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Mathematics and Physics
The End of True Second Order Dirac Hydrogen Equation
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
The original Coulomb field radial Dirac equation is essentially a first order differential system of
two-component wave functions. The second order differential equation of the original wave function
comp onent directly converted from the first order differential equation set is called the true second
order Dirac equation. Relativistic quantum mechanics usually ignores the physical meaning of wave
function and only focuses on the energy eigenvalue. So, the main expectation of solving the true second
order Dirac equation of hydrogen-like atoms is that the Dirac energy level formula is the eigenvalue of
the equation. Here I derive two true second order Dirac equations that are mutually independent in
form but actually constrained by common energy parameters, and then use the traditional boundary
conditions of hydrogen-like atoms to solve the true second order Dirac equation. The conclusion drawn
from this is not exactly the same as the traditional understanding. 1) The formal solution of the true
second order Dirac equation satisfying the traditional boundary conditions takes the Dirac hydrogen
level formula as the energy eigensolution, which seems to meet the expectation; 2) However, when the
radial quantum number is 0, regardless of the value of the angular quantum number, the complete
expression of the wave function as the exact solution of the equation diverges at the coordinate origin,
which does not meet the traditional boundary conditions, which means that the universe is collapsed
and does not conform to the fact of the universe structure. From this it is concluded that the Dirac
energy level formula is only the formal eigenvalue of the true second order Dirac equation that does
not conform to the physical meaning. This announced the end of the expectation of using traditional
b oundary conditions to solve the true second order Dirac equation to naturally obtain the Dirac
energy level formula. This result will promote the re-study of the exact solution of the original Dirac
equation.
Keywords: Unitary principle, Dirac equation, Inevitable solution, Pseudo solution, Wave function
divergence, Energy of imaginary number.
PACS number(s): 02.30.Gp—Special functions; 02.30.Hq—Ordinary differential equations;
03.30.+p—Sp ecial relativity; 03.65.Pm—Relativistic wave equations; 03.65.Ge—Solutions of wave
equations: bound states; 32.10.Fn—Fine and hyperfine structure; 33.15.Pw—Fine and hyperfine
structure.
1 Introduction
The relativistic Dirac equation
[1-8]
is a four-order ma-
trix wave equation in three-dimensional rectangular co-
ordinate system. It is essentially a differential equation
system composed of four first order differential equa-
tions. The great influence of the Dirac equation has
given birth to various specious of equations with the
name of the Dirac equation, such as the teratogenic
first order Dirac equation group and its correspond-
ing teratogenic second order Dirac equation
[9-12]
, which
have been terminated
[13]
. Perhaps because of the dif-
ficulty in mathematical processing, several generations
of physicists have avoided the true second order Dirac
equation directly converted from the original Dirac e-
quation, and instead put forward a variety of puzzling
and fundamentally different heteromorphous second or-
der Dirac equation
[14-23]
or heteromorphous first order
Dirac equation
[24-30]
, or even the title Dirac equation
[31]
with unclear mathematical process, to piece together the
energy level formula of hydrogen-like atom that meets
the expectation. What about the solution of the true
second order Dirac equation transformed from the orig-
inal first order Dirac equation?
From a mathematical point of view, people can write a
differential equation at will to discuss its solution. How-
ever, the Dirac equation is not written randomly, and its
construction idea is beyond that of the previous wave e-
quation. If we do not discuss the applicable scope of
the operator replacement rule of mechanical quantities,
the matrix structure of the Dirac equation is quite cre-
ative and charming. However, from the physical point
of view, the construction principle of the equation used
for the specific physical model must conform to the ba-
sic laws of physics, and the real solution must be able to
correctly describe the physical laws. In modern physics,
when the true solution of the equation does not conform
to the physical phenomenon, it often appears to distort
)Citation: Dongfang, X. D. The End of True Second Order Dirac Hydrogen Equation. Mathematics & Nature 2, 202205 (2022).
202205-2 X. D. Dongfang The End of True Second Order Dirac Hydrogen Equation
the mathematical calculation to obtain the formal solu-
tion to replace the true solution, and even independent-
ly define the operation process that does not conform
to the mathematical operation rules, so that the results
are consistent with the experimental observation results,
and then some people provide the unreal observation re-
sults that cannot be repeated, to confuse the false with
the true, and to add fuel to the flames. Just like the
gravitational wave of spiral double stars, the data obvi-
ously do not conform to the famous equation
[32-34]
, but
it has been confirmed by propaganda. The expected so-
lution of the terminated deformed Dirac equation is not
the inevitable solution of the equation. The correctness
of such papers for mass production can be checked by
using Dongfang’s unitary principle, thus presenting its
true background.
