linear recurrence relationship system de-
termined by the first-order teratogenic simplified Dirac
equation system for the hydrogen.
The Dirac equation is a four-order matrix first-order
differential equation system of four-component wave
function. When applied to the hydrogen atom, after
being transformed by special operation independent of
mathematics, the last thing to be dealt with is the radial
Dirac equation of two-component wave function in the
following form,
c
⇀
α · ˆp +
1 0
0 −1
mc
2
−
Ze
2
4πε
0
r
ψ
1
ψ
2
= E
ψ
1
ψ
2
(1)
where
⇀
α is the Dirac matrix, ˆp = −i~∇, ~ = h/2π with
the plank constant h, c the velocity of light in a vacuum,
and mthe rest mass of electron.
⇀
α · ˜p is defined by the
Dirac algebra
⇀
α · ˜p =
0 −i
i 0
−i~
∂
∂r
−
i~
r
+
i~
r
1 0
0 −1
ˆκ
(2)
Where the angular quantum number κ =
±1, ±2, ±3, ··· . The two-comp onent wave func-
Mathematics & Nature (2022) Vol. 2 No. 1 202202-3
tion is usually expressed in the following form,
ψ
1
ψ
2
=
1
r
F (r)
G (r)
(3)
The Coulomb interaction energy of a point nucleus and
a particle of charge −e is V = −e
2
r. Substituting (2)
and (3) into (1) yields the first-order differential equation
of the newly introduced component functions F (r) and
G (r), which is called the first-order radial teratogenic
simplified Dirac equation system,
dG
dr
+
κ
r
G −
E + m
0
c
2
~c
+
Zα
r
F = 0
dF
dr
−
κ
r
F +
E − m
0
c
2
~c
+
Zα
r
G = 0
(4)
where α = e
2
~c ≈ 1/137 is the fine-structure constant,
m
0
is the rest mass of the electron, cthe velocity of light
in vacuum, ~ = h/2π and h is the Plank constant; E is
the energy eigenvalue parameter. The expectant solu-
tion of this equation includes two terms: energy eigen-
value and eigenfunction. It should be noted that the
symbols in this equation set listed in different textbooks
are inconsistent, but this does not affect the demonstra-
tion process and conclusion of this paper.
Generations of researchers have worked hard to find
different methods to solve the Dirac equation for the hy-
drogen atom, including introducing function transforma-
tion and constructing new first-order and second-order
similar equations. There is a so-called “decoupling” so-
lution. First, function transformation is introduced to
transform the original Dirac equation into a first-order
differential equation set about new variables, then in-
to a second-order differential equation for solution, and
then a second-order differential equation that does not
meet expectations is removed. Various transformation
equations with transformation functions
[29]
are similar
or even identical in nature. The typical teratogenic sim-
plified Dirac equation process of hydrogen like atom is as
follows. Introducing linear combination function trans-
formation
ρ =
2
m
2
0
c
4
− E
2
~c
r
G (ρ) =
m
0
c
2
+ Ee
−
ρ
2
[ϕ
1
(ρ) + ϕ
2
(ρ)]
F (ρ) =
m
0
c
2
− Ee
−
ρ
2
[ϕ
1
(ρ) − ϕ
2
(ρ)]
(5)
and substituting this into equation (4) reads the equa-
tions,
dϕ
1
dρ
−
1 −
ZαE
~cλρ
ϕ
1
+
κ
ρ
+
Zαm
0
c
2
~cλρ
ϕ
2
= 0
dϕ
2
dρ
−
ZαE
~cλρ
ϕ
2
+
κ
ρ
−
Zαm
0
c
2
~cλρ
ϕ
1
= 0
(6)
where γ =
√
κ
2
− Z
2
α
2
, ν = 0, 1, 2, ··· , n. This is the
recursion relationship group that the coefficients of the
two series should meet, which is called the teratogenic
simplified recursion relationship group of the Dirac e-
quation. The decoupling method only selects one of the
recurrence relationships and deletes the other recurrence
relationship that cannot determine the eigenvalues. In
formal logic, Dirac formula (3) is obtained. So, does the
recurrence relationship group (7) really have an eigen-
solution? In the following, the machine method and the
mathematical deduction method are respectively used
to prove that the eigensolution of the teratogenic sim-
plified first-order differential equation system (6) of the
recurrence relationship group (7) does not exist.
ϕ
1
= ρ
γ
∞
ν=0
α
ν
ρ
ν
, ϕ
2
= ρ
γ
∞
ν=0
β
ν
ρ
ν
Inserting them into equation (6) and comparing the co-
efficients of similar terms yield two recurrence relations
ν + γ +
ZαE
~cλ
α
ν
+
κ +
Zαm
0
c
2
~cλ
β
ν
= α
ν−1
κ −
Zαm
0
c
2
~cλ
α
ν
+
ν + γ −
ZαE
~cλ
β
ν
= 0
(7)
where γ =
√
κ
2
− Z
2
α
2
, ν = 0, 1, 2, ··· , n. This is a
system of recurrence relations, which is called the ter-
atogenic simplified Dirac system of recurrence relations.
Solving this recurrence relation system is as cumbersome
as solving the recurrence relations of two second-order
differential equations. In the decoupling solution, the
latter is selected and a second-order differential equation
is selectively removed, and the eigenvalue (3) is obtained
in formal logic. This actually implies that the linear re-
currence relation group (7) has an eigensolution and the
eigenvalue is also Dirac formula (3). So does recurrence
relation group (7) really have an eigensolution? In the
following, the eigenvalues of recurrence relations (7) and
therefore the eigensolutions of the teratogenic simplified
first-order differential equations (6) are proved by the
machine method and mathematical deduction method
respectively do not exist.
3 Machine proof of no solution for ground
state and first excitation states
Wolfram Mathematica and other operational pro-
grams can well handle linear and quasilinear equations.
The existence of eigensolutions of quasilinear equations
(7) can be proved by Wolfram Mathematica program.
