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Mathematics and Physics
Ground State Solution of Dongfang Modified Dirac Equation
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
In order to deal with the mathematical contradiction that the Dirac wave function does not meet
the definite solution condition, an effective and reasonable solution is to replace the traditional rough
b oundary condition with the precise boundary condition in which the nuclear radius is written. The
exact solution of the hydrogen-like atom self consistent Dirac equation satisfying the exact boundary
conditions has subversive physical significance. It also shows a new mathematical point of view, that
is, the boundary parameters become one of the eigensolutions of the equation, and the solutions of the
equation may be completely different due to the slight difference of the boundary conditions. Dongfang
mo dified Dirac hydrogen equation replaces the angular quantum number defined by the illogical Dirac
electron theory with the intrinsic angular quantum number determined by the exact solution of the
equation. Here I further study of the ground state solution of the modified Dirac equation which
satisfies the exact boundary conditions. The results show that the ground state of the modified Dirac
equation for hydrogen-like atoms is a triple degenerate state with three intrinsic angular momentum
and three intrinsic wave functions; the intrinsic angular momentum of the ground state is neither the
angular momentum constructed by the anti-logic of Dirac electron theory nor the angular momentum
of Schr¨odinger equation. The two comp onents of one of the intrinsic wave functions of the ground
state are linearly related. The existence of the exact solution of the intrinsic ground state and the
essential difference between the intrinsic ground state energy level and the Dirac ground state energy
level further illustrate that the angular momentum constructed by the so-called Dirac algebra is not a
corollary of scientific logic, and the Dirac equation cannot explain the fine structure of hydrogen-like
atoms. It is only one of the most puzzling representatives of modern physics as the basic equation
of quantum field theory. However, because Dirac equation contains rich mathematical problems and
unique pro cessing technology, it will help the development of mathematical theory to incorporate it
into mathematics textbooks as a new mathematical model.
Keywords: Dongfang unitary principle; Modified Dirac equation; Quantum radius of atomic nu-
cleus; Intrinsic angular quantum number.
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
Physics uses mathematical language to describe nat-
ural phenomena and reveal the laws of motion of vari-
ous substances in nature. Physics cannot be separated
from equations. The equations of physics must always
be the inference of the basic laws of physics and can-
not be constructed at will; the special solution of the
physical equation must conform to the physical meaning.
Mathematical equations can be constructed arbitrarily.
Mathematical theory pays attention to the integrity of
the general solution of mathematical equations, and the
meaning of the solution is often ignored. There are two
kinds of phenomena in modern physics. One is incor-
rect equations or incorrect solutions of equations, which
are used to explain experimental phenomena that can-
not be explained by classical physics
[1-17]
, and then are
generally accepted. The other is the falsification, simu-
lation and even computer-generated data used to verify
the inference of incorrect equations, and then received
high praise
[18-26]
.
It is the mainstream of modern physics to build phys-
ical theory according to expectations, and it is also the
most fundamental reason that theoretical physics is in
trouble and puzzles physics readers. The simplest exam-
ple is that there has never been a meson in the atom-
ic nucleus, but the meson theory of Yukawa’s nuclear
force, which completely distorts the mathematical rules
and is filled with uncorrectable calculation errors
[27, 28]
,
is recognized as a major breakthrough in physical theo-
ry. In recent 100 years, it has been widely publicized as
an important theory to describe the interaction between
nucleons, and it is not allowed to question and test, but
continues to develop. The formal solution of the Dirac
equation of hydrogen like atom that does not meet the
conditions of a definite solution is much more complex.
)Citation: Dongfang, X. D. Ground State Solution of Dongfang Modified Dirac Equation. Mathematics & Nature 2, 202208 (2022).
202208-2 X. D. Dongfang Ground State Solution of Dongfang Modified Dirac Equation
Dirac energy level formula implies imaginary energy, and
the divergence of Dirac wave function at the coordinate
origin means the collapse of the universe, which is com-
pletely inconsistent with objective facts. Therefore, the
use of Dirac energy level formula to explain the fine sp ec-
tral structure of the hydrogen like atom is purely a mat-
ter of grafting. It is thought-provoking that this kind of
carefully constructed abnormal logic has its own char-
m because of the novelty of the floating light, which has
been continuously promoted and developed as one of the
outstanding achievements in physics.
