MATHEMATICS & NATURE
Mathematics,Physics, Mechanics & Astronomy
Random Vol. 2 Sn: 202209
https://mathnature.github.io
c
⃝ Mathematics & Nature–Free Media of Eternal Truth, China, 2022 https://orcid.org/0000-0002-3644-5170
.
Article
.
Mathematics and Physics
The End of Dirac Hydrogen Equation in One Dimension
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
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 sp ecific physical mo del 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 numb er 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.
Keywords: Dongfang unitary principle; One-dimensional hydrogen atom; Dirac equation; Imagi-
nary number energy; Existence and uniqueness theorems for solutions of differential equations.
PACS number(s):
03.65.Pm—Relativistic wave equations; 03.65.Ge—Solutions of wave equation-
s: bound states; 02.30.Gp—Special functions; 02.30.Hq—Ordinary differential equations; 02.30.Jr—
Partial differential equations; 02.60.Lj—Ordinary and partial differential equations; boundary value
problems; 32.10.Fn—Fine and hyperfine structure.
1 Introduction
Testing the reliability of mathematical computing pro-
cesses in physics and the data provided by some well-
known experimental reports has led to a large number
of breakthrough discoveries that have subverted tradi-
tional knowledge, the most important of which is the
widespread misconceptions in modern physics. Rather
than continue to deify the beautiful and illusory stories
that have been painstakingly written in the past, it is
better to confront these facts directly, which is the fun-
damental reason why physics theory cannot achieve new
breakthroughs.
The end
[1, 2]
of Yukawa’s nuclear meson theory
[3-5]
,
the end
[6]
of the Klein Gordon equation for Coulomb
field
[7-13]
, the end
[14, 15]
of the teratogenicity theory for
the Dirac hydrogen equation
[16-32]
, the end
[33]
of the in-
duced second order Dirac equation for hydrogen, the end
[34]
of the isomeric second order Dirac equation for hydro-
gen, the end
[35]
of the expected solution for the standard
second order Dirac equation, the challenge solution
[36]
for the Dirac equation for hydrogen, the negation of the
neutron state solution
[37]
and ground state solution
[38]
for
the modified Dirac equation for hydrogen to the Dirac
energy level formula, these new discoveries have made
)Citation: Dongfang, X. D. The End of Dirac Hydrogen Equation in One Dimension. Mathematics & Nature 2, 202209 (2022).
The mathematical calculations of all papers on the Dirac equation of one-dimensional hydrogen atom published in a large number
of famous journals are distorted, and the so-called exact solutions given there are all pseudo solutions. This paper does not comment
specifically on the pseudoscience theory unanimously advocated by famous journals one by one and only gives the inevitable mathematical
solution and the final conclusion of the Dirac equation for one-dimensional hydrogen atom.
202209-2 X. D. Dongfang The End of Dirac Hydrogen Equation in One Dimension
it clear to us that relativistic quantum mechanics has
evolved into a distorted logic that uses formal math-
ematics to piece together expectations, while relevant
experimental reports have all claimed to verify the con-
clusions of actually distorted logic. Therefore, inferences
that conform to standard mathematical operation rules
are forced to be abandoned and denied because they do
not meet expectations. From this perspective, modern
physics not only hides many intentional or unintention-
al errors, but also hides serious ethical issues. From
the physical perspective of whether an equation truly
describes natural laws, we have to doubt the reliability
of the Dirac equation. However, from a purely mathe-
matical perspective, where differential equations can be
written at will, the ingenious construction of Dirac equa-
tions and the introduction of interesting mathematical
problems that are often unforeseen in writing differential
equations still has great charm.
Applying Dongfang unitary principle can reveal more
and fatal logical contradictions hidden in modern physic-
s. The Dongfang unitary principle points out: There
is a definite transformation relationship between differ-
ent metrics describing the natural law, and the natural
law itself does not change due to the selection of dif-
ferent metrics. When the mathematical expression of
natural laws under different metrics is transformed into
one metric, the result must be the same as the inher-
ent form under this metric, 1=1, meaning the trans-
formation is unitary.