The following are the steps and conclusions of Wolfram
Mathematica to prove that the equation system (7) has
no solution. Substitute letters E → ε for machine recog-
nition, and substitute E → ε and λ =
m
2
0
c
4
− E
2
~c
into the recurrence relationship group (7), so as to ob-
tain the form expressed by four kinds of undetermined
parameters α
ν−1
, α
ν
, β
ν
and ε,
202202-4 X. D. Dongfang The End of Teratogenic Simplified Dirac Hydrogen Equations
ν +
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
ν
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
ν
= α
ν−1
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
ν
+
ν +
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
ν
= 0
(8)
The recurrence relationship group (7) or (8) is derived
from the teratogenic simplified Dirac equation (6), and
is not equivalent to the linear recurrence relationship
derived from the series solution of the original Dirac e-
quation.
Now let’s prove that the eigensolution of the cor-
responding ground state of the recurrence relationship
group (8) does not exist. Using the method of proof
to the contrary, if the eigensolution exists, then the
ground state solution must exist, and the recurrence re-
lationship group can be interrupted to n = 0, that is,
α
1
= α
2
= ··· = 0,β
1
= β
2
= ··· = 0. Noting that
α
−1
= α
−2
= ··· = 0, β
−1
= β
−2
= ··· = 0, when
shilling ν = 0 is substituted into the equation system (8),
the ground state solution requires α
0
̸= 0 and β
0
̸= 0,
and two homogeneous equations are obtained,
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
0
= 0
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
0
+
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
0
= 0
Then let ν = 1 be substituted into the first equation of
equation (8). The ground state solution requires α
1
= 0
and β
1
= 0, and then another particular equation is ob-
tained
α
0
= 0
This particular equation is easily ignored. The above
three equations give a set of contradictory equations.
We should have easily judged whether the solution of
this equation does not exist. However, various deforma-
tions hide this conclusion and are given solutions that
meet expectations, indicating that their expected solu-
tions are unreasonable and have been strongly attacked.
Such issues should have a standardized and systemat-
ic scientific procedure to reduce unnecessary disputes.
This is the reason for recommending machine certifi-
cation. The format for solving this equation set with
Mathematica is
Solve[{
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
0
== 0,
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
0
+
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
0
== 0,
α
0
= 0}, {α
0
, β
0
, ε}]
(9)
Run the calculation after outputting the above format
statement in Mathematica, and the output result is:
{{α
0
− >0,β
0
− >0}}. If the first term of the first two
equations is simplified from the last equation, a formal
energy eigenvalue can be obtained, which is equivalen-
t to the formal solution of the second equation, but it
does not meet the first equation. The results show that
the eigenvalues of the ground-state energy ε, i.e. E, do
not exist, while the recurrence relationship group (8) has
only trivial solutions, so the teratogenic simplified Dirac
equation group (6) has only trivial solutions. In modern
physics, once meaningless equations have been written
with some mathematical skills to meet the expectations
of some form of solution, physics readers usually focus
on the success of prediction, and rarely test whether the
relevant mathematical process is logical. This illogical
mathematical technique has been widely cited and pop-
ularized, which has created strong pressure for more re-
searchers to follow so as to achieve the expected goal
efficiently. The progress and development of physical
theory in this mode is very difficult.
In the same way, it can also be proved that the
first order transformation Dirac differential equations
(6) has no first excited state solution. Let the recur-
rence relationship group (8) break to n = 1, that is,
α
2
= α
3
= ··· = 0, β
2
= β
3
= ··· = 0, and in princi-
ple, the wave function of the first excited state requires
α
1
̸= 0, β
1
̸= 0. Noting that α
−1
= α
−2
= ··· = 0,
β
−1
= β
−2
= ··· = 0, substitute shilling ν = 0 into the
equation system (8), and use α
−1
= 0 to obtain
Mathematics & Nature (2022) Vol. 2 No. 1 202202-5
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
0
= 0
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
0
+
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
0
= 0
Then substitute ν = 1 into equation system (8) to get
1 +
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
1
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
1
= α
0
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
1
+
1 +
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
1
= 0
Then substitute ν = 2 into equation system (8), and use
α
2
= β
2
= 0 to obtain another particular equation
α
1
= 0
This special equation is destructive. The above five e-
quations give the equations to determine the ground s-
tate solution, which determines that the first excited s-
tate solution does not exist. Using the machine proof
method, the Mathematica solution format of this equa-
tion system is
Solve[{
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
0
== 0,
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
0
+
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
0
== 0,
1 +
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
1
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
1
== α
0
,
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
1
+
1 +
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
1
== 0,
α
1
= 0}, {α
0
, β
0
, α
1
, β
1
, ε}]
(10)
Running the calculation after outputting the above for-
mat statement in Mathematica, and there is no output
result for a long time. The reason may be that the energy
parameter is not solvable. Remove the energy parame-
ter from the required solving parameters and modify the
format of solving equations to
Solve[{
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
0
== 0,
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
0
+
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
0
== 0,
1 +
κ
2
− Z
2
α
2
+
Zαε
m
2
0
c
4
− ε
2
α
1
+
κ +
Zαm
0
c
2
m
2
0
c
4
− ε
2
β
1
== α
0
,
κ −
Zαm
0
c
2
m
2
0
c
4
− ε
2
α
1
+
1 +
κ
2
− Z
2
α
2
−
Zαε
m
2
0
c
4
− ε
2
β
1
== 0,
α
1
= 0}, {α
0
, β
0
, α
1
, β
1
}]
(11)
The calculation output is: {{α
0
− >0, β
0
− >0, α
1
− >0,
β
1
− >0}}. This indicates that the parameter represent-
ing energy cannot be solved, or even if the root of the
energy parameter is obtained, it is a formal solution and
does not conform to the physical meaning. The wave
function does not exist, and the energy parameters can
be assigned randomly. The process of pro cessing the
equation is different, and the formal solution obtained
202202-6 X. D. Dongfang The End of Teratogenic Simplified Dirac Hydrogen Equations
may be different, but the formal solution has no arbi-
trary meaning. Therefore, the energy eigenvalue of the
solution of the first excited state does not exist, while
the equation system has only trivial solutions, and the
recurrence relationship system (8) thus the differential
equation system (6) has no nontrivial solutions.