There is a famous saying that facts sp eak louder than
words. Perhaps this is the fundamental reason why
experimental data claiming major discoveries in mod-
ern physics are often unrepeatable. Systematic error
correction often leads to the distortion of experimen-
tal data, which later developed into a large number
of tampering, and even replaced by simulation experi-
mental data and even fabricated data. Readers usual-
ly cannot question experimental reports that claim sig-
nificant observational data. However, There is a defi-
nite transformation relationship between different met-
rics describing the natural law, and the natural law itself
does not change due to the selection of different met-
rics. When the mathematical expression of natural laws
under different metrics is transformed into one metric,
the result must be the same as the inherent form un-
der this metric, 1=1, meaning the transformation is u-
nitary,Chen:2011,Dongfang:2021. In short, the result of
the transformation of the mathematical form of the nat-
ural law in different metrics to the same metric is u-
nique. The discovery of this basic principle, called the
Dongfang unitary principle, which is generally applica-
ble to testing the logic of the theory and the reliability
of experimental reports, makes things very simple. For
example, the conclusion of the unitary principle test of
the reliability of LIGO’s gw150914 signal is irreversible:
even if the original spiral binary star merger event that
assumes that LIGO’s predetermined signal is not repro-
ducible really exists, the detailed data of the gw150914
signal released by LIGO just proves that the described
binary stars failed to achieve the final merger because
of the loss of key Peugeot spikes; Because spiral binaries
have not b een successfully merged, if they exist, they
will continue to radiate equal frequency signals for a long
time; However, LIGO cannot receive this kind of equal
frequency late signal that can be repeatedly observed,
so it violates the unitary principle; It can be seen that
the so-called spiral binary gravitational wave experimen-
t report is obviously an elaborate fairy tale. It can be
predicted that those who adhere to scientific truth and
those who safeguard academic lies will form two camps
in the field of theoretical physics in the next 100 years.
Although the unique advantages of the degree system en-
sure that the team of the latter is very large, the truth
will one day awaken the conscience of young scholars.
The basic assumptions of modern physics usually do
not conform to the unitary principle and are obvious-
ly unreliable. A large number of completely distort-
ed conclusions in modern physics were covered up and
whitewashed by false calculations and fabricated exper-
imental reports. I have proved that the so-called Ein-
stein principle of invariance of the speed of light is pure-
ly a mathematical magic definition
[30]
, listed examples
of quantum number morbid equation that prove that
quantum mechanics destroys the law of conservation of
energy, proposed the angular motion law
[31]
, explained
the limitations of the quantum mechanics operator prin-
ciple through the angular motion law operator evolu-
tion equation group
[32]
, and analyzed the macroscopic
quantized exact equation contained in the GW150914
signal of LIGO
[33-35]
. Using the unitary principle to
systematically test the development theory of relativity
and quantum mechanics, I ended Yukawa’s nuclear me-
son theory
[36, 37]
, the Klein-Gordon equation of Coulom-
b field
[38]
, the teratogenic simplified Dirac equation
[39]
,
the isomeric second-order Dirac equation
[40]
, the expect-
ed solution of the real second-order Dirac equation
[41]
,
and then gave the challenging solution of the hydrogen-
like atom Dirac equation
[42]
, and negated the angular
momentum value defined by the illogical Dirac electron
theory and gives the neutron state solution of the mod-
ified Dirac hydrogen equation
[43]
. No matter how slan-
dered, strangled or pretended to be ignored by main-
stream physicists and mainstream scientific media, any
research achievements devoted to exposing false calcula-
tions and false experimental reports in modern physic-
s and correcting correctable calculations
[44-46]
will pro-
mote the profound revolution of theoretical physics cal-
culation and experimental test, the false calculation and
its conclusion will eventually be replaced by the true cal-
culation and conclusion, and the fake observation data
experimental report will eventually be completely aban-
doned. This is the great motive power for someone to
always adhere to the truth.
