MATHEMATICS & NATURE
Mathematics,Physics, Mechanics & Astronomy
Random Vol. 2 Sn: 202203
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
Dongfang Solution of Induced Second Order Dirac Equations
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Solving the radial Dirac equation of the hydrogen atom, it usually follows the treatment method of
Schr¨odinger 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
pro cess 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
comp onent 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 equa-
tion; Dongfang solution; Existence and uniqueness of Solution.
PACS number(s): 03.65.Pm—Relativistic wave equations; 03.65.Ge—Solutions of wave equations:
b ound states; 32.10.Fn—Fine and hyperfine structure; 33.15.Pw—Fine and hyperfine structure.
1 Introduction
How much influence does the Dirac equation
[1-7]
have
in the field of physics and even mathematics? Google
academic search Dirac equation obtained more than 1.2
million pieces of information, and search second-order
Dirac equation obtained more than 860000 pieces of in-
formation, and this data has been increasing. How-
ever, the vast majority of articles on the Dirac equa-
tion published in various prestigious journals is vague
and useless
[8-10]
, or is fundamental errors and cannot be
corrected
[11-20]
. This exposes the defects of the academic
system in which a few people control the scientific field.
The original Dirac equation is a matrix equation,
that is, a system of first order partial differential equa-
tions. However, the Dirac equation is rarely correctly
understood
[21-26]
. The evidence to support this view is
that those questions that should have clear answers have
been ignored. For example, according to the conditions
of the Dirac matrix, using the relativistic momentum en-
ergy relationship and the operator principle of quantum
mechanics to construct the first order wave equation, is
the result unique to the Dirac equation? Does changing
the sign of the Dirac matrix affect the physical mean-
ing and the exact solution of the Dirac equation? Is the
Darwin
[27]
and Gordon
[28]
solutions of the Dirac equation
of the Coulomb field, which the textbook advocates, rea-
sonable? Just like Wong and Yeh
[16, 29]
, is it scientific to
change the sign of mass in the Dirac equation and create
a mixed operator to construct the so-called second-order
Dirac equation? Is there any other solution method for
the Dirac equation to give different exact solutions
[30]
?
How many processes and conclusions of handling Dirac
equation that do not conform to mathematical rules are
covered up? Does the radial momentum operator and
angular quantum number defined by Dirac’s theory sat-
isfy the mathematical and physical inference of causali-
ty? From these important problems that have not been
)Citation: Dongfang, X. D. Dongfang Solution of Induced Second Order Dirac Equations. Mathematics & Nature 2, 202203 (2022).
*The author once published the article “Equivalent solution of radial Dirac-Coulomb equation” signed as Rui Chen, but did not explain
the intention of giving the authentic second-order Dirac equation of Coulomb field because of succumbing to the bad hidden rules of
theoretical physics that it is not allowed to point out the errors of famous theories and famous pap ers. Now change the title of the article,
modify the abstract and conclusion of the article, indicate the authors point of view, and republish this article.
202203-2 X. D. Dongfang Dongfang Solution of Induced Second Order Dirac Equations
paid attention to in the past hundred years, the treat-
ment of the Dirac equation in physics is not reliable,
and even there are second-order Dirac equations with d-
ifferent forms. The Dirac equation is also bound to the
definition of the Dirac Sea and antimatter, etc.
[31-37]
. In
fact, they are not necessarily causal.
The formal solutions of the first and second order de-
formed Dirac equations, which are mathematically very
simple to deal with, obtained by introducing the function
transformation, do not satisfy the first order deformed
Dirac and are therefore terminated
[38]
. This prompted
me to examine the theory of Dirac equation in depth.
It is difficult to test all kinds of Dirac equation one by
one in the research of limited forces. Many theories have
been successfully published for recognition only by virtue
of the fame of the Dirac equation. In fact, they are not
real Dirac equation theories. We know that in math-
ematics, we can write a partial differential equation or
a system of partial differential equations randomly and
study its solution. But in physics, we cannot arbitrarily
construct a wave equation without physical model and
mechanical laws. The construction of any quantum me-
chanical wave equation that conforms to scientific logic
will mean a major revolution or progress in quantum the-
ory. Therefore, those nominal Dirac equations that are
not real but just borrowed from Dirac’s fame have no
physical significance, even though the pieced expected
solutions there are often interpreted in a dazzling way.
Because of the characteristics of differential, when
dealing with Schr ¨o dinger equation, Klein-Gordon e-
quation and the Dirac equation of Coulomb-like field,
the radial wave function or the radial wave function is
usually expressed as a new variable divided by the radi-
al independent variable, thus transforming the wave e-
quation into a new variable differential equation. When
dealing with the Dirac equation, we usually imitate this
method, and take the quotient of two new variables rep-
resented by the upper and lower components of the two-
component wave function and the indep endent variables,
thus obtaining the induced first order Dirac equation
system. The first order differential equations can be
transformed into two second order differential equation-
s, which are the induced second order Dirac equation-
s of hydrogen-like atoms. This paper focuses on the
mathematical process, discusses the exact solution of
the induced second-order Dirac equation of hydrogen-
like atoms, and gives the Dongfang eigensolution with
new mathematical meaning.
2 Induced first-order Dirac hydrogen equa-
tion
Dongfang unitary principle
[39-42]
has a very powerful
logic verification function. Using the unitary principle
to test quantum mechanics, we have found problems and
conclusions that have not been found in the past. The
variability of the relative speed of light
[41]
, the unsolved
morbid equation of quantum number
[42]
, the Com quan-
tum equation of LIGO signal
[43-45]
, the modification of
the basic equation of molecular dynamics
[46]
, the opera-
tor evolution equation group of angular motion law
[47]
,
the end of the Yukawa nuclear force meson theory
[48]
, the
end of the Klein-Gordon equation of Coulomb field
[49]
and the end of the teratogenic simplified Dirac hydro-
gen equation
[38]
should have a profound impact on the
development of physics.
When solving the radial Schr¨odinger equation of hy-
drogen atom, the wave function is usually expressed as
a new unknown function divided by the radial indepen-
dent variable, so that the second order differential e-
quation for solving the original radial wave function is
transformed into solving the second order equation for
the new unknown function. This seems to be for conve-
nience, but it doesn’t bring much convenience. However,
this habit has been followed by the radial Dirac equation
used to deal with hydrogen model. Generally, the two
components of the Dirac wave function are expressed
as two new variables divided by the radial independent
variables, so that the original radial Dirac equation of
the hydrogen atom is transformed into a first-order e-
quation group ab out two new variables, which is called
the induced first-order Dirac equation group. The cor-
responding two second-order differential equations are
called induced second-order Dirac differential equation-
s. The purpose of naming “inducted equation” is to
avoid confusing similar equations with different physical
meanings.