Dongfang’s unitary principle
[35-37]
, which is general-
ly applicable to the logical test of natural and social
sciences, abstracts a very simple logic. There is a def-
inite transformation relationship between different met-
rics describing the natural law, and the natural law itself
does not change due to the selection of different metrics.
When the mathematical expression of natural laws un-
der different metrics is transformed into one metric, the
result must be the same as the inherent form under this
metric, 1=1, meaning the transformation is unitary. We
have made some important discoveries by using the uni-
tary principle to test the logic of modern physics
[38-43]
,
and will make more and more important breakthroughs.
The discovery of the law of angular motion
[44]
will make
the unitary principle test of the basic principle of the
operator construction of the wave equation of quantum
mechanics fruitful. This is because the describing an-
gular motion law is a system of equations composed of
multiple equations, and the corresponding operator evo-
lution equations constitute many different metrics of the
same physical model, which leads to the conclusion that
is difficult to find in the past is subversive enough prob-
lems. For example, is there any solution to the various
equations of the evolution equations of angular motion
law operator? Is the solution of each operator evolution
equation consistent? How does the bound state model
relate to the quantized energy? What are the eigenso-
lutions of different operator evolution equations? The
statistical interpretation of wave functions obviously re-
quires that they meet the principle of normalization, so
they are the same. Therefore, the statistical interpre-
tation of wave function is also challenged. There are
sufficient reasons to question the construction principle
of the wave equation. The Dirac equation is also con-
structed by the method of constructing a wave equation
with operators. So, is the Dirac equation unique?
The solution of the Dirac equation of the hydrogen
atom seems to be perfect, and the result is considered
to describe the fine spectral structure of the hydrogen
atom. The Dirac equation is defined as having the causal
relationship with the spin of antimatter and particle due
to some additional explanations. However, when we try
to demonstrate such causal relationships one by one, we
know that some descriptions are popular science exag-
gerations that cannot be proved by scientific logic. It
is a waste to comment one by one on the many second
order Dirac equations generated by the first order Dirac
hydrogen equation with a huge halo overhead. It is suffi-
cient to have the terminated teratogenic Dirac equation
as a representative. Now we know that the real purpose
behind the definition of a new term ”decoupling” for the
teratogenic second order Dirac equation is to delete one
solution that does not conform to the expected energy
value and retain the other solution that meets the expec-
tation, thus covering up the irreconcilable contradiction
between the two solutions. The so-called ”decoupling” is
by no means a scientific principle but is cunningly used
as a scientific principle. This is just a profile of the false
prosperity of modern physics.
Here we discuss the true second order Dirac equation
which is directly transformed from the original first or-
der radial differential equation of the hydrogen atom. It
has two second order differential equations. The bound-
ary conditions for transforming the first order differential
equations into the second order differential equations re-
main unchanged. We will prove that these two second
order Dirac differential equations have the same quan-
tized energy eigenvalues as the first order Dirac differen-
tial equation, which is consistent with the Dirac energy
level formula, and seems to meet expectations. However,
the solution of the equation determined by the bound-
ed boundary conditions of the wave function does not
completely meet the boundary conditions. When the
radial quantum number is 0, it diverges at the coordi-
nate origin, which means that the universe is collapsed
and does not conform to the fact of the universe struc-
ture. This announced the end of the expectation of solv-
ing the true second order Dirac equation satisfying the
traditional boundary conditions to naturally obtain the
Dirac energy level formula of hydrogen-like atoms.
2 Conclusions and comments
The mathematical treatment process of the main e-
quations of relativistic quantum mechanics was tested
by the unitary principle, and the Yukawa nuclear force
meson theory was ended, the relativistic Klein-Gordon
equation of Coulomb field was ended, the teratogenic
first-order Dirac equation group and its corresponding
teratogenic second-order Dirac equation were ended, and
the isomeric second-order Dirac equation of Coulomb
field was also ended. Here I end the expected solutions
of the two true second-order differential equations trans-
formed from the original first-order radial Dirac equa-
tions of the Coulomb field or hydrogen-like atom.
The solution of the true second-order Dirac equation
of the Coulomb field satisfying the traditional bound-
Mathematics & Nature (2022) Vol. 2 No. 1 202205-3
ary conditions is similar to the solution of the first-order
Dirac equation system, and the energy level formula is
the same, which seems to meet the expectations. How-
ever, the wave function as the expected solution hides
the case of divergence at the origin of coordinates and
does not meet the boundary conditions, so the expected
solution is only a pseudosolution. All kinds of second-
order Dirac equations introduced by famous scientific
journals are meaningless in fact. Relevant calculations
distort mathematical operation rules or physical logic.