The interpretation of the Dirac equation to the quan-
tum mechanical wave equation of the fine structure of
the hydrogen-like atom spectrum is flashy. However, if
we take the Dirac equation as a pure mathematical prob-
lem and try to find its reasonable part to the maximum,
we can find interesting mathematical problems that have
not been found in the past. The main conclusions are
that, in order to ensure that the Dirac equation and its
solution are mathematically self-consistent, it is neces-
sary to replace the rough boundary condition without
considering the size of the atomic nucleus with the exact
boundary condition written in the size of the atomic nu-
cleus, and the angular quantum number must be taken
as the intrinsic parameter of the Dirac equation, which is
determined by the exact solution of the equation rather
than constructed by the Dirac algebra theory. In this
way, the neutron state solution obtained by dealing with
the Dirac equation of the hydrogen-like atom is the true
intrinsic solution. This is a special case of the challenge
Mathematics & Nature (2022) Vol. 2 No. 1 202208-3
solution of the Dirac equation. The accuracy of the ener-
gy level formula of the challenge solution is equivalent to
the accuracy of the Bohr energy level, without regard to
the fine structure of the atomic spectrum. So what are
the intrinsic ground state solution and intrinsic excited
state solution of the Dirac equation? This paper discuss-
es the intrinsic ground state solution of the Dongfang
modified Dirac equation for hydrogen-like atoms that
satisfies the exact boundary conditions and determines
the intrinsic ground state angular quantum number of
the Dirac equation. The result again proves that the
angular quantum number constructed by Dirac algebra
theory is invalid.
2 Conclusions and comments
It is generally believed that the Dirac equation suc-
cessfully integrates quantum mechanics and special rel-
ativity, opens up the field of relativistic quantum me-
chanics, predicts the existence of antimatter, explains
the fine structure of hydrogen atom spectrum, promotes
the development of electromagnetic theory to quantum
electrodynamics, lays the foundation of quantum field
theory, and promotes the development of particle physic-
s and high-energy physics. The Dirac equation seems to
have been one of the most important basic equations in
physics
[47-55]
. However, the Dirac wave function and the
Dirac energy level formula are one of the formal solu-
tions of the Dirac equation of hydrogen like atoms that
do not meet the predetermined rough boundary condi-
tions, hiding logical disasters such as the collapse of the
universe and virtual energy that do not conform to the
facts. These problems make Dirac electron theory one of
the typical representatives of the unprecedented success
of modern physics in explaining experimental phenome-
na through the non-real solutions of equations.
There are two solutions to solve the difficulties caused
by the angular quantum number constructed by Dirac
electron theory, such as the traditional solution of Dirac
equation does not meet the conditions of definite solution
and virtual energy. One is tantamount to rebuild the
wave equation, and the other is to reprocess the Dirac
equation. At present, I choose the latter scheme, hoping
to find the most reasonable logic for the Dirac equa-
tion.. Considering the size of the atomic nucleus, any
physical quantity describing the hydrogen like atomic
system takes the space outside the atomic nucleus as
the domain of definition, writes the radius of the atomic
nucleus into the b oundary conditions, replaces the tra-
ditional rough boundary conditions that do not consider
the size of the atomic nucleus with the precise bound-
ary conditions, and re solves the Dirac equation of the
hydrogen like atom. We obtain the quantized intrinsic
energy formula with an accuracy equivalent to the Bohr
energy level, which is the challenge solution of the Dirac
equation.
Relativistic quantum mechanics only pays attention
to the acquisition of quantized energy formula but has
always ignored the specific form of wave function. When
we attempt to obtain the wave function of the ener-
gy state corresponding to the higher principal quantum
number, the challenge solution of the Dirac equation of
the hydrogen like atom satisfying the exact boundary
conditions presents the existence of the solution. We
need to take the angular quantum number as one of the
eigensolutions of the equation, or the solution of the e-
quation does not exist. Therefore, I introduce the un-
determined angular quantum number C to replace the
angular quantum number κ = ±1, ±2, ± 3, · · · defined
by Dirac electron theory, thereby modifying the Dirac
equation of hydrogen like atoms. The modified Dirac
equation has neutron state solution. The neutron state
solution gives the minimum nontrivial angular quantum
number of the hydrogen like atom, which is not the result
of quantum mechanics in the past. At the same time,
it also gives the radius of the hydrogen like atom. For
hydrogen atoms, the neutron state solution of the mod-
ified Dirac equation seems to describe the structure of
neutrons. In this paper, we further study the ground s-
tate solution of the modified Dirac equation for hydrogen
like atoms. It is clear that the ground state has three in-
trinsic angular quantum numbers and the ground state
energy is a triple degenerate state. The results of the
neutron state solution and the intrinsic ground state so-
lution of the modified Dirac equation show that the un-
determined angular quantum number is complex, and we
don’t know what the value of the undetermined angular
quantum number of various excited states is. However,
the existing conclusions have once again proved that the
Dirac equation cannot explain the fine structure of hy-
drogen like atom spectrum. The Dirac electron theory
breaks the rule that angular quantum number of quan-
tum mechanics is one of the eigensolutions of the equa-
tion, and it is unreasonable to artificially construct the
angular quantum number. The three intrinsic angular
quantum numbers of the ground state are strictly math-
ematical conclusions. We need further study to reveal
its physical significance.