Many documents claim to have constructed a unique
second-order Dirac hydrogen equation and obtained an
ideal solution. The construction of the so-called second-
order Dirac equation raises the puzzling question: why
claim to construct the second-order Dirac equation but
discard the second-order Dirac equation derived from
the Dirac equation system? I have treated the induced
second-order Dirac differential equation as the original
second-order Dirac equation for a long time. In fact,
there are some differences between them. From a math-
ematical point of view, the original Dirac equation and
the induced second-order Dirac equation of hydrogen can
be selected as two metrics respectively. According to
Dongfang unitary principle, the quotient of the wave
function divided by the radial independent variable as
the exact solution of the induced second-order Dirac e-
quation should be reduced to the exact solution of the
Dirac equation. We need to give the exact solution of
the induced second-order Dirac equation to confirm this
conclusion. For this reason, here we first derive the in-
duced second-order Dirac hydrogen equation.
In the Dirac electron theory, the four-order matrix
Dirac equation of the original four-component wave
function is applied to the hydrogen atom. After a series
of logical processing that cannot be proved by mathe-
matical theory, it is transformed into the second-order
radial Dirac equation of the two-component wave func-
tion,
Mathematics & Nature (2022) Vol. 2 No. 1 202203-3
−i~c
d
dr
+
1
r
0 −i
i 0
−
~cκ
r
0 1
1 0
+ mc
2
1 0
0 −1
ψ
1
ψ
2
=
E +
α~c
r
ψ
1
ψ
2
(1)
Where α = e
2
~c ≈ 1/137 is the fine-structure constan-
t, m is the static mass of the electron, c is the speed
of light in vacuum, ~ = h/2π is the reduced Planck
constant, and E is the energy eigenvalue parameter,
κ = ±1, ±2, ··· is the angular quantum number de-
fined by Dirac’s hydrogen atom theory through a set
of operation rules independent of mathematics, r is the
size of the radial vector and the independent variable in
the equation, and ψ
1
and ψ
2
are the two unknown com-
ponents of the two-component wave function. Expand
equation (1)
dψ
2
dr
+
1 + κ
r
ψ
2
−
mc
2
− E
~c
−
α
r
ψ
1
= 0
dψ
1
dr
+
1 − κ
r
ψ
1
−
mc
2
+ E
~c
+
α
r
ψ
2
= 0
(2)
Usually, two new functions are introduced to replace the
two components of the wave function,
ψ
1
ψ
2
=
r
−1
F (r)
r
−1
G (r)
(3)
Its first derivative is
dψ
1
dr
=
1
r
dF
dr
−
1
r
2
F,
dψ
2
dr
=
1
r
dG
dr
−
1
r
2
G (4)
Substitute (3) and (4) for equation (2) to obtain the
first-order differential equations
[1-3, 5, 6]
forF and G,
dG
dr
+
κ
r
G −
mc
2
− E
~c
−
α
r
F = 0
dF
dr
−
κ
r
F −
mc
2
+ E
~c
+
α
r
G = 0
(5)
This system of equations is not the original wave func-
tion components ψ
1
and ψ
2
, but the differential equa-
tions of the new variables F (r) and G (r). It is called
the induced first-order Dirac radial equations of the hy-
drogen atom. For the convenience of calculation, param-
eter c
1
, c
2
, a and dimensionless independent variable
parameter ρ are usually introduced. The definition is as
follows,
c
1
=
mc
2
+ E
~c
, c
2
=
mc
2
− E
~c
, a =
√
c
1
c
2
, ρ = ar (6)
Substitute (6) into (5), and the two equations become,
dG
dρ
+
κ
ρ
G −
c
2
a
−
α
ρ
F = 0
dF
dρ
−
κ
ρ
F −
c
1
a
+
α
ρ
G = 0
(7)
This is the abbreviation of the induced first-order Dirac
radial equations of the hydrogen atom.
3 Looking back on the solution of the first
order Dirac equation system
This section re-solves the induced first-order Dirac e-
quations (7) to get rid of the bad habit of relying too
much on literature search, citing classics, showing erudi-
tion and bypassing the necessary calculations. First find
the general solution of equation (7), and then use the
definite solution condition to determine the special solu-
tion. The usual way is to transplant the definite solution
condition of Schr¨odinger equation to the Dirac equation.
The definite solution condition is selected as the natural
boundary condition that the wave function is bounded
in the whole space, so the specific form of the natural
boundary condition of Dirac radial wave function is
lim
ρ→0
ψ
1
ψ
2
=
0
0
,
ψ
1
(0 < ρ < ∞)
ψ
2
(0 < ρ < ∞)
̸=
±∞
±∞
, lim
ρ→∞
ψ
1
ψ
2
=
0
0
(8)
From this, it can be inferred that the conditions for determining the solution of the induced first-order Dirac radial
equations (7) are
lim
ρ→0
F
G
=
0
0
,
F (0 < ρ < ∞)
G (0 < ρ < ∞)
̸=
±∞
±∞
, lim
ρ→∞
F
G
=
0
0
(9)
Referring to the solution of Schr¨odinger equation of hy-
drogen atom, take the solution F → e
−ρ
and G → e
−ρ
of
the second-order asymptotic equation d
2
F
dρ
2
−F → 0
and d
2
G
dρ
2
− G → 0 when ρ → ∞ as the weighting
function, and make the solution of equation (7) in the
form of,
F = e
−ρ
f (ρ) , G = e
−ρ
g (ρ) (10)
Substitute (10) into equation (7) to obtain the first-order
differential equations of f and g,
dg
dρ
−
1 −
κ
ρ
g −
c
2
a
−
α
ρ
f = 0
df
dρ
−
1 +
κ
ρ
f −
c
1
a
+
α
ρ
g = 0
(11)
Find the series solution of f and g, but the boundary
condition (11) requires that both series be interrupted
202203-4 X. D. Dongfang Dongfang Solution of Induced Second Order Dirac Equations
as polynomials,
f =
n
ν=0
b
ν
ρ
s+ν
, g =
n
ν=0
d
ν
ρ
s+ν
(12)
Substitute (12) into (10) to get,
n
ν=0
αb
ν
+ (s + ν + κ) d
ν
− d
ν−1
−
c
2
a
b
ν−1
ρ
s+ν−1
= 0
n
ν=0
(s + ν − κ) b
ν
− αd
ν
−
c
1
a
d
ν−1
− b
ν−1
ρ
s+ν−1
= 0
Therefore, the coefficients of two polynomials (12) satis-
fy the recurrence relationship group,
αb
ν
+ (s + ν + κ) d
ν
−
c
2
a
b
ν−1
− d
ν−1
= 0
(s + ν − κ) b
ν
− αd
ν
− b
ν−1
−
c
1
a
d
ν−1
= 0
(13)
Let ν = 0, notice b
−1
= d
−1
= 0, and get the linear
homogeneous equations for b
0
and d
0
,
αb
0
+ (s + κ) d
0
= 0
(s − κ) b
0
− αd
0
= 0
The necessary and sufficient condition for the system of
the above linear homogeneous equations to have nontriv-