According to the unitary principle, since the expected
solution of the standard second-order Dirac equation is
only a pseudo-solution, the expected solution of all oth-
er second-order Dirac equations also belongs to pseudo-
solution, unless the constructed second-order Dirac e-
quation has nothing to do with the original Dirac first-
order differential equation system, but only uses Dirac’s
fame to increase the appeal of the article. The end of
the expectation of the true second-order Dirac equation
means the end of the distorted mathematical logic and
reasoning of the so-called second-order Dirac equation
and the so-called higher-order Dirac equation
[47]
.
It is puzzling that all kinds of incoherent formal so-
lutions of the real and false Dirac equation contain the
same so-called fine structure energy level expectation.
The Dirac equation has the challenge of thinking not
only in physical logic but also in mathematical logic,
so it is worthy of further study. However, the focus
of this paper is to determine the standard form of the
true second-order Dirac equation. The purpose is to
develop the research of relevant mathematical theories
to directly deal with such second-order differential e-
quations, thus revealing the conclusions of mathemat-
ics and physics that are rarely known. Are there any
solutions to the two true second-order Dirac equations
of hydrogen-like atoms that conform to the mathemati-
cal and physical meanings? What ends here is only the
expected solution of the true second-order Dirac equa-
tion, not the second-order Dirac equation itself. Dirac’s
hydrogen atom theory only focuses on the energy level
formula, which has caused many authors to follow suit.
In fact, even if an unreasonable wave equation is con-
structed so that some term in the series solution part is
zero, the same so-called quantized energy formula can be
obtained. Although the essence of quantum mechanics
is the quantized energy formula, obtaining the quantum
energy formula through wave equation may not confor-
m to the unitary principle. A theory that conforms to
the unitary principle may not be correct, but a theo-
ry that does not conform to the unitary principle must
be wrong. Therefore, one can foresee how the unitary
principle will bring about changes in scientific theory.
The expected solution of the first-order Dirac equa-
tion system of a hydrogen-like atom, which is respected
by the standard course, is recognized as accurately de-
scribing the spectral fine structure of the hydrogen atom.
How reliable is the accepted conclusion? We can use the
unitary principle to test it in depth and get a conclusion.
It is one of the basic principles that quantum mechanics
has not been strictly proved to construct wave equation-
s by replacing mechanical quantity with operators and
acting on wave functions. Attempts to prove or deny
this principle will have unexpected results. Dongfang’s
law of angular motion puts forward requirements for the
scope of application of this basic principle of quantum
mechanics. The angular motion law of the same physi-
cal model is a set of equations containing multiple equa-
tions. If the principle of constructing wave equations by
quantum mechanics operators is universally established,
a large number of equations representing the angular
motion law will lead to a large number of wave equa-
tions of the same physical model. The solutions of these
equations are not necessarily consistent. Therefore, the
angular motion law has subversive significance for the
unitary test of quantum mechanics.
As a physical equation, Dirac equation needs to un-
dergo a comprehensive logical test. However, in the field
of mathematics, differential equations can be construct-
ed at will. After constructing the first order differen-
tial equation, transforming it into the second order dif-
ferential equation will produce enough additional roots.
According to the unitary principle, the solution of the
second order differential equation from the first order
differential equation system should contain the special
solution of the first order differential equation system,
otherwise the conversion calculation from the first or-
der differential equation to the second order differential
equation will be prohibited, which will make the mathe-
matical rules not conform to the unitary principle. Here,
the original first-order Dirac radial differential equation-
s of the hydrogen atom are transformed into standard
second-order Dirac equations. So, can we obtain the
expected solution of the first order differential equation
system from the second order differential equation? The
mutual transformation and exact solution of the first or-
der differential equation and the second order differen-
tial equation must conform to the unitary principle, oth-
erwise it will constitute a mathematical paradox. The
conclusions of many mathematical problems in physics,
especially many calculations in theoretical physics, are
not reliable or questionable, and may require the par-
ticipation of mathematicians to be reasonably corrected
and improved.
202205-4 X. D. Dongfang The End of True Second Order Dirac Hydrogen Equation
1 Dirac, P. A. Forms of relativistic dynamics. Reviews of Mod-
ern Physics 21, 392 (1949).