Introducing the intrinsic angular quantum number to
modify the Dirac equation is a but not the only way to
deal with the hidden logical contradiction of the equa-
tion. Can we modify Dirac equation according to the
construction of the Dirac equation? Using the Dong-
fang unitary principle, we can systematically prove that
the theory of relativity is not tenable. Obviously, only
introducing the undetermined intrinsic angular quantum
number to correct the Dirac equation based on the rel-
ativistic momentum and energy relationship is not the
ultimate answer to improve the quantum theory. But
from a mathematical point of view, relativity only in-
creases the speed of light factor on the basis of Newto-
nian mechanics, which raises a question: does the non-
relativistic theory of high-speed motion exist? Is it pos-
202208-4 X. D. Dongfang Ground State Solution of Dongfang Modified Dirac Equation
sible for this problem to lead to the discovery of the
wave equation similar to the Dirac equation? Of course,
there will always be great resistance to subversive re-
vision of famous theories. Then within the framework
of relativity, what kind of anti mathematical problem-
s are hidden in the algebraic theory of defining angu-
lar quantum numbers in Dirac electron theory, and how
Dirac electron theory goes beyond mathematical rules
to transform the equation of four component wave func-
tion into two-component wave function equation, which
need further research. The discovery and solution of ev-
ery relevant new problem may lead to the revision of
the Dirac equation again, and even lead to completely
different conclusions. Therefore, it is necessary to test
the construction logic of the Dirac equation for the two-
component wave function.
The neutron state solution and the intrinsic ground
state solution of the Dongfang modified Dirac hydrogen
equation exceed the customary concepts of quantum me-
chanics. This pioneering new conclusion indicates that
it seems that every additional quantum number to solve
the intrinsic solution of the modified Dirac equation or
the special solution of the similar equation that satis-
fies the exact boundary will become an inexhaustible
source of writing for theoretical physics or mathematic-
s researchers. However, with each additional quantum
number, the difficulty of accurate solution will increase.
The contribution of this kind of work on mathematics
may be greater than that of physics.
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Latest findings
The End of Dirac Hydrogen Equation in One Dimension
The angular quantum number and radial momentum operator of Dirac's electron theory belongs to the formal mathematical definition, rather than the result of standard mathematical calculation. Whether such formal mathematical definition that changes the fundamental nature of physical logic is reasonable or not can be judged according to whether the definitions of the radial momentum operator and angular momentum eigenvalue are consistent with the calculation results of standard mathematics. Selecting a specific physical model can find the exact answer to the problem, which forces us to seriously deal with the similarities and differences between the formal mathematical definition and the standard mathematical calculation results. Here I discuss the one-dimensional Dirac equation model of hydrogen-like atom and give the standard mathematical calculation conclusion: the one-dimensional Dirac equation has four non-equivalent first-order differential equation systems; the four first-order differential equation systems can also be converted into a system of differential equations composed of two second-order differential equations with the same energy parameter. The mathematical process of accurately solving the general solution of the second-order differential equations is beyond the existing mathematical basis. However, it can be proved that the exact solution of the ground state and a specified excited state of the first-order differential equations does not exist, It means that the first-order Dirac differential equations and the corresponding second-order differential equations of the hydrogen-like atom in the one-dimensional case have no energy quantized eigensolutions. The expression of the radial momentum operator here has a clear conclusion that is different from the definition of Dirac electron theory. The one-dimensional Dirac hydrogen equation is actually a special case of zero angular momentum. The specific zero angular momentum is intentionally avoided by Dirac electron theory because its existence exposes the serious non-consistency of the Dirac equation. This proves that a large number of mathematical calculations on the exact solution of the one-dimensional Dirac equation of the hydrogen atom, which is respected by various scientific documents, belong to the spurious calculation of the expected results. The one-dimensional Dirac equation is thus ended.