ial solutions is that the coefficient determinant is zero, so
the index value satisfying the boundary condition (11)
is obtained,
s =
κ
2
− α
2
(14)
Let ν = n and ν = n+1 respectively. Noting thatb
n+1
=
d
n+1
= 0, the last four equations are obtained from (13),
αb
n
+ (s + n + κ) d
n
−
c
2
a
b
n−1
− d
n−1
= 0
(s + n − κ) b
n
− αd
n
− b
n−1
−
c
1
a
d
n−1
= 0
−
c
2
a
b
n
− d
n
= 0
−b
n
−
c
1
a
d
n
= 0
According to the parameter definition in (6), a
2
= c
1
c
2
,
the last two equations are linearly related. The first two
equations must be equivalent. From these four equation-
s, the eigenvalue equation for determining the quantized
energy formula is obtained,
a =
α (c
1
− c
2
)
2 (s + n)
(15)
Where n = 0, 1, 2, ··· is the radial quantum number,
and κ = ±1, ±2, ··· is the angular quantum number de-
fined by Dirac’s hydrogen atom theory. Substitute the
parameter definition in (6), and finally obtain the Dirac
energy level formula of hydrogen atom
E =
mc
2
1 +
α
2
(
n+
√
κ
2
−α
2
)
2
(16)
The above reproduces the calculation process of the
Dirac energy level formula of the hydrogen atom, rather
than just citing the existing conclusions of a large num-
ber of documents. It is generally considered that doing
so in the paper is unnecessary because it is a repetition of
the existing work. However, the fact is that only by re-
peatedly and independently deducing the existing theory
can we truly grasp the essence of this theory and discover
major issues that have not been noticed in history. The
Dirac energy level formula has been rapidly recognized
and widely spread because it predicts the fine structure
of the hydrogen atom spectrum. Combining the above
steps, the exact solution of the induced first-order Dirac
radial equations (7) for hydrogen is determined by (10),
(12), (13), (14) and (16). If the characteristic quantity
of the hydrogen atom is used to represent the defined
parameter (6), the energy eigenvalue needs to be substi-
tuted into (6), but the expression form of each parameter
thus obtained is not concise. It is advisable to retain the
energy parameter E and eliminate the parameters c
1
, c
2
and a. Here, the exact solution of the first-order Dirac
induced by hydrogen is merged as follows,
F = e
−ρ
n
ν=0
b
ν
ρ
√
κ
2
−α
2
+ν
, G =e
−ρ
n
ν=0
d
ν
ρ
√
κ
2
−α
2
+ν
αb
ν
+
κ
2
−α
2
+κ+ν
d
ν
−
mc
2
−E
mc
2
+E
b
ν−1
−d
ν−1
=0
κ
2
−α
2
−κ+ν
b
ν
−αd
ν
−b
ν−1
−
mc
2
+E
mc
2
−E
d
ν−1
=0
E =
mc
2
1+
α
2
(
n+
√
κ
2
−α
2
)
2
, (n =0, 1, ··· , κ = ±1, ±2, ···)
(17)
4 Induced second-order Dirac hydrogen e-
quation
The Dirac equation has great influence, which has led
to the emergence of many second-order Dirac equations.
We have ended the transformation Dirac equation that
decomposes the wave function, and the corresponding
transformation second order Dirac equation is also nat-
urally ended. It is puzzling that for nearly 100 years, the
second order differential equation transformed from the
first order differential equation system has been avoided.
There may be some reasons behind this.
Now consider the second-order Dirac equation trans-
formed by the induced first-order Dirac equation system
(7) for the hydrogen. The expressions for F and G are
obtained from the first and second equations of (7) re-
spectively,
F =
aρ
c
2
ρ − αa
dG
dr
+
κa
c
2
ρ − αa
G
G =
aρ
c
1
ρ + αa
dF
dρ
−
κa
c
1
ρ + αa
F
(18)
Their first derivatives are,
Mathematics & Nature (2022) Vol. 2 No. 1 202203-5
dF
dρ
=
aρ
c
2
ρ − αa
d
2
G
dρ
2
+
d
dρ
aρ
c
2
ρ − αa
+
κa
c
2
ρ − αa
dG
dρ
+
d
dρ
κa
c
2
ρ − αa
G
=
aρ
c
2
ρ − αa
d
2
G
dρ
2
+
[κc
2
ρ − α (1 + κ) a] a
(ρc
2
− αa)
2
dG
dρ
−
κc
2
a
(ρc
2
− aα)
2
G
dG
dρ
=
aρ
c
1
ρ + αa
d
2
F
dρ
2
+
d
dρ
aρ
c
1
ρ + αa
−
κa
c
1
ρ + αa
dF
dρ
−
d
dρ
κa
c
1
ρ + αa
F
=
aρ
c
1
ρ + αa
d
2
F
dρ
2
−
[κc
1
ρ − α (1 − κ) a] a
(aα + c
1
ρ)
2
dF
dρ
+
κc
1
a
(c
1
ρ + αa)
2
F
(19)
Use (18) and (19) to eliminate the term containing G in (7) the first equation and the term containing F in the
second equation, one gets,
aρ
aα + ρc
1
d
2
F
dρ
2
+
a
2
α
(aα + ρc
1
)
2
dF
dρ
+
α
ρ
+
aκc
1
(aα + ρc
1
)
2
−
aκ
2
aαρ + ρ
2
c
1
−
c
2
a
F = 0
aρ
c
2
ρ − αa
d
2
G
dρ
2
−
a
2
α
(aα − ρc
2
)
2
dG
dρ
+
−
α
ρ
−
c
1
a
−
aκc
2
(aα − ρc
2
)
2
+
aκ
2
aαρ − ρ
2
c
2
G = 0
(20)
This is a system of second-order differential equations constrained by the energy parameter E written in c
1
, c
2
and A. Multiply the first equation of equation (20) with (aα + ρc
1
)
2
and the second equation of equation (20) with
(aα − ρc
2
)
2
, and get the following form
αρ
2
+
c
1
a
ρ
3
d
2
F
dρ
2
+ αρ
dF
dρ
+
α
3
− ακ
2
ρ
ρ −
κ
2
− κ − 2α
2
c
1
+ α
2
c
2
a
ρ +
αc
2
1
a
2
−
2αc
1
c
2
a
2
ρ
2
−
c
2
1
c
2
a
3
ρ
3
F = 0
αρ
2
−
c
2
a
ρ
3
d
2
G
dρ
2
+ αρ
dG
dρ
+
α
3
− ακ
2
ρ
ρ +
α
2
c
1
+
κ
2
+ κ − 2α
2
c
2
a
ρ −
2αc
1
c
2
a
2
−
αc
2
2
a
2
ρ
2
+
c
1
c
2
2
a
3
ρ
3
G = 0
(21)
Then use the relation a =
√
c
1
c
2
of the parameters defined in formula (6) to simplify, and obtain the second-order
induced Dirac equation of the hydrogen atom:
αρ
2
+
c
1
a
ρ
3
d
2
F
dρ
2
+ αρ
dF
dρ
+
α
α
2
− κ
2
+
κ − κ
2
+ 2α
2
c
1
− α
2
c
2
a
ρ + α
c
2
1
a
2
− 2
ρ
2
−
c
1
a
ρ
3
F = 0
αρ
2
−
c
2
a
ρ
3
d
2
G
dρ
2
+ αρ
dG
dρ
+
α
α
2
− κ
2
+
κ
2
+ κ − 2α
2
c
2
+ α
2
c
1
a
ρ + α
c
2
2
a
2
− 2
ρ
2
+
c
2
a
ρ
3
G = 0
(22)
5 The S state Dirac wave function satisfies the induced second-order Dirac equation
The induced second order Dirac equation (22) of the hydrogen atom is transformed from the induced first order
Dirac equation (7) of the hydrogen atom. The solution of the first order equation (23) should satisfy the second
order equation (22). However, it is too troublesome to determine the coefficients of the series one by one to give the
specific wave function. This may be the main reason why Dirac quantum theory and all the literature on the Dirac
equation only focus on the quantized energy formula and avoid the wave function.