4 Logical contradictions of unreal solutions
of ground state and arbitrary energy state
The above machine proves that the teratogenic sim-
plified second-order Dirac equation system (6) has no
ground state solution and the first excited state solu-
tion. It also clearly shows that the teratogenic simplified
second-order Dirac equation has no other eigensolution-
s. The formal solutions given in relevant literature are
pseudosolutions. The differential equation without solu-
tion is given pseudo-solution, and its operation process
must hide the logical contradiction that violates the u-
nitary principle and form a mathematical paradox. We
have always recommended the unitary principle to re-
searchers. The unitary principle is the most effective
logical principle to test the wrong old theory and estab-
lish the correct new theory.
Now let’s discuss the reason why there is no solution
to the recurrence relation system satisfied by the coef-
ficients of the formal series solution of the first order
Dirac equation. First, the mathematical contradiction
implied in the eigenvalues of the teratogenic simplified
Dirac recurrence relation group (7) in the ground state
case will be revealed. It is assumed that the formal
ground state solution of the teratogenic simplified Dirac
equation (6) takes the form φ
01
= α
0
ρ
√
κ
2
−Z
2
α
2
and
φ
02
= β
0
ρ
√
κ
2
−Z
2
α
2
, substituting it into the system of
equations (6) yields the following system of algebraic e-
quations that the undetermined parameters α
0
, β
0
and
E
0
satisfy
κ
2
− Z
2
α
2
+
ZαE
0
m
2
0
c
4
− E
2
0
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− E
2
0
β
0
= 0
κ −
Zαm
0
c
2
m
2
0
c
4
− E
2
0
α
0
+
κ
2
− Z
2
α
2
−
ZαE
0
m
2
0
c
4
− E
2
0
β
0
= 0
α
0
= 0
(12)
Admittedly, this system of recurrence relations can also
be obtained by making ν = 0 and ν = 1 in the recur-
rence relation group (8), where α
−1
= 0. Clearly, the
third formula α
0
= 0 is just a negation to all unreal
solutions, because α
0
= 0 must read β
0
= 0, indicat-
ing that the ground state of the teratogenic simplified
Dirac equation does not exist. However, relevant litera-
ture avoids the result of α
0
= 0 caused by β
0
= 0 and
transfers to the general case of discussing any energy s-
tate, thus forming a formal mathematical logic. For the
ground state case, the reasoning carried out by this for-
mal mathematical logic is equivalent to substituting the
third formula of (12) into the first and second formulas,
because it is easy to create a false image to obtain the
formal non-trivial solution, and there should be β
0
̸= 0
and then
α
0
= 0, β
0
̸= 0
κ +
Zαm
0
c
2
m
2
0
c
4
− E
2
0
β
0
= 0
κ
2
− Z
2
α
2
−
ZαE
0
m
2
0
c
4
− E
2
0
β
0
= 0
(13)
Dirac theory has defined κ = ±1, ±2, ±3, ··· through
special mathematical steps. This system of equations
has two incompatible solutions. From the second equa-
tion and the third equation,
m
2
0
c
4
− E
2
0
= −
Zαm
0
c
2
κ
m
2
0
c
4
− E
2
0
=
ZαE
0
√
κ
2
− Z
2
α
2
(14)
Obviously, the two different results do not conform to
the unitary principle. In addition, the first result itself
destroys the unitary principle because the different def-
inition domains of positive and negative values of Dirac
angular quantum numbers also lead to contradictory val-
ue domains. The conclusion that destroys the unitary
principle constitutes a mathematical paradox of 1 ̸= 1.
If we mechanically eliminate the root sign on the left of
the equation without thinking, further calculation will
lead to more contradictions. However, under the decou-
pling operation of the teratogenic simplified Dirac equa-
tion theory, the first solution corresponding to the imag-
inary number energy that is not tenable is deleted, and
only the second formal solution of (14) is taken, giving
the same energy eigenvalue as the Dirac formula, which
has been widely recognized by the academic community
and has been continuously developed. Middle school s-
tudents all know that this solution is completely wrong.
Why was it considered a deliberate attack on outstand-
ing research achievements to criticize such serious and
low-level mathematical mistakes in the early years and
submit them to famous academic journals? Defining the
Mathematics & Nature (2022) Vol. 2 No. 1 202202-7
term “decoupling” to take the second of the two results
in (14) and convert it into the Dirac energy level formu-
la seems to be a small but fatal mathematical error! In
fact, the first two equations of equation system (12) for-
m a formal linear homogeneous equation system, which
satisfies the necessary and sufficient conditions for the
existence of nontrivial solutions,
√
κ
2
−Z
2
α
2
+
ZαE
0
√
m
2
0
c
4
−E
2
0
κ+
Zαm
0
c
2
√
m
2
0
c
4
−E
2
0
κ−
Zαm
0
c
2
√
m
2
0
c
4
−E
2
0
√
κ
2
−Z
2
α
2
−
ZαE
0
√
m
2
0
c
4
−E
2
0
=0
(15)
Only the non-trivial solutions of α
0
and β
0
given by the
first two equations in (12) are denied by the last equation
in (12). This contradiction is summed up in the follow-
ing concise form {α
0
̸= 0, β
0
̸= 0} ∩ {α
0
= 0}. This is
why the machine calculation (9) cannot solve the un-
known number z that expresses the energy eigenvalue
parameter E
0
. One often pays attention to that if the
formula of energy agrees with the Dirac formula, but
not attaches importance to that if the logic is correct.
In fact, α
0
= 0 in the equations (12) implies that the
factor of the wave function (5) for the ground state is
inadvertently written as the form
G
0
(ρ) =
m
0
c
2
+ Ee
−ρ/2
ϕ
02
(ρ)
F
0
(ρ) = −
m
0
c
2
− Ee
−ρ/2
ϕ
02
(ρ)
(16)
As we all know, the Dirac equation for the hydrogen
atom has no such solution. In other words, Dirac wave
function cannot be decomposed in this way.