[39-43]
Mathematical equations are
one of the best tools for accurately describing the natu-
ral laws. According to Dongfang unitary principle, the
results of the transformation of mathematical forms of
natural laws from different metrics to the same metric
must be unique. For example, the rectangular coordi-
nate system, the cylindrical coordinate system, and the
spherical coordinate system naturally constitutes three
different metrics of wave equation theory. The wave e-
quations of hydrogen atoms are handled in spherical co-
ordinates. What new things does processing in cylindri-
cal coordinates bring to mathematics? I have been us-
ing the Dongfang unitary principle to test complex and
huge physical systems and some mathematical theories,
and have made significant progress. Although attempts
to publish these research results have encountered un-
precedented resistance and failed in the past 40 years,
today, with the advent of the Internet, any researcher
has been able to freely publish their research and accept
the test of readers. Truth, although often insignificant
to individuals and groups who only pursue or maintain
fame and fortune, has infinite charm for those who love
it.
Since the birth of quantum mechanics, the mathe-
matical essence of quantum mechanics has been deeply
hidden. This is why it is difficult to solve those most
fundamental quantum mechanical problems such as the
morbid
[42]
equation of quantum numbers. Establish-
ing wave equations and solving them accurately are the
essence of quantum mechanics. Handling wave equa-
tions without considering the exact boundary condition-
s of atomic nucleus size and the boundary conditions
written in atomic nucleus size constitute two metrics of
wave equation theory. The unitary principle requires
that wave equations either have consistent solutions un-
der two boundary conditions, or make a choice between
different exact solutions under two boundary conditions.
We systematically tested the Dirac theory of hydrogen
atoms using the unitary principle and obtained two clear
conclusions: 1) The exact solution of the Dirac equation
of hydrogen atoms under rough boundary conditions im-
plies difficulties that cannot be eliminated, such as wave
function divergence at the coordinate origin and imagi-
nary energy, and therefore belongs to a formal solution;
2) The exact solution of the Dirac equation for hydrogen
atoms under precise boundary conditions is an inevitable
solution, which eliminates the difficulties of wave func-
tion divergence and imaginary energy. However, the ac-
curacy of the obtained energy level formula is compara-
ble to that of the Bohr energy level formula, which does
not meet expectations. This requires looking for reasons
from the origin of the establishment of wave equations,
rather than writing seemingly profound qualitative ex-
planations that can satisfy some people’s wishes.
The mainstream solutions of the Klein-Gordon
Coulomb field equation, coupled second-order Dirac hy-
drogen equation, and Dirac hydrogen equation of rela-
tivistic quantum mechanics has all been ended. Here,
the Dirac equation for one-dimensional hydrogen atoms
is further ended. If we choose three-dimensional wave
equation and the one-dimensional wave equation as t-
wo metrics to test the theory of quantum mechanics
wave equation, then according to the unitary principle,
the exact solution of three-dimensional wave equation
must contain the exact solution of the corresponding
one-dimensional wave equation. I discussed in detail the
Dirac equation for one-dimensional hydrogen atoms. Its
exact solution cannot be obtained from the Dirac equa-
tion for three-dimensional hydrogen atoms, and the ex-
act solution does not exist. This declared the end of the
Dirac equation for one-dimensional hydrogen atoms.
2 Conclusions and comments
I have listed the standard one-dimensional Dirac equa-
tions for hydrogen like atoms and combined them into six
systems of one-dimensional first order Dirac differential
equation. Some one-dimensional first order Dirac dif-
ferential equations have virtual coefficients, while some
one-dimensional first order Dirac equations have real co-
efficients. Six one-dimensional first order Dirac differen-
tial equation systems correspond to two one-dimensional
second order Dirac equations. Then it is proved in turn
that the one-dimensional second order Dirac equation
Mathematics & Nature (2022) Vol. 2 No. 2 202209-3
has no expected solution, while the one-dimensional first
order Dirac equation with imaginary and real coefficients
has no expected solution, declaring the end of the one-
dimensional Dirac hydrogen equation. The expression of
the one-dimensional radial angular motion operator nat-
urally occurring in the one-dimensional Dirac equation is
contrary to the definition of the radial motion operator
in the Dirac electron theory. It should be noted that oth-
er articles on the one-dimensional Dirac equation
[59-84]
actually have a wrong understanding of one-dimensional
Dirac equation.
If we review the literature after independently com-
pleting the main calculations of the Dirac equation for
one-dimensional hydrogen atoms, we will find that the
independently derived conclusions are inconsistent with
the conclusions of the papers on the relativistic wave e-
quation for one-dimensional hydrogen atoms that have
been published continuously since a long time ago. How-
ever, the process of independent calculation is rigorous!