Here we write the specific form of the so-called S state wave function when the series is interrupted to the radial
quantum number n = 0, and explain that the solution of the induced first-order Dirac equation system satisfies the
induced second-order Dirac equation. When n = 0, the simplest form of Dirac energy level formula of S state is
obtained from energy level formula (16),
E
0
= mc
2
1 −
α
2
κ
2
(23)
Substituting (23) into (6) gives the defined parameters. Take n = 0, substitute (14) into (12) and then into (10),
and the specific forms of the components of the two wave functions are only different coefficients. The results are as
202203-6 X. D. Dongfang Dongfang Solution of Induced Second Order Dirac Equations
following,
c
01
a
0
=
κ
α
1+
1−
α
2
κ
2
,
c
02
a
0
=
κ
α
1−
1−
α
2
κ
2
c
2
01
a
2
0
=
α
2
κ
2
1+
1−
α
2
κ
2
2
,
c
2
02
a
2
0
=
κ
2
α
2
1−
α
2
κ
2
−1
2
F
0
=b
0
ρ
√
κ
2
−α
2
e
−ρ
, G
0
=d
0
ρ
√
κ
2
−α
2
e
−ρ
(24)
The S state solution of the first order induced Dirac equation (24) must satisfy the S state form of the second order
induced Dirac equation
αρ
2
+
c
01
a
0
ρ
3
d
2
F
0
dρ
2
+ αρ
dF
0
dρ
+
α
3
− ακ
2
−
κ
2
− κ − 2α
2
c
01
+ α
2
c
02
a
0
ρ +
αc
2
01
a
2
0
− 2α
ρ
2
−
c
01
a
0
ρ
3
F
0
= 0
αρ
2
−
c
02
a
0
ρ
3
d
2
G
0
dρ
2
+ αρ
dG
0
dρ
+
α
3
− ακ
2
+
α
2
c
01
+
κ
2
+ κ − 2α
2
c
02
a
0
ρ −
2α −
αc
2
02
a
2
0
ρ
2
+
c
02
a
0
ρ
3
G
0
= 0
(25)
Calculate the derivatives of the S state wave function in (24), and the results are listed as follows
dF
0
dρ
= e
−ρ
κ
2
− α
2
− ρ
ρ
√
κ
2
−α
2
−1
b
0
dG
0
dρ
= e
−ρ
κ
2
− α
2
− ρ
ρ
√
κ
2
−α
2
−1
d
0
d
2
F
0
dρ
2
= e
−ρ
ρ
√
κ
2
−α
2
−2
κ
2
− α
2
−
κ
2
− α
2
− 2
κ
2
− α
2
ρ + ρ
2
b
0
d
2
G
0
dρ
2
= e
−ρ
ρ
√
κ
2
−α
2
−2
κ
2
− α
2
−
κ
2
− α
2
− 2
κ
2
− α
2
ρ + ρ
2
d
0
(26)
Substitute (24) and (26) into the left side of the first equation and the second equation of (25) respectively, and get
αρ
2
+
c
01
a
0
ρ
3
d
2
F
0
dρ
2
+αρ
dF
0
dρ
+
α
3
−ακ
2
−
κ
2
−κ−2α
2
c
01
+α
2
c
02
a
0
ρ+
αc
2
01
a
2
0
−2α
ρ
2
−
c
01
a
0
ρ
3
F
0
=
αρ
2
+
κ
α
1+
1−
α
2
κ
2
ρ
3
e
−ρ
ρ
√
κ
2
−α
2
−2
κ
2
−α
2
−
κ
2
−α
2
−2
κ
2
−α
2
ρ+ρ
2
+αρe
−ρ
κ
2
−α
2
−ρ
ρ
√
κ
2
−α
2
−1
+
α
3
−ακ
2
−
κ
2
−κ−2α
2
mc
~
1+
1−
α
2
κ
2
+α
2
1−
1−
α
2
κ
2
ρ
κ
α
+
α
κ
2
α
2
1+
1−
α
2
κ
2
2
−2α
ρ
2
−
κ
α
1+
1−
α
2
κ
2
ρ
3
ρ
√
κ
2
−α
2
e
−ρ
b
0
=−b
0
e
−ρ
ρ
1+
√
κ
2
−α
2
1+
1−
α
2
κ
2
κ
α
κ
2
−α
2
−κ
(1+2ρ)+α
1−2
1−
α
2
κ
2
κ+2
κ
2
−α
2
+2ρ
=−b
0
e
−ρ
ρ
1+
√
κ
2
−α
2
κ+
κ
2
−α
2
√
κ
2
−α
2
−κ
α
(1+2ρ)+α
1−2
κ
2
−α
2
+2
κ
2
−α
2
+2ρ
=0
(27)
Mathematics & Nature (2022) Vol. 2 No. 1 202203-7
αρ
2
−
c
02
a
0
ρ
3
d
2
G
0
dρ
2
+αρ
dG
0
dρ
+
α
3
−ακ
2
+
α
2
c
01
+
κ
2
+κ−2α
2
c
02
a
0
ρ−
2α−
αc
2
02
a
2
0
ρ
2
+
c
02
a
0
ρ
3
G
0
=
αρ
2
−
κ
α
1−
1−
α
2
κ
2
ρ
3
e
−ρ
ρ
√
κ
2
−α
2
−2
κ
2
−α
2
−
κ
2
−α
2
−2
κ
2
−α
2
ρ+ρ
2
+αρe
−ρ
κ
2
−α
2
−ρ
ρ
√
κ
2
−α
2
−1
+
α
3
−ακ
2
+
α
2
mc
~
1+
1−
α
2
κ
2
+
κ
2
+κ−2α
2
1−
1−
α
2
κ
2
κ
α
ρ
−
2α−α
1−
α
2
κ
2
−1
2
κ
2
α
2
ρ
2
+
1−
1−
α
2
κ
2
κ
α
ρ
3
ρ
√
κ
2
−α
2
e
−ρ
d
0
=−d
0
e
−ρ
ρ
1+
√
κ
2
−α
2
1−
α
2
κ
2
−1
κ
α
κ+
κ
2
−α
2
(1+2ρ)+α
1−2
1−
α
2
κ
2
κ+2
κ
2
−α
2
+2ρ
=−d
0
e
−ρ
ρ
1+
√
κ
2
−α
2
κ
2
−α
2
−κ
κ+
√
κ
2
−α
2
α
(1+2ρ)+α
1−2
κ
2
−α
2
+2
κ
2
−α
2
+2ρ
=0
(28)
The above calculation process is reserved to facilitate the reader to test the mathematical logic. Scientists should
often calculate famous theories independently, rather than go out of the mode that has always relied on a large
number of interpretations based on memory and replication. Although the traditional mode can make individuals
quickly succeed, imperfect and even erroneous theories continue to continue, eventually making some scientific theories
develop into religions.