Quantum mechanics accurately solves the wave equa-
tion satisfied by the bound state quantum system to
obtain the energy eigenvalues. However, the so-called
decoupling method is essentially just to avoid the two
contradictory formal solutions and choose the one that
meets the expectation. Decoupling and removing any e-
quation in the second order teratogenic simplified Dirac
equation set is not in line with scientific logic. Why
can all the literature consistently retain the equation
that meets the expectation? In short, there are enough
mathematical paradoxes hidden in the logic of the pseu-
doeigensolution (14) of the ground state of the variable
Dirac equation, which has been covered up by the math-
ematical deformation process and must be explicitly de-
nied now.
If a theory hides the basic contradiction such as 1 =
−1 or 1 ̸= 1, it must be wrong from beginning to end.
Now let’s prove that there is no arbitrary energy state so-
lution for the first-order transformation Dirac equations.
An incorrect theory often gives some specious deduction
that cannot be distinguished from the correct theory, its
mathematical errors are often ignored by readers. For
the system of recurrence relations (7), one often pays at-
tention to the general case to find the eigen-solution to
the system of differential equations (6) for the n-excited
state,
ϕ
1
= α
0
ρ
γ
+ α
1
ρ
γ+1
+ ··· + α
n−1
ρ
γ+n−1
+ α
n
ρ
γ+n
ϕ
2
= β
0
ρ
γ
+ β
1
ρ
γ+1
+ ··· + β
n−1
ρ
γ+n−1
+ β
n
ρ
γ+n
(17)
Formally, inserting (17) into (6) and using γ =
√
κ
2
− Z
2
α
2
as well as λ =
m
2
0
c
4
− E
2
~c, one ob-
tains the system of linear algebraic equation for five un-
determined parameters α
0
, β
0
, α
1
, β
1
and E
1
, it is
κ
2
− Z
2
α
2
+
ZαE
2
m
2
0
c
4
− E
2
n
α
0
+
κ +
Zαm
0
c
2
m
2
0
c
4
− E
2
n
β
0
= 0
κ −
Zαm
0
c
2
m
2
0
c
4
− E
2
n
α
0
+
κ
2
− Z
2
α
2
−
ZαE
2
m
2
0
c
4
− E
2
n
β
0
= 0
1 +
κ
2
− Z
2
α
2
+
ZαE
2
m
2
0
c
4
− E
2
n
α
1
+
κ +
Zαm
0
c
2
m
2
0
c
4
− E
2
n
β
1
= α
0
κ −
Zαm
0
c
2
m
2
0
c
4
− E
2
n
α
1
+
1 +
κ
2
− Z
2
α
2
−
ZαE
2
m
2
0
c
4
− E
2
n
β
1
= 0
.
.
.
n +
κ
2
− Z
2
α
2
+
ZαE
2
m
2
0
c
4
− E
2
n
α
n
+
κ +
Zαm
0
c
2
m
2
0
c
4
− E
2
n
β
n
= α
n−1
κ −
Zαm
0
c
2
m
2
0
c
4
− E
2
n
α
n
+
n +
κ
2
− Z
2
α
2
−
ZαE
2
m
2
0
c
4
− E
2
n
β
n
= 0
α
n
= 0
(18)
202202-8 X. D. Dongfang The End of Teratogenic Simplified Dirac Hydrogen Equations
Of course, this system of equations can also be obtained
directly from the system of linear recurrence relations
(7). The first and second equations requires that the
determinant of the coefficient equal zero, this has no
problem. However, combining the last three formulas of
(17) gives
κ +
Zαm
0
c
2
m
2
0
c
4
− E
2
n
β
n
= α
n−1
n +
κ
2
− Z
2
α
2
−
ZαE
2
m
2
0
c
4
− E
2
n
β
n
= 0
(19)
in order to obtain the formal non-trivial solution, let
β
1
̸= 0, it must order that the coefficients before β
n
e-
quals zero, n +
√
κ
2
− Z
2
α
2
−ZαE
2
m
2
0
c
4
− E
2
= 0,
producing the eigenvalues of the energy levels for the
n-excited state
E
n
=
m
0
c
2
1 +
Z
2
α
2
(
n+
√
κ
2
−Z
2
α
2
)
2
(20)
It very coincidentally gives the Dirac energy level for-
mula. However, this result does not make the equa-
tion group (18) tenable. The inverse recurrence oper-
ation will give the conclusion that eitherβ
0
= 0 and
α
−i
̸= 0 (i = 1, 2, 3, ···) contradicting with the formula
(17) or β
0
= 0 and α
−1
= 0 becoming the ground state
negative solution (13). Therefore, there is no arbitrary
energy state solution for the first-order teratogenic sim-
plified Dirac equations. Relativistic quantum mechanics
usually only pays attention to fitting the expected ener-
gy level formula, but does not pay attention to whether
the solution of the equation can meet the equation itself,
which is an urgent problem to be solved.