At this point, we began to change our cognition and no-
ticed that the calculations in many papers actually se-
riously distorted the rules of mathematical operations.
Compared to our learning habits of trying to memorize
mathematical steps in textb ooks and then dealing with
some related exercises, closing textbooks to learn a new
physics model and strictly following mathematical op-
eration rules to independently deduce and then reading
textbooks will always yield more and significantly dif-
ferent results. The latter allows us not to be distracted
by generally accepted conclusions that are not actually
true when trying to discover the essence of a problem,
and in particular allows us to maintain the advantage of
independent thinking to discover those undiscovered but
crucial subversive conclusions. The educational system
determines the criteria for talent selection, and a large
number of talented people with strong memories have
been given opportunities. As a result, textbooks and
well-known mainstream journals have been cloned from
generation to generation. The development of physic-
s today, including numerous experimental reports and
so-called achievements in experimental engineering, is
fraught with errors and lies that we cannot discern and
clarify using ordinary research methods.
Whether the Dirac equation has truly achieved bril-
liant achievements in physics is not the most important
issue. According to Dongfang’s unitary principle, the
most important question is why the Dirac equation and
Klein Gordon equation based on the same relativistic
momentum and energy relationship are so different, re-
gardless of whether the relativistic momentum relation-
ship can be proven to be actually not true. Even assum-
ing that there is a causal relationship between the Dirac
matrix and the intrinsic spin of the electron that cannot
be proven, the wave equation constructed by the Dirac
matrix is actually a set of partial differential equations.
So, why can’t we directly construct a first order differen-
tial equation system that describes the hydrogen atom,
or can we construct other different first order partial d-
ifferential equations? The conclusion of every question
is important. Then, what are the essential differences
between the first order partial differential equations and
the second order partial differential equations, so that
we have to choose Dirac’s first order partial differential
equations and abandon the second order partial differen-
tial equations when dealing with the so-called relativis-
tic effects of hydrogen atoms? Whether viewed from
a physical or mathematical perspective, the problems
posed by the Dirac equation may b e enough to change
the traditional cognition of mathematicians and physi-
cists. Regarding the Dirac equation for hydrogen atoms,
the three-dimensional Dirac equation still has a mathe-
matically meaningful formal eigensolution. So why is the
one-dimensional Dirac equation forced to end because it
has no formal eigensolution due to complex number en-
ergy? In fact, one-dimensional wave equations belong to
the special case where the traditional angular quantum
number l = 0 in quantum mechanics, while the angular
quantum number defined by Dirac is κ = ±1, ±2, · · · ,
excluding the traditional zero angular quantum number,
which just avoids the serious logic difficulties hidden in
one-dimensional equations. The intrinsic solution of the
three-dimensional Dirac equation cannot contain the in-
trinsic solution of the one-dimensional Dirac equation,
which is a logical contradiction in itself.
Numerous papers on the Dirac equation
[59-84]
for one-
dimensional hydrogen atoms advocated by various sci-
entific journals hide false calculations, absurd logic, and
erroneous conclusions. The readers do not have to
comment on intentional or unintentional errors one by
one, but only to explore the correct handling and infer-
ence of the one-dimensional Dirac equation for hydro-
gen atoms. Here, I prove that the Dirac equation for
one-dimensional hydrogen atoms does not have an in-
trinsic solution for energy quantization, thereby ending
the Dirac equation for one-dimensional hydrogen atoms.
It is well known that the energy quantized form eigenso-
lutions of the Dirac equation for three-dimensional hy-
drogen atoms are in line with expectations. According
to Dongfang’s unitary principle, differences in the con-
clusions of mathematical equations of one-dimensional,
two-dimensional, and three-dimensional wave equation-
s of the same physical model cannot lead to essential
differences between the existence and non-existence of
quantized energy eigensolutions. Therefore, the termi-
nation of the Dirac equation for one-dimensional hydro-
gen atoms has an important impact on the systematic
testing of the theory of Dirac for hydrogen atoms.