When the radial quantum number n = 1, the exact solution of the induced first-order Dirac equation system (7)
also satisfies the second-order induced Dirac equation (22), but the calculation process of the test is lengthy and
cumbersome, which is omitted here. The larger the radial quantum number, the more cumbersome the calculation.
I have never tested whether the exact solution of the first-order Dirac equation system (7) satisfies the second-order
induced Dirac equation (22) when n = 2, 3, 4, ···, but I am sure that there will be no negative conclusion. Readers,
especially physicists who rely too much on reciting conclusions and the qualitative logic of modern physics and rarely
calculate independently, can try to verify. Every time such verification is completed, there will be new gains.
6 Dongfang solution of induced second-order Dirac hydrogen equation
If only the energy eigenvalues is concerned, it is possible to treat the formal solution of the wave equation that
cannot be tested as the true solution, and the formal solution may be the false solution of the equation. The
coefficients of equation (22) are all polynomials, and the results obtained by using the approximate solution to
construct the weighted function are consistent with those obtained by using the generalized optimal differential
equation theorem
[50, 51]
. The former is familiar to us. When ρ → ∞, the approximate second order equation derived
from the approximate equations of the two equations in (22) is the same, d
2
F
dρ
2
− F → 0, d
2
G
dρ
2
− G → 0.
The solution of the approximate equation satisfying the boundary condition is F → e
−ρ
, G → e
−ρ
. It is inferred
that the solution of equation (22) has the same form as that of equation (10),
F = e
−ρ
f (ρ) , G = e
−ρ
g (ρ) (29)
Their first and second derivative is respectively,
dF
dρ
= e
−ρ
df
dρ
− f
,
d
2
F
dρ
2
= e
−ρ
d
2
f
dρ
2
− 2
df
dρ
+ f
dG
dρ
= e
−ρ
dg
dρ
− g
,
d
2
G
dρ
2
= e
−ρ
d
2
g
dρ
2
− 2
dg
dρ
+ g
202203-8 X. D. Dongfang Dongfang Solution of Induced Second Order Dirac Equations
Substitute them into equation (22) to obtain the second order differential equation of f and g,
αρ
2
+
c
1
a
ρ
3
e
−ρ
d
2
f
dρ
2
− 2
df
dρ
+ f
+ αρe
−ρ
df
dρ
− f
+
α
α
2
− κ
2
+
κ − κ
2
+ 2α
2
c
1
a
− α
2
c
2
a
ρ + α
c
2
1
a
2
− 2
ρ
2
−
c
1
a
ρ
3
e
−ρ
f = 0
αρ
2
−
c
2
a
ρ
3
e
−ρ
d
2
g
dρ
2
− 2
dg
dρ
+ g
+ αρe
−ρ
dg
dρ
− g
+
α
α
2
− κ
2
+
κ
2
+ κ − 2α
2
c
2
a
+ α
2
c
1
a
ρ + α
c
2
2
a
2
− 2
ρ
2
+
c
2
a
ρ
3
e
−ρ
g = 0
(30)
By combining similar terms, the above equations are simplified as following,
αa
2
+ ac
1
ρ
d
2
f
dρ
2
+
αa
2
ρ
− 2αa
2
− 2ac
1
ρ
df
dρ
+
α
α
2
− κ
2
a
2
ρ
2
−
αa
2
−
2α
2
+ κ − κ
2
ac
1
+ α
2
ac
2
ρ
+ α
c
2
1
− a
2
f = 0
αa
2
− ac
2
ρ
d
2
g
dρ
2
+
αa
2
ρ
− 2αa
2
+ 2ac
2
ρ
dg
dρ
+
α
α
2
− κ
2
a
2
ρ
2
−
a
αa +
2α
2
− κ − κ
2
c
2
− α
2
c
1
ρ
+ α
c
2
2
− a
2
g = 0
(31)
It should be noted that the two equations of equation (31) are constrained by the same energy parameter, so they are
not completely independent of each other, but are implicit second-order differential equations. Let the form of the
interrupted series solution of the two function components F = e
−ρ
f (ρ) and G = e
−ρ
g (ρ) satisfying the boundary
condition (9) to be,
F = e
−ρ
n
ν=0
b
ν
ρ
√
κ
2
−α
2
+ν
, G = e
−ρ
n
ν=0
d
ν
ρ
√
κ
2
−α
2
+ν
(32)
Substitute it into equation (31) to get,
α +
c
1
a
ρ
n
ν=0
(s + ν) (s + ν − 1) b
ν
ρ
s+ν−2
+
α
ρ
− 2α − 2
c
1
a
ρ
n
ν=0
(s + ν) b
ν
ρ
s+ν−1
+
α
α
2
− κ
2
ρ
2
−
1
ρ
α −
2α
2
+ κ − κ
2
c
1
a
+ α
2
c
2
a
+ α
c
2
1
a
2
− 1
n
ν=0
b
ν
ρ
s+ν
= 0
α −
c
2
a
ρ
n
ν=0
(s + ν) (s + ν − 1) d
ν
ρ
s+ν−2
+
α
ρ
− 2α + 2