5 General proof of no solution for terato-
genic simplified Dirac recurrence relations
Here let’s discuss the general solution of the corre-
sponding recurrence relation group (7), so as to prove
the nonexistence of the real solution of the first-order ter-
atogenic simplified Dirac equation. The process of find-
ing the solution of the coupled recurrence relationship
is usually complex, but for two series (17) as the formal
solution, the formal recurrence relationship group corre-
sponding to the series coefficient can be transformed into
two uncoupled recurrence relationships, and the eigen-
value solution method is very simple. Directly elimi-
nating the coefficient β
ν
in the equation (7) gives the
uncoupled recurrence relationship of the coefficient α
ν
in the power series ϕ
1
,
α
ν
=
ν + γ −
ZαE
~cλ
(ν + γ)
2
− κ
2
+ Z
2
α
2
α
ν−1
(21)
Then eliminating the coefficient α
ν
in the equation (7)
gives another uncoupled recurrence relations for the co-
efficient β
ν
in the power series ϕ
2
. The second equation
in (7) can be written in the following form
α
ν
= −
~c (ν + γ) λ − ZαE
~cκλ − Zαm
0
c
2
β
ν
(22)
consequently
α
ν−1
= −
~c (ν − 1 + γ) λ − ZαE
~cκλ − Zαm
0
c
2
β
ν−1
(23)
substituting (22) and (23) into the first equation (7)
reads
β
ν
=
ν − 1 + γ −
ZαE
~cλ
(ν + γ)
2
− κ
2
+ Z
2
α
2
β
ν−1
(24)
Consequently, the system of recurrence relations (7) is
equivalent to two uncoupled recurrence relations of first
order(21) and (24). Because having the same energy
eigenvalue parameters, they compose a system of equa-
tions
α
ν
=
ν + γ −
ZαE
~cλ
(ν + γ)
2
− κ
2
+ Z
2
α
2
α
ν−1
β
ν
=
ν − 1 + γ −
ZαE
~cλ
(ν + γ)
2
− κ
2
+ Z
2
α
2
β
ν−1
(25)
On the surface, these two recurrence relations are in-
dependent of each other. But in fact, they contain the
same parameter E representing energy, so they are in-
separable. However, in their respective equations, the
eigenvalues must meet the recurrence law of each self.
It is supposed that the linear recurrence relations ter-
minate at the term of ν = n, namely, α
n
̸= 0, β
n
̸= 0
and α
n+1
= α
n+2
= ··· = 0, β
n+1
= β
n+2
= ··· = 0.
Substituting for equations (25) and using the sign λ =
m
2
0
c
4
− E
2
~c, we obtain
E
n
=
m
0
c
2
1 +
Z
2
α
2
(
n+1+
√
κ
2
−Z
2
α
2
)
2
E
n
=
m
0
c
2
1 +
Z
2
α
2
(
n+
√
κ
2
−Z
2
α
2
)
2
(26)
where κ = ±1, ±2, ··· and n = 0, 1, 2, ···, two formu-
la are similar, nevertheless are actually different. When
looking from a mathematical point of view, two formu-
la have two different eigenvalues sets, the one and only
eiegenvalues parameter of the same wave equation must
satisfy two different eigenvalues sets, it can but choose
their intersection. However, when looking from a physi-
cal point of view, the intersection delete energy of ground
state, falling short of the natural law. In fact, the energy
eigenvalues of the ground state given by the first formu-
la is just the energy eigenvalues of the first excited state
given by the second formula, this is very inexplicable.
Mathematics & Nature (2022) Vol. 2 No. 1 202202-9
For the same quantum system satisfying the Dirac wave
equation, different solution method produce two differ-
ent energy eigenvalues sets, the formulas (26) ever arose
the profound misconception. There are some antagonis-
tic points of view considering that making substitution
n + 1 → n for the first formula it is just the second
formula. However, as n = 0, this substitution implies
0 + 1 → 0. As the subscript of the coefficient of series,
the natural number n cannot be allowed to make such
substantiation. The different two results of the equa-
tions (26) denotative the solutions of the same energy
eigenvalue parameter, it should order
m
0
c
2
1 +
Z
2
α
2
(
n+1+
√
κ
2
−Z
2
α
2
)
2
?
=
m
0
c
2
1 +
Z
2
α
2
(
n+
√
κ
2
−Z
2
α
2
)
2
(27)
implying 1 = 0, which destroys the existence and unique-
ness theorem of the solution and does not accord with
the unitary principle. Therefore, it is not reasonable and
effective to choose any of formal solutions that are con-
tradictory. Teratogenic Simplified Dirac equation has no
solution. There are too many interpretations, false qual-
itative descriptions and even pure sophistry in modern
physics, which can be exposed and refuted by the unitary
principle, and it is the best choice to give a conclusion
through mathematical calculation.
The theoretical creators and promoters of the terato-
genic simplified Dirac equation based on the teratogenic
simplified Dirac wave function derive two contradicto-
ry second-order differential equations of hydrogen, and
then use the term “decoupling” to understate and dilute
the reader’s attention. Remove one equation whose for-
mal solution does not meet the expected solution and
retain another differential equation whose formal solu-
tion meets the expected solution, and obtain the Dirac
energy level formula, which is only a false solution, the
obtained wave functions and energy eigenvalues do not
actually satisfy the teratogenic simplified Dirac equa-
tion. The mathematical logic proof of the existence of
the solution of the equation is strict. Machine proof
only gives us enlightenment to find the reason for the
result of machine proof. The machine proved that the
teratogenic simplified Dirac equation has only trivial so-
lutions. However, if the equation system is simplified
first, the machine processing of the simplified equation
system may also output the formal solution that only
satisfies one equation in the equation system and does
not satisfy other equations, which will give us the illu-
sion that the formal solution is the real solution. This
is the reason why the logic proof is further given when
the Dirac equation of transformation has been proved to
be no solution by machine. It must be pointed out that
the exact solution of the original second-order Dirac e-
quation with the exact boundary conditions
[30, 31]
is the
more important issues to be discussed.