2 Dirac, P. A. M. Relativistic quantum mechanics. Proceedings
of the Royal Society of London. Series A, Containing Pa-
pers of a Mathematical and Physical Character 136, 453-464
(1932).
3 Dirac, P. A. M. The principles of quantum mechanics. (Ox-
ford university press, 1981).
4 Chirgwin, B. & Flint, H. Dirac’s Equation for the Neutron
and Proton. Nature 155, 724-724 (1945).
5 Feynman, R. P. Space-time approach to non-relativistic quan-
tum mechanics. Reviews of modern physics 20, 367 (1948).
6 Thaller, B. The dirac equation. (Springer Science & Business
Media, 2013).
7 Gross, F. Relativistic quantum mechanics and field theory.
(John Wiley & Sons, 1999).
8 Greiner, W. Relativistic quantum mechanics. Vol. 2
(Springer, 2000).
9 Waldenstro/m, S. On the Dirac equation for the hydrogen
atom. American Journal of Physics 47, 1098-1100 (1979).
10 Ciftci, H., Hall, R. L. & Saad, N. Iterative solutions to the
Dirac equation. Physical Review A 72, 022101 (2005).
11 Alhaidari, A. Solution of the Dirac equation with position-
dependent mass in the Coulomb field. Physics Letters A 322,
72-77 (2004).
12 Alhaidari, A. Solution of the Dirac equation with position-
dependent mass in the Coulomb field. Physics Letters A 322,
72-77 (2004).
13 Dongfang, X. D. The End of Teratogenic Simplified Dirac
Hydrogen Equations. Mathematics & Nature 2, 012 (2022).
14 Sukumar, C. Supersymmetry and the Dirac equation for a
central Coulomb field. Journal of Physics A: Mathematical
and General 18, L697 (1985).
15 Karwowski, J. & Kobus, J. The dirac second order equation
and an improved quasirelativistic theory of atoms. Interna-
tional journal of quantum chemistry 30, 809-819 (1986).
16 Kobus, J., Karwowski, J. & Jask´olski, W. Matrix elements of
rq for quasirelativistic and Dirac hydrogenic wavefunctions.
Journal of Physics A: Mathematical and General 20, 3347
(1987).
17 Martin, I. & Karwowski, J. Quantum defect orbitals and the
Dirac second order equation. Journal of Physics B: Atomic,
Molecular and Optical Physics 24, 1539 (1991).
18 Esposito, G. & Santorelli, P. Qualitative properties of the
Dirac equation in a central potential. Journal of Physics A:
Mathematical and General 32, 5643 (1999).
19 Fischer, C. F. & Zatsarinny, O. A B-spline Galerkin method
for the Dirac equation. Computer Physics Communications
180, 879-886 (2009).
20 Kruglov, S. Modified Dirac equation with Lorentz invariance
violation and its solutions for particles in an external mag-
netic field. Physics Letters B 718, 228-231 (2012).
21 Yesiltas.
¨
O. Second order confluent supersymmetric approach
to the Dirac equation in the cosmic string spacetime. The
European Physical Journal Plus 135, 1-15 (2020).
22 Alhaidari, A. Dirac equation with coupling to 1/r singular
vector potentials for all angular momenta. Foundations of
Physics 40, 1088-1095 (2010).
23 Egrifes, H. & Sever, R. Bound states of the Dirac equation
for the PT-symmetric generalized Hulth´en potential by the
Nikiforov–Uvarov method. Physics Letters A 344, 117-126
(2005).
24 Laporte, O. & Uhlenbeck, G. E. Application of spinor analy-
sis to the Maxwell and Dirac equations. Physical Review 37,
1380 (1931).
25 Kalnins, E. G., Miller Jr, W. & Williams, G. C. Matrix op-
erator symmetries of the Dirac equation and separation of
variables. Journal of mathematical physics 27, 1893-1900
(1986).
26 Hostler, L. Relativistic Coulomb Sturmian matrix elements
and the Coulomb Green’s function of the second-order Dirac
equation. Journal of mathematical physics 28, 2984-2989
(1987).
27 Dyall, K. G. Interfacing relativistic and nonrelativistic meth-
ods. I. Normalized elimination of the small component in the
modified Dirac equation. The Journal of chemical physics
106, 9618-9626 (1997).
28 Manakov, N. & Zapriagaev, S. Solution of the Dirac-Coulomb
problem by the second order Dirac equation approach. Phys-
ica Scripta 1997, 36 (1997).