c
2
a
ρ
n
ν=0
(s + ν) d
ν
ρ
s+ν−1
+
α
α
2
− κ
2
ρ
2
−
1
ρ
α +
2α
2
− κ − κ
2
c
2
a
− α
2
c
1
a
− α
1 −
c
2
2
a
2
n
ν=0
d
ν
ρ
s+ν
= 0
(33)
Combine similar terms to obtain the recurrence relationship satisfied by the coefficients of the two series,
n
ν=0
α
(s + ν)
2
+
α
2
− κ
2
b
ν
−
2 (s + ν − 2)
c
1
a
− α
c
2
1
a
2
− 1
b
ν−2
+
2α
2
+ κ − κ
2
+ (s + ν − 2) (s + ν − 1)
c
1
a
− α
2
c
2
a
− α (2s + 2ν − 1)
b
ν−1
ρ
s+ν−2
= 0
n
ν=0
α
(s + ν)
2
+
α
2
− κ
2
a
2
d
ν
+
2 (s + ν − 2)
c
2
a
− α
1 −
c
2
2
a
2
d
ν−2
−
2α
2
− κ − κ
2
+ (s + ν − 2) (s + ν − 1)
c
2
a
− α
2
c
1
+ α (2s + 2ν − 1)
d
ν−1
ρ
s+ν−2
= 0
(34)
Mathematics & Nature (2022) Vol. 2 No. 1 202203-9
Let ν = 0, note that b
−1
= b
−2
= ··· = 0, d
−1
= d
−2
= ··· = 0, and b
0
̸= 0, d
0
̸= 0. The above two recurrence
relations give the same index equation,
α
s
2
+
α
2
− κ
2
a
2
= 0
Solve this equation, eliminate the negative ro ots that do not meet the boundary conditions, and take the positive
root. The result is consistent with (14),
s =
κ
2
− α
2
(35)
Substitute this result into (34), and the recurrence relationship group satisfied by the coefficients of the two series is
specifically expressed as,
α
2ν
κ
2
− α
2
+ ν
2
b
ν
−
2
κ
2
− α
2
+ ν − 2
c
1
a
− α
c
2
1
a
2
− 1
b
ν−2
+
2 + α
2
+ κ − 3ν + (2 ν − 3)
κ
2
− α
2
+ ν
2
c
1
a
− α
2
c
2
a
− α
2ν − 1 + 2
κ
2
− α
2
b
ν−1
= 0
α
2ν
κ
2
− α
2
+ ν
2
d
ν
+
2
κ
2
− α
2
+ ν − 2
c
2
a
− α
1 −
c
2
2
a
2
d
ν−2
−
2 + α
2
− κ − 3ν + (2 ν − 3)
κ
2
− α
2
+ ν
2
c
2
a
− α
2
c
1
a
+ α
2
κ
2
− α
2
+ 2ν − 1
d
ν−1
= 0
(36)
In (36), let ν = n + 2, note that the expected solution requires p
n+1
= p
n+2
= ··· = 0, q
n+1
= q
n+2
= ··· = 0, and
p
n
̸= 0, q
n
̸= 0, so two eigenvalue equations are obtained, which should be equivalent because they describe the same
energy parameter,
2 (s + n) ac
1
− α
c
2
1
− a
2
b
n
= 0
2 (s + n) ac
2
− α
a
2
− c
2
2
d
n
= 0
Restore with defined parameters (6),
2 (s + n)
√
m
2
c
4
− E
2
~c
mc
2
+ E
~c
− α
mc
2
+ E
~c
2
−
mc
2
− E
mc
2
+ E
~
2
c
2
= 0
2 (s + n)
√
m
2
c
4
− E
2
~c
mc
2
− E
~c
− α
mc
2
− E
mc
2
+ E
~
2
c
2
−
mc
2
− E
~c
2
= 0
After simplification, the two eigenvalue equations are the same, namely,
(s + n)
m
2
c
4
− E
2
− αE = 0
Solve this equation and use (35) or (14) to obtain the Dirac energy level formula shown in (16),
E =
mc
2
1 +
α
2
(
n+
√
κ
2
−α
2
)
2
(37)
Where n = 0, 1, 2, ··· is the radial quantum numb er, and κ = ±1, ±2, ··· is the angular quantum number defined by
Dirac’s hydrogen atom theory.
The induced second-order Dirac equation (22) of hydrogen atom is actually a set of second-order equations con-
strained by the same energy parameter, and its eigensolutions are determined by (29), (32), (35), (36) and (37). The
exact solutions of the induced second-order Dirac equations for the hydrogen with the retention of energy parameter
202203-10 X. D. Dongfang Dongfang Solution of Induced Second Order Dirac Equations
E and elimination of parameters c
1
, c
2
and a are summarized as follows
F =e
−ρ
n
ν=0
b
ν
ρ
√
κ
2
−α
2
+ν
, G=e
−ρ
n
ν=0
d
ν
ρ
√
κ
2
−α
2
+ν
α
2ν
κ
2
−α
2
+ν
2
b
ν
−
2
κ
2
−α
2
+ν −2
mc
2
+E
mc
2
−E
−
2αE
mc
2
−E
b
ν−2
+
2+α
2
+κ−3ν +(2ν −3)
κ
2
−α
2
+ν
2
mc
2
+E
mc
2
−E
−α
2
mc
2
−E
mc
2
+E
−α
2ν −1+2
κ
2
−α
2
b
ν−1
=0
α
2ν
κ
2
−α
2
+ν
2
d
ν
+
2
κ
2
−α
2
+ν −2
mc
2
−E
mc
2
+E
−
2αE
mc
2
+E
d
ν−2
−
2+α
2
−κ−3ν +(2ν −3)
κ
2
−α
2
+ν
2
mc
2
−E
mc
2
+E
−α
2
mc
2
+E
mc
2
−E
+α
2
κ
2
−α
2
+2ν −1
d
ν−1
=0
E =
mc
2
1+
α
2
(
n+
√
κ
2
−α
2
)
2
, (n=0, 1, ··· , κ = ±1, ±2, ···)
(38)
This is an expression group, named Dongfang solution of
the induced second-order Dirac equation of the hydrogen
atom.