6 Deception of Decoupling second order
teratogenic simplified Dirac Equation
In modern physics, there is a widespread phenomenon
of seriously distorting mathematics and creating pseudo-
scientific achievements. Now, the decoupling technique
of the distorted simplified second-order Dirac equation is
taken as an example to illustrate the deceptive nature of
the physical theory that distorts mathematics. Terato-
genic simplified second-order Dirac equation is derived
from teratogenic simplified first-order Dirac equation set
(6). Write the replacement forms of the two functions
according to the equation set (6),
ϕ
1
=
1
κ −
Zαm
0
c
2
~cλ
−ρ
dϕ
2
dρ
+
ZαE
~cλ
ϕ
2
ϕ
2
=
1
κ +
Zαm
0
c
2
~cλ
−ρ
dϕ
1
dρ
+
ρ −
ZαE
~cλ
ϕ
1
(28)
Differential each expression once and get
dϕ
1
dρ
=
−ρ
d
2
ϕ
2
dρ
2
+
ZαE
~cλ
− 1
dϕ
2
dρ
κ −
Zαm
0
c
2
~cλ
dϕ
2
dρ
=
−ρ
d
2
ϕ
1
dρ
2
+
ρ −
ZαE
~cλ
− 1
dϕ
1
dρ
+ ϕ
1
κ +
Zαm
0
c
2
~cλ
(29)
Substitute (28) and (29) into the teratogenic simplified
first-order Dirac equations (6) to yield two teratogenic
simplified second-order Dirac equations,
ρ
2
d
2
ϕ
1
dρ
2
−
ρ
2
− ρ
dϕ
1
dρ
−
κ
2
+
Z
2
α
2
~
2
λ
2
E
2
c
2
−m
2
0
c
2
+
1−
ZαE
~cλ
ρ
ϕ
1
=0
ρ
2
d
2
ϕ
2
dρ
2
−
ρ
2
− ρ
dϕ
2
dρ
−
κ
2
+
Z
2
α
2
~
2
λ
2
E
2
c
2
− m
2
0
c
2
−
ZαE
~cλ
ρ
ϕ
2
= 0
(30)
According to the boundary conditions, the wave function
is b ounded, and the series solution of the two deformed
simplified second-order equations must be interrupted to
a polynomial of any term. However, the two Dirac equa-
tions are constrained by the same energy parameter E,
and they are actually a system of second-order differen-
tial equations. And the solutions of the two equations
must also satisfy the first-order equations (6) and the
original first-order Dirac equations (4). Therefore, the
highest power terms of the interrupted series solutions
of the two equations are the same, respectively set as
φ
1
=
n
ν=0
b
ν
ρ
s+ν
, φ
2
=
n
ν=0
d
ν
ρ
s+ν
Substitute them into the two equations of (30) to get
202202-10 X. D. Dongfang The End of Teratogenic Simplified Dirac Hydrogen Equations
n
ν=0
(s + ν)
2
− κ
2
−
Z
2
α
2
~
2
λ
2
E
2
c
2
− m
2
0
c
2
b
ν
−
s + ν −
ZαE
~cλ
b
ν−1
ρ
s+ν
= 0
n
ν=0
(s + ν)
2
− κ
2
−
Z
2
α
2
~
2
λ
2
E
2
c
2
− m
2
0
c
2
d
ν
−
s + ν − 1 −
ZαE
~cλ
d
ν−1
ρ
s+ν
= 0
Then one gets a recursive relationship group composed of two recursive relationships that appear to be independent
of each other but are actually constrained by the same energy parameter,
(s + ν)
2
− κ
2
−
Z
2
α
2
~
2
λ
2
E
2
c
2
− m
2
0
c
2
b
ν
−
s + ν −
ZαE
~cλ
b
ν−1
= 0
(s + ν)
2
− κ
2
−
Z
2
α
2
~
2
λ
2
E
2
c
2
− m
2
0
c
2
d
ν
−
s + ν − 1 −
ZαE
~cλ
d
ν−1
= 0
(31)
In the recursive relationship group (31), let ν = 0, and
note that b
−1
= d
−1
= 0, while b
0
̸= 0 and d
0
̸= 0,
so the two equations have the same index equation,
s
2
−
κ
2
+
Z
2
α
2
E
2
~
2
c
2
λ
2
−
Z
2
α
2
m
2
0
c
2
~
2
λ
2
= 0. Take the positive
root that conforms to the bounded boundary condition
of the wave function from the two roots,
s =
κ
2
+
Z
2
α
2
E
2
~
2
c
2
λ
2
−
Z
2
α
2
m
2
0
c
2
~
2
λ
2
(32)
Then, in the recursion relation group (31), let v = n + 1,
and notice that b
n+1
= d
n+1
= 0, while b
n
̸= 0 and
d
n
̸= 0, so two inconsistent eigenvalue equations are ob-
tained,
s + n + 1 −
ZαE
~cλ
b
n
= 0
s + n −
ZαE
~cλ
d
n
= 0
(33)
where λ =
m
2
0
c
4
− E
2
~c. Middle school students are
good at dealing with such equations. No matter what
value E takes, it is impossible to satisfy two of these e-
quations at the same time, which means that the wave
function thus obtained cannot satisfy the first-order e-
quations (6) and the original Dirac equations (4). A
large number of related mathematical calculations are
invalid. However, the theoretical creator of the terato-
genic simplified Dirac equation defined a term “decou-
pling”, deleted the first equation of (30) and retained
the second equation, thus logically listing only the sec-
ond recursive relationship of (31), and then taking only
the second eigenequation of (33), it seems that the Dirac
energy level formula of hydrogen can be reasonably ob-
tained. This pseudo-science operation is very deceptive.
If the “decoupling” technique can be incorporated into
mathematics, then all the equations will be changed in-
to one equation. Physicists who deny or even slander
the above reasoning and conclusions should ask middle
school students how to read so as not to make the lowest
level of mathematical mistakes and should not indulge
in their dazzling degrees and various titles all day long.
Famous journals only publish gossip articles with un-
clear logic, calculation errors and fabricated observation
data, but they always refuse to make breakthroughs in
correct discovery. Can editors of famous scientific jour-
nals such as Nature set an example, resign collectively,
and hand over the disposal of those journals with the
high reputation to the Great Mouse team, so as to save
sacred science and rebuild academic ethics?
7 Conclusions and comments
Did one
[32-36]
ever note that the Dirac hydrogen wave
function was practically unable to b e separated into the
form (5) and its similar forms? According to the unitary
principle, no matter what transformation is introduced
to transform the Dirac equation into the teratogenic sim-
plified Dirac equation satisfied by the new function, if
the determined new function is replaced back into the
introduced transformation, the result must be the same
as the original solution of the Dirac equation. The ex-
pected solution of the original Dirac equation cannot be
decomposed into the form of (5), so it is incorrect to
introduce function transformation (5). The formal solu-
tion of the transformation equation obtained has neither
mathematical nor physical significance. Therefore, we
do not need to carry out so many calculations and use
the unitary principle to directly draw the conclusion that
the teratogenic simplified solution is a pseudo-solution of
Dirac’s hydrogen equation. From this, we can also ap-
preciate the great role of the unitary principle.