From the Bohr model of the hydrogen atom, to the
Schrodinger equation of the hydrogen atom, to the Dirac
equation of the hydrogen atom, to quantum computa-
tion and quantum information, behind the flourishing
phenomena presented by quantum mechanics, there are
not only a large number of principled errors in cognition
and computation, but also lies hidden by gorgeous math-
202209-4 X. D. Dongfang The End of Dirac Hydrogen Equation in One Dimension
ematical forms. Scientific theories and exp erimental re-
ports can tolerate computational flaws and cognitive er-
rors, but they should not tolerate lies that intentionally
distort mathematics and fabricate observational data to
create physical achievements. Famous scientific journals
should take the lead in ending the behavior of distorting
mathematical calculations and fabricating observational
data, and making real contributions to the development
of strict scientific theories, rather than trying to stifle
those great discoveries
[85-87]
.
1 Dongfang, X. D. Nuclear Force Constants Mapped by Yukawa
Potential. Mathematics & Nature 1, 202109 (2021).
2 Dongfang, X. D. The End of Yukawa Meson Theory of Nu-
clear Forces. Mathematics & Nature 1, 202110(2021).
3 Yukawa, H. On the Interaction of Elementary Particles I.
Proc. Phys. Math. Soc. Jpn. 17, 48 (1935).
4 Yukawa, H. & Sakata, S. On the interaction of elementary
particles II. Proceedings of the Physico-Mathematical Society
of Japan. 3rd Series 19, 1084-1093 (1937).
5 Yukawa, H. Models and methods in the meson theory. Re-
views of Modern Physics 21, 474 (1949).
6 Dongfang, X. D. The End of Klein-Gordon Equation for
Coulomb Field. Mathematics & Nature 2, 202201 (2022).
7 Kanth, A. R. & Aruna, K. Differential transform method
for solving the linear and nonlinear Klein–Gordon equation.
Computer Physics Communications 180, 708-711 (2009).
8 Klein, O. Quantentheorie und f¨unfdimensionale Relativ-
it¨atstheorie. Zeitschrift f¨ur Physik 37, 895-906 (1926).
9 Gordon, W. Der comptoneffekt nach der schr¨odingerschen
theorie. Zeitschrift f¨ur Physik 40, 117-133 (1926).
10 Scott, A. C. A nonlinear Klein-Gordon equation. American
Journal of Physics 37, 52-61 (1969).
11 Burt, P. & Reid, J. Exact solution to a nonlinear Klein-
Gordon equation. Journal of Mathematical Analysis and Ap-
plications 55, 43-45 (1976).
12 Detweiler, S. Klein-Gordon equation and rotating black holes.
Physical Review D 22, 2323 (1980).
13 Feshbach, H. & Villars, F. Elementary relativistic wave me-
chanics of spin 0 and spin 1/2 particles. Reviews of Modern
Physics 30, 24 (1958).
14 Greiner, W. Relativistic Quantum Mechanics: Wave Equa-
tions, Springer, 2000, 3rd Edition, pp225.
15 Dongfang, X. D. The End of Teratogenic Simplified Dirac Hy-
drogen Equations. Mathematics & Nature 2, 202202 (2022).
16 Dirac, P. A. M. The quantum theory of the electron. Proceed-
ings of the Royal Society of London. Series A, Containing
Papers of a Mathematical and Physical Character 117, 610-
624 (1928).
17 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).
18 Dirac, P. A. M. The principles of quantum mechanics, 4th
ed. (Clarendon Press, Oxford, 1962, pp270).
19 Schiff, L. I., Quantum Mechanics, (McGraw Hill Press, New
York, 1949, pp326).
20 Bjorken, J. D., Drell, S.D. Relativistic Quantum Mechanics,
(McGraw–Hill, New York, 1964, pp52. 60)
21 Dirac, P. A. M. Relativistic wave equations. Proceedings of
the Royal Society of London. Series A-Mathematical and
Physical Sciences 155, 447-459 (1936).
22 Weinberg, S. Gravitation and cosmology: principles and ap-
plications of the general theory of relativity. (1972).
23 Ohlsson, T. Relativistic quantum physics: from advanced
quantum mechanics to introductory quantum field theory.
(Cambridge University Press, 2011).
24 Biedenharn, L. C. Remarks on the relativistic Kepler prob-
lem, Phys Rev., 126, 845 (1962).
25 Wong, M. K. F. & Yeh, H. Y. Simplified solution of the Dirac
equation with a Coulomb potential, Phys. Rev. D 25, 3396
(1982).
26 Su, J. Y. Simplified solution of the Dirac-Coulomb equation,
Phys. Rev. A 32, 3251 (1985).