How to name the research results is a topic beyond
the topic. Historically, after independent research re-
sults of professional researchers are published, their s-
tudents or friends or followers may name them in the
newly published promotion articles. I have made a lot of
breakthroughs. In the past 40 years, I have continuous-
ly submitted contributions to famous journals at home
and abroad, but few journals have published these pa-
pers. Later, when I checked the literature, I found that
other authors later published the same results in influ-
ential journals as in the manuscript. The usual saying
is that many people around the world are studying the
same breakthrough topic, and others have published it,
but you have not published it. However, the physical
and mathematical logic of the articles that published
the same results violated the unitary principle. Over
the past few decades, many manuscripts submitted to
famous academic journals such as Nature and Physical
Review have been stifled. This long experience makes it
impossible for me to continue to listen to the so-called a-
cademic ethics of mainstream academic journals. When
I propagandize breakthrough discoveries, I often get s-
landered and abused. Some mainstream scholars even
joined forces to write lengthy libel letters with online
signatures and send them to the departments where I
work, harassing my work and threatening my life. Ev-
ery researcher hopes that his hard research achievements
of decades will be recognized. Since no prestigious a-
cademic journal has accepted any of my breakthrough
research achievements for decades, how to name my re-
search achievements now is naturally not bound by the
hidden rules of mainstream academia. Of course, attack-
ers can also name each research result according to their
wishes.
The spread of truth, of course, depends on the at-
tention, reproduction, interpretation and development
of theories by many researchers. However, it is a virtue
to respect the work of the original creator. A real re-
searcher, a researcher who loves the truth, and a re-
searcher who really adheres to the moral code will not
concentrate on blaming the original author for naming
the research results. When reading the expression group
(38), his attention will be attracted to the question of
whether the exact solution (17) of the first order induced
Dirac equation group (7) of the hydrogen atom is equiv-
alent to the exact solution of the second order induced
Dirac equation group (22), thus meeting the Dongfang
normalization principle. There are great differences be-
tween them in form. Section 5 tests the consistency of
the two when n = 0 and introduces the positive results
when n = 1. So, what is the process and result of the
test for any radial quantum number n > 2?
7 Conclusions and comments
The Dirac equation has great mathematical charm,
resulting in the generation of many second-order Dirac e-
quations. But some are unreasonable, and some are even
wrong. It takes too much time and energy to test various
theories one by one. The teratogenic simplified Dirac
equation has been ended, including the corresponding
second-order Dirac equation
[11-15]
. Here, the induced
second-order Dirac equation of the hydrogen atom is
derived, and the solution of the induced second-order
Dirac equation is given, which is a neutral result. The i-
somorphic second-order Dirac equation and the isomeric
second-order Dirac equation will be discussed later, and
then the first-order Dirac equation will be reprocessed
from a new perspective. We are opening a bright win-
Mathematics & Nature (2022) Vol. 2 No. 1 202203-11
dow to let the world gradually see the right and wrong
of the whole Dirac quantum theory and its derivative
theory.
It is not easy to test the exact solution (17) of the first
order Dirac equation system (7) or the exact solution of
the second order induced Dirac equation system (22). It
is complex enough to just test whether the solution of
the induced first order Dirac equation system satisfies
the two induced second order differential equations. For
each kind of Dirac equation, it is very important to dis-
tinguish those subtle differences that are imperceptible.
Although some differences are subtle and even difficult
to find, they may contain principled problems. In fact,
they cause the theory to deviate too far from the nat-
ural law and have to distort the logical reorganization
to make the inference conform to the expectation. For
example, construct the so-called complete conservative
exchange observable value
[7]
. There are essential differ-
ences in the theories of various named Dirac hydrogen
equations, but all claim to obtain the same expected so-
lution. Have we ever thought ab out studying the causal
relationship between them? On the surface, completely
different wave equations of the same physical model can
have the same energy eigenvalue set. This seems to par-
tially conform to Dongfang unitary principle. However,
conforming to the unitary principle is only a necessary
condition, not a sufficient condition, for the theory to
hold.
The test of the unitary principle of the induced
second-order Dirac equation has two meanings: 1) The
two second-order differential equations form a system of
equations subject to the constraints of the common ener-
gy parameters, so they are not completely independent.
The calculation results of the energy eigensolutions are
consistent with the unitary principle, but this does not
mean that the problem is over; 2) After the energy eigen-
solutions of the two second-order differential equations
are proved to be the same, the wave functions as the so-
lutions of the respective equations show independence.
They can have indep endent normalization coefficients.
Compared with the two-component wave functions of
the original Dirac equation, the conclusion does not con-
form to the unitary principle, which poses a challenge to
the physical meaning of the multi-component wave func-
tions of Dirac. The meaning of Dirac multi-component
wave function is not clear. The traditional normalized
definition of wave function is only a guess that has not
been strictly proved. Is the Bonn statistical interpreta-
tion of wave function unique and reasonable? Finding
the ultimate answer to such questions will promote the
great changes in physical theory and the development of
mathematical theory.
1 Dirac, P. A. M. The quantum theory of the electron. Pro-
ceedings of the Royal Society of London. Series A, Contain-
ing Papers of a Mathematical and Physical Character 117,
610-624 (1928).
2 Dirac, P. A. M. The quantum theory of the electron. Part
II. Proceedings of the Royal Society of London. Series A,
Containing Papers of a Mathematical and Physical Charac-
ter 118, 351-361 (1928).
3 Dirac, P. A. M. The principles of quantum mechanics. (Ox-
ford university press, 1981).
4 Thaller, B. The dirac equation. (Springer Science & Business
Media, 2013).
5 Greiner, W. Relativistic quantum mechanics. Vol. 2
(Springer, 2000).
6 Schiff, L. I. Quantum Mechanics 3rd. New York: M cGraw-
Hill (1968).
7 Zeng, J. Y. Quantum Mechanics II. 611-620 (Beijing: Science
Press, 1997).
8 Rubinow, S. I. & Keller, J. B. Asymptotic solution of the
Dirac equation. Physical Review 131, 2789 (1963).
9 Schoop, L. M. et al. Dirac cone protected by non-symmorphic
symmetry and three-dimensional Dirac line node in ZrSiS.
Nature communications 7, 11696 (2016).
10 Kastler, D. The Dirac operator and gravitation. Communi-
cations in Mathematical Physics 166, 633-643 (1995).
11 Martin, P. C. & Glauber, R. J. Relativistic theory of radiative
orbital electron capture. Physical Review 109, 1307 (1958).
12 Biedenharn, L. C. Remarks on the relativistic Kepler prob-
lem. Physical Review 126, 845 (1962).
13 Hostler, L. Coulomb Green’s functions and the Furry approx-
imation. Journal of Mathematical Physics 5, 591-611 (1964).
14 Auvil, P. R. & Brown, L. M. The relativistic hydrogen atom:
A simple solution. American Journal of Physics 46, 679-681
(1978).
15 Scadron, M. D. & Scadron, M. D. Nonrelativistic Perturba-
tion Theory. Advanced Quantum Theory and Its Applica-
tions Through Feynman Diagrams, 159-179 (1979).
16 Wong, M. & Yeh, H.-Y. Simplified solution of the Dirac equa-
tion with a Coulomb potential. Physical Review D 25, 3396
(1982).