The nonexistence of the real solution of the terato-
genic simplified Dirac equation should be a very simple
mathematical problem and the similar problems implicit
in theoretical physics are far more than the formal so-
lution of the teratogenic simplified Dirac equation. The
abnormal treatment of the Dirac equation is a typical
example that the solution of modern physical equation
violates the rules of a mathematical operation. The in-
ference of the formal theory of ultra-small micro mo-
tion law and ultra large macro motion law described by
modern physics is often similar to the inference of the
Mathematics & Nature (2022) Vol. 2 No. 1 202202-11
stifled correct theory, and the distinction between the
two is limited in experimental observation, because the
two types of inferences have the same order of magnitude
of accuracy. Modern physics research is always keen on
the experiments of major projects while neglecting log-
ical tests. Experimental observations without causality
or even generated data have been publicized to verify the
theory that is actually wrong
[37-39]
. This is the funda-
mental reason for the stagnant development of theoreti-
cal physics in recent decades. Most of the calculation of
the teratogenic simplified solution is invalid rather than
wrong, which means that the calculation of each step
is in accordance with the mathematical rules except for
the “decoupling”. However, the phenomenon of distort-
ing mathematics, concealing errors and fabricating ob-
servation data in modern physics is very serious. The
reason why I have analyzed in detail the acts of invalid
calculation, incorrect calculation, conclusive falsification
and experimental report falsification in modern physics
is that I hope that the disclosure of cunning pseudosci-
entific acts will no longer be subjected to unprovoked
attacks and slanders. Present the correct calculation re-
sults and conclusions to the world, and the future main-
stream ethical scientists will objectively comment. In
the future, the scientific world would not have been as
dark as in the past and now.
Since 1985, we have revealed and corrected some
mathematical errors implied in theoretical physics. In a
few published articles in Chinese
[40]
and English
[41]
, the
description of such problems is relatively euphemistic,
and there is no clear explanation that some calculations
are wrong, which may be a reason for failing to attract
attention. To reveal some problems existing in theoret-
ical physics, we usually need to make many aspects of
argumentation and more than ten kinds of calculation-
s. Only when the results are consistent can we give a
conclusion. These problems are found in testing rele-
vant logic with the unitary principle, some are qualita-
tive tests, and some are quantitative tests. However,
it will take quite a long time to widely test and cor-
rect the mathematical errors and specious conclusions
implied in theoretical physics. Using Wolfram Mathe-
matica to test the absence of the solution of teratogenic
simplified Dirac recursive relation set, which created a
precedent for the machine to test the theoretical physic-
s conclusion, and pointed out a bright way to correc-
t mathematical errors implied in theoretical physics
[42]
.
If the computer’s negation of the teratogenic simplified
Dirac equation is still in doubt, then people should find
the reason why the computer proves that the teratogenic
simplified Dirac equation has no solution. This will not
only give a final conclusion to this problem, but also
will test the systematic program of theoretical physical
logic by machine, and even produce a universal method
similar to the machine proof of geometric theorem
[43, 44]
,
which can be used to widely test physical theory. Of
course, now the most urgent need for theoretical physics
is to face up to the fact that because simple mathematics
has been ignored, physical theory may have been creat-
ing the desired results in some incorrect logic for nearly
a hundred years, and therefore missed those correct the-
ories and important scientific inferences
[45]
.
1 Dirac, P. A. M., The Quantum Theory of the Electron, Proc.
R. Soc. London, Ser. A 117, 610-624 (1928), 118 (779),
351-361 (1928).
2 Dirac, P. A. M., The principles of quantum mechanics, 4th
ed. (Clarendon Press, Oxford, 1962, pp270).
3 Schiff, L. I., Quantum Mechanics, (McGraw Hill Press, New
York, 1949, pp326).
4 Bjorken, J. D., Drell, S.D. Relativistic Quantum Mechanics,
(McGraw–Hill, New York, 1964, pp52.-60)
5 Dirac, P. A. M. Relativistic wave equations. Proceedings of
the Royal Society of London. Series A-Mathematical and
Physical Sciences 155, 447-459 (1936).
6 Ohlsson, T. Relativistic quantum physics: from advanced
quantum mechanics to introductory quantum field theory.
(Cambridge University Press, 2011).
7 Weinberg, S. Gravitation and cosmology: principles and ap-
plications of the general theory of relativity. (1972).
8 Dongfang, X. D. The End of Klein-Gordon Equation for
Coulomb Field. Mathematics & Nature 2, 202201 (2022).
9 Dongfang, X. D. On the Relativity of the Speed of Light.
Mathematics & Nature 1, 202101 (2021).
10 Dongfang, X. D. The Morbid Equation of Quantum Numbers.
Mathematics & Nature 1, 202102 (2021).
11 Chen, R. Unitary principle and real solution of Dirac equa-
tion, Arxiv prepring arXiv: 0908.4320 (2009).
12 Dongfang, X. D. Dongfang Angular Motion Law and Opera-
tor Equations. Mathematics & Nature 1, 202105 (2021).
13 Brualdi, R. A., Introductory Combinatories, (Pearson Edu-
cation, Inc., 5 Edition, 2004).
14 Chen, R. The Optimum Differential Equations. Chinese
Journal of Engineering Mathematics, 91, 82-86(2000).
15 Chen, R. The Uniqueness of the Eigenvalue Assemblage for
Optimum Differential Equations, Chinese Journal of Engi-
neering Mathematics 20, 121-124(2003).
16 C. G. Darwin, The Wave Equations of the Electron, Proc. R.
Soc. London, Ser. A 118, 654 (1928).
17 Gordon, W.
¨
Uber den Stoß zweier Punktladungen nach der
Wellenmechanik. Zeitschrift f¨ur Physik 48, 180-191 (1928).
18 P. A. M. Dirac, The principles of Quantum Mechanics,
Clarendon Press, Oxford, 1958, pp270.
19 L. C. Biedenharn, Remarks on the relativistic Kepler prob-
lem, Phys Rev., 126, 845 (1962).