29 Thylwe, K.-E. Amplitude-phase methods for analyzing the
radial Dirac equation: calculation of scattering phase shifts.
Physica Scripta 77, 065005 (2008).
30 Thylwe, K.-E. A new amplitude-phase method for analyzing
scattering solutions of the radial Dirac equation. Journal of
Physics A: Mathematical and Theoretical 41, 115304 (2008).
31 Gerritsma, R. et al. Quantum simulation of the Dirac equa-
tion. Nature 463, 68-71 (2010).
32 Dongfang, X. D. Relativistic Equation Failure for LIGO Sig-
nals. Mathematics & Nature 1, 202103 (2021).
33 Dongfang, X. D. Dongfang Com Quantum Equations for
LIGO Signal. Mathematics & Nature 1, 202106 (2021).
34 Dongfang, X. D. Com Quantum Proof of LIGO Binary Merg-
ers Failure. Mathematics & Nature 1, 202107 (2021).
35 Dongfang, X. D. On the relativity of the speed of light. Math-
ematics & Nature 1, 202101 (2021).
36 Dongfang, X. D. The Morbid Equation of Quantum Numbers.
Mathematics & Nature 1, 202102 (2021)
37 Dongfang, X. D. Dongfang Modified Equations of Molecular
Dynamics. Mathematics & Nature 1, 202104 (2021).
38 Dongfang, X. D. Dongfang Modified Equations of Electro-
magnetic Wave. Mathematics & Nature 1, 202108 (2021).
39 Dongfang, X. D. Nuclear Force Constants Mapped by Yukawa
Potential. Mathematics & Nature 1, 202109 (2021).
40 Dongfang, X. D. Dongfang Solution of Induced Second Order
Dirac Equations. Mathematics & Nature 2, 202203 (2022).
41 Dongfang, X. D. The End of Yukawa Meson Theory of Nu-
clear Forces. Mathematics & Nature 1, 202110(2021).
42 Dongfang, X. D. The End of Klein-Gordon Equation for
Coulomb Field. Mathematics & Nature 2, 202201 (2022).
43 Dongfang, X. D. The End of Isomeric Second Order Dirac Hy-
drogen Equations. Mathematics & Nature 2, 202204 (2022).
44 Dongfang, X. D. Dongfang Angular Motion Law and Opera-
tor Equations. Mathematics & Nature 1, 202105 (2021).
45 Chen, R. The Optimum Differential Equations. Chinese
Journal of Engineering Mathematics, 91, 82-86(2000).
46 Chen, R. The Uniqueness of the Eigenvalue Assemblage for
Optimum Differential Equations, Chinese Journal of Engi-
neering Mathematics 20, 121-124(2003).
47 Li, S.-C. & Li, X.-G. High-order compact methods for the
nonlinear Dirac equation. Computational and Applied Math-
ematics 37, 6483-6498 (2018).
Latest findings
Dongfang Challenge Solution of Dirac Hydrogen Equation
In quantum mechanics, the Schrödinger equation is used to describe the bound state system, and the exact solution of the equation needs to be determined by boundary conditions. However, the size of an atomic nucleus is usually not considered, and the boundary condition that the proposed wave function should meet is only a rough form. The rough boundary condition causes the S-state wave function of the exact solution of the Klein-Gordon equation and Dirac equation of the so-called relativistic quantum mechanics describing the bound state of the Coulomb field to diverge at the coordinate origin and makes the hydrogen-like atom with the nuclear charge number Z>137 appear unreal virtual energy. The divergence of the wave function means that the probability density of the electron or meson appearing near the nucleus will increase rapidly. What it predicts is the untrue conclusion that the hydrogen-like atom in S-state will collapse rapidly into a neutron-like atom. Considering that the atomic nucleus has a certain radius, the exact boundary conditions that the wave function should satisfy are given here, and then the Dirac equation of hydrogen-like atom is solved again, and a new exact solution without wave function divergence and virtual energy is obtained. Surprisingly, the exact boundary condition makes the angular quantum number naturally regress to the eigenvalue determined by the exact solution of the equation. It has no contribution to the quantized energy, which means that the angular quantum number constructed by Dirac electron theory is denied. Unlike the Dirac energy level formula, the new energy level formula corresponding to the solution of the Dirac equation for the hydrogen-like atom satisfying the exact boundary conditions has no so-called fine structure, and its accuracy is equivalent to the accuracy of the Bohr energy level. The exact boundary condition solution poses a serious challenge to the Dirac relativistic quantum mechanics, which is called the challenging solution of the Dirac equation for the hydrogen atom.