17 Su, J.-Y. Simplified solutions of the Dirac-Coulomb equation.
Physical Review A 32, 3251 (1985).
18 Barut, A. & Zanghi, N. Classical model of the Dirac electron.
Physical Review Letters 52, 2009 (1984).
19 Bialynicki-Birula, I., Gornicki, P. & Rafelski, J. Phase-space
structure of the Dirac vacuum. Physical Review D 44, 1825
(1991).
20 Wu, S.-Q. Separability of a modified Dirac equation in a
five-dimensional rotating, charged black hole in string the-
ory. Physical Review D 80, 044037 (2009).
21 De Vries, E. & Van Zanten, A. The Dirac matrix group
and Fierz transformations. Communications in Mathemat-
ical Physics 17, 322-342 (1970).
22 Good Jr, R. Properties of the Dirac matrices. Reviews of
Modern Physics 27, 187 (1955).
23 Macfarlane, A. Dirac matrices and the Dirac matrix descrip-
tion of Lorentz transformations. Communications in Mathe-
matical Physics 2, 133-146 (1966).
24 Green, H. Dirac matrices, teleparallelism and parity conser-
vation. Nuclear Physics 7, 373-383 (1958).
25 Poole Jr, C. P. & Farach, H. A. Pauli-Dirac matrix genera-
tors of Clifford algebras. Foundations of Physics 12, 719-738
(1982).
26 Talman, J. D. Minimax principle for the Dirac equation.
Physical review letters 57, 1091 (1986).
27 Darwin, C. G. The wave equations of the electron. Proceed-
ings of the Royal Society of London. Series A, Containing
Papers of a Mathematical and Physical Character 118, 654-
202203-12 X. D. Dongfang Dongfang Solution of Induced Second Order Dirac Equations
680 (1928).
28 Gordon, W.
¨
Uber den Stoß zweier Punktladungen nach der
Wellenmechanik. Zeitschrift f¨ur Physik 48, 180-191 (1928).
29 Wong, M. & Yeh, H.-Y. Exact solution of the Dirac-Coulomb
equation and its application to bound-state problems. II. In-
teraction with radiation. Physical Review A 27, 2305 (1983).
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 Ambjørn, J., Greensite, J. & Peterson, C. The axial anomaly
and the lattice Dirac sea. Nuclear Physics B 221, 381-408
(1983).
32 Perry, R. J. Effects of the Dirac sea on finite nuclei. Physics
Letters B 182, 269-273 (1986).
33 Kurasawa, H. & Suzuki, T. Effects of the Dirac sea on the
Coulomb sum value for electron scattering. Nuclear Physics
A 490, 571-584 (1988).
34 Quinn, H. R. & Witherell, M. The asymmetry between mat-
ter and antimatter. Physics Today 56, 30 (2003).
35 Sizov, R. A. Dirac’s “holes” are the true Antielectrons and
real particles of Antimatter. Journal of Modern Physics 6,
2280 (2015).
36 Jentschura, U. D. Gravitationally coupled Dirac equation for
antimatter. Physical Review A 87, 032101 (2013).
37 Chardin, G. & Manfredi, G. Gravity, antimatter and the
Dirac-Milne universe. Hyperfine Interactions 239, 1-13
(2018).
38 Dongfang, X. D. The End of Teratogenic Simplified Dirac Hy-
drogen Equations. Mathematics & Nature 2, 202202 (2022).
39 Chen, R. Limitations of conservation laws in fractionally in-
ertial reference frames.Physics Essays 24, 413-418 (2011).
40 Chen, R. Correction to Solution of Dirac Equation. arXiv
preprint arXiv:0908.4320 (2009).
41 Dongfang, X. D. On the relativity of the speed of light. Math-
ematics & Nature 1, 202101 (2021).
42 Dongfang, X. D. The Morbid Equation of Quantum Numbers.
Mathematics & Nature 1, 202102 (2021).
43 Dongfang, X. D. Relativistic Equation Failure for LIGO Sig-
nals. Mathematics & Nature 1, 202103 (2021).
44 Dongfang, X. D. Dongfang Com Quantum Equations for
LIGO Signal. Mathematics & Nature 1, 202106 (2021).
45 Dongfang, X. D. Com Quantum Proof of LIGO Binary Merg-
ers Failure. Mathematics & Nature 1, 202107 (2021).
46 Dongfang, X. D. Dongfang Modified Equations of Molecular
Dynamics. Mathematics & Nature 1, 202104 (2021).
47 Dongfang, X. D. Dongfang Angular Motion Law and Opera-
tor Equations. Mathematics & Nature 1, 005 (2021).
48 Dongfang, X. D. The End of Yukawa Meson Theory of Nu-
clear Forces. Mathematics & Nature 1, 202110(2021).
49 Dongfang, X. D. The End of Klein-Gordon Equation for
Coulomb Field. Mathematics & Nature 2, 202201 (2022).
50 Chen, R. The Optimum Differential Equations. Chin. J.
Engin. Math 17, 82-86 (2000).
51 Chen, R. The uniqueness of the eigenvalue assemblage for
optimum differential equations. Chin. J. Engin. Math 20,
121-124 (2003).
Latest findings
The End of Isomeric Second Order Dirac Hydrogen Equations
Biedenharn and Wong wrote the Dirac equation in the form that the combination of the differential term and the function term is equal to zero, then changed the positive and negative sign of the mass term and removed the wave function to extract a mixing operator, and then used this mixing operator to act on the first-order Dirac equation. The resulting second-order equation is called the isomeric second-order Dirac equation. Because the equations in mathematical sense can be constructed arbitrarily, the isomeric second-order Dirac equation can exist as a pure differential equation. However, as the wave equation of quantum mechanics, the isomeric second-order Dirac equation advocated by famous journals seriously lacks scientific basis and destroys mathematical principles, and the processing of isomeric second-order Dirac equation is completely false calculation. Here, the real heterogeneous second-order Dirac equation is first derived, and it is proved that it is a system of equations composed of four non-solvable second-order partial differential equations of four wave function components. Then it is proved that the highly respected second-order Dirac equation of single-component wave function isomerism is forged, and the Dirac energy level formula of hydrogen atom pieced together is only a prop to cover up the above lies. Then it is proved that the most ideal isomeric second-order radial Dirac equation of hydrogen atom is also an unsolvable differential equation system composed of at least two partial differential equations, which further illustrates the fraud of the highly respected isomeric second-order Dirac equation of single-component wave function. Finally, it is proved that the mixed operator method for constructing the isomeric second-order Dirac equation destroys the unitary principle and leads to many confused and uncertain conclusions. The results of these rigorous calculations declare the end of the heterogeneous second-order Dirac equation and the mixed operator method itself used for the construction of heterogeneous wave equations.