20 M. K. F. Wong, H. Y. Yeh, Simplified solution of the Dirac
equation with a Coulomb potential, Phys. Rev. D 25, 3396
(1982).
21 J. Y. Su, Simplified solution of the Dirac-Coulomb equation,
Phys. Rev. A 32, 3251 (1985).
22 Hakan Ciftci, Richard L. Hall, and Nasser Saad, Iterative
solutions to the Dirac equation, Phys.Rev. A 72, 022101
(2005).
23 M. Pudlak, R. Pincak, V. A. Osipov, Low-energy electronic
states in spheroidal fullerenes, Phys. Rev. B, 74, 235435
(2006).
24 A. B. Voitkiv, B. Najjari, J. Ullrich, Excitation of heavy hy-
202202-12 X. D. Dongfang The End of Teratogenic Simplified Dirac Hydrogen Equations
drogenlike ions in relativistic collisions, Phys. Rev. A, 75,
062716 (2007).
25 F. Ferrer, H. Mathur, T. Vachaspati, G. Starkman, Zero
modes on cosmic strings in an external magnetic field, Phys.
Rev. D, 74, 025012 (2006).
26 N. Boulanger, P. Spindel, F. Buisseret, Bound states of Dirac
particles in gravitational fields, Phys. Rev. D, 74, 125014
(2006).
27 H. W. Crater, C. Y. Wong, P. Van Alstine, Tests of two-body
Dirac equation wave functions in the decays of quarkonium
and positronium into two photons, Phys. Rev. D, 74, 054028
(2006).
28 W. Greiner, Relativistic Quantum Mechanics: Wave Equa-
tions, Springer, 2000, 3rd Edition, pp225.
29 Fischer, C. F. & Zatsarinny, O. A B-spline Galerkin method
for the Dirac equation. Computer Physics Communications
180, 879-886 (2009).
30 Chen, R. New exact solution of Dirac-Coulomb equation with
exact boundary condition. International Journal of Theoret-
ical Physics 47, 881-890 (2008).
31 Alonso, V., De Vincenzo, S. & Mondino, L. On the bound-
ary conditions for the Dirac equation. European Journal of
Physics 18, 315 (1997).
32 Alhaidari, A. D., Bahlouli, H., Al-Hasan, A., Abdelmonem,
M. S., Relativistic scattering with a spatially dependent ef-
fective mass in the Dirac equation, Phys. Rev. A 75, 062711
(2007).
33 Korol, V. A., Kozlov, M. G., Relativistic corrections to the
isotope shift in light ions, Phys. Rev. A 76, 022103 (2007).
34 Amaro, J. E., Barbaro, M. B., Caballero, J. A., Donnelly, T.
W., Udias, J. M., Final-state interactions and superscaling
in the semi-relativistic approach to quasielastic electron and
neutrino scattering, Phys. Rev. C 75, 034613 (2007).
35 Alberto, P., Castro, A. S., Malheiro, M., Spin and pseudospin
symmetries and the equivalent spectra of relativistic spin -1/2
and spin -0 particles, Phys. Rev. C 75, 047303 (2007).
36 Lamata, L., Le´on, J., Sch¨atz, T., Solano, E., Dirac Equation
and Quantum Relativistic Effects in a Single Trapped Ion,
Phys. Rev. Lett. 98, 253005(2007).
37 Dongfang, X. D. The End of Yukawa Meson Theory of Nu-
clear Forces. Mathematics & Nature, 1, 202110(2021).
38 Dongfang, X. D. Relativistic Equation Failure for LIGO Sig-
nals. Mathematics & Nature 1, 202103 (2021).
39 Dongfang, X. D. Com Quantum Proof of LIGO Binary Merg-
ers Failure. Mathematics & Nature, 1, 202107 (2021).
40 Chen, R. The Problem of Initial Value for the Plane Trans-
verse Electromagnetic Mode. Acta Phys. Sin-CH ED 49
(15), 2518 (2000).
41 Dongfang, X. D. Dongfang Modified Equations of Electro-
magnetic Wave. Mathematics & Nature, 1, 202108 (2021).
42 Christiansen, T. Isophasal, isopolar, and isospectral
Schr¨odinger operators and elementary complex analysis.
American journal of mathematics 130, 49-58 (2008).
43 Wu, W. T. Basic principles of mechanical theorem proving in
elementary geometries. Journal of automated Reasoning 2,
221-252 (1986).
44 Wu, W. T. Mechanical theorem proving in geometries: Basic
principles, Springer (1994).
45 Dongfang, X. D. Dongfang Com Quantum Equations for
LIGO Signal. Mathematics & Nature, 1, 202106 (2021).
Latest findings
Dongfang Solution of Induced Second Order Dirac Equations
Solving the radial Dirac equation of the hydrogen atom, it usually follows the treatment method of Schrödinger equation of the hydrogen atom and expresses the two-component wave function as two new variables divided by the radial independent variables, thus transforming the equation into the induced first-order Dirac equation system. According to the induced first-order Dirac equations, two induced second order Dirac equations constrained by the same energy parameter can be obtained. Here, I study the eigensolutions of the induced second-order Dirac equation system of the hydrogen atom, and draw several unusual conclusions. The exact solution of the first-order induced Dirac equation system of hydrogen atom satisfies the induced second-order Dirac equation, but the complexity of correlation checking increases with the increase of the radial quantum number, and even the checking process of energy states with small radial quantum numbers is very complicated; The induced second-order Dirac equation of hydrogen atom has an eigensolution, and its energy eigenvalue is the same as that of the induced first-order Dirac equation; Different from the constraint of the coefficients of the two wave function components of the first-order Dirac equation system, the exact solutions of the two induced second-order equations are independent of each other, which means that the coefficients of the two component functions have their own normalized coefficients. The independence of the component function of the second order equation poses a new challenge to the physical meaning of the multi-component wave function of Dirac theory.
Keywords: Dirac equation; Induced first-order Dirac equation; Induced second-order Dirac equation; Dongfang solution; Existence and uniqueness of Solution.