27 Ciftci, H., Hall, R. L. & Saad, N. Iterative solutions to the
Dirac equation, Phys.Rev. A 72, 022101 (2005).
28 Pudlak, M., Pincak, R. & Osipov, V. A. Low-energy electron-
ic states in spheroidal fullerenes, Phys. Rev. B, 74, 235435
(2006).
29 Voitkiv, A. B., Najjari, B. & Ullrich, J. Excitation of heavy
hydrogenlike ions in relativistic collisions, Phys. Rev. A, 75,
062716 (2007).
30 Ferrer, F., Mathur, H., Vachaspati, T. & Starkman, G. Zero
modes on cosmic strings in an external magnetic field, Phys.
Rev. D, 74, 025012 (2006).
31 Boulanger, N., Spindel, P. & Buisseret, F. Bound states of
Dirac particles in gravitational fields, Phys. Rev. D, 74,
125014 (2006).
32 Crater, H. W., Wong, C. Y. & Alstine, P. V. Tests of two-
body Dirac equation wave functions in the decays of quarko-
nium and positronium into two photons, Phys. Rev. D, 74,
054028 (2006).
33 Dongfang, X. D. Dongfang Solution of Induced Second Order
Dirac Equations. Mathematics & Nature 2, 202203 (2022).
34 Dongfang, X. D. The End of Isomeric Second Order Dirac Hy-
drogen Equations. Mathematics & Nature 2, 202204 (2022).
35 Dongfang, X. D. The End of Expectation for True Second
Order Dirac Equation. Mathematics & Nature 2, 202205
(2022).
36 Dongfang, X. D. Dongfang Challenge Solution of Dirac Hy-
drogen Equation. Mathematics & Nature 2, 202206 (2022).
37 Dongfang, X. D. Neutron State Solution of Dongfang Mod-
ified Dirac Equation. Mathematics & Nature 2, 202207
(2022).
38 Dongfang, X. D. Ground State Solution of Dongfang Modified
Dirac Equation. Mathematics & Nature 2, 202208 (2022).
39 Chen, R. Limitations of conservation laws in fractionally in-
ertial reference frames. Physics Essays 24, 413-419(2011).
40 Chen, R. Correction to Solution of Dirac Equation(Unitary
Principle and Real Solution of Dirac Equation). arXiv.org:
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. Dongfang Angular Motion Law and Opera-
tor Equations. Mathematics & Nature 1, 202105 (2021).
44 Einstein, A. On the electrodynamics of moving bodies. An-
nalen der physik 17, 891-921 (1905).
45 Einstein, A. Relativity: The Special and the General Theory-
100th Anniversary Edition. (Princeton University Press,
2019).
46 Minkowski, H. Die Grundgleichungen f¨ur die elektromag-
netischen Vorg¨ange in bewegten K¨orpern. Nachrichten
von der Gesellschaft der Wissenschaften zu G¨ottingen,
Mathematisch-Physikalische Klasse 1908, 53-111 (1908).
47 Einstein, A. in The Meaning of Relativity, 54-75 (Springer,
1922).
48 Einstein, A. The theory of relativity. (Springer, 1950).
Mathematics & Nature (2022) Vol. 2 No. 2 202209-5
49 Bergmann, P. G. Introduction to the Theory of Relativity.
(Courier Corporation, 1976).
50 Pauli, W. Theory of relativity. (Courier Corporation, 2013).
51 Dirac, P. A. M. General theory of relativity. Vol. 14 (Prince-
ton University Press, 1996).
52 Einstein, A. Die Feldgleichungun der Gravitation. Sitzungs-
ber. K. Preuss. Akad. Wiss., 844-847 (1915).
53 Einstein, A. The foundation of the general theory of relativ-
ity. Annalen Phys. 14, 769-822 (1916).
54 Zhang, Y. Z. Special relativity and its experimental founda-
tion. Vol. 4 (World Scientific, 1997).
55 Ehlers, J. Contributions to the relativistic mechanics of con-
tinuous media. General Relativity and Gravitation 25, 1225-
1266 (1993).
56 Janssen, M. & Mecklenburg, M. From classical to rela-
tivistic mechanics: Electromagnetic models of the electron.
(Springer, 2006).
57 Chen, R. The Optimum Differential Equations. Chinese
Journal of Engineering Mathematics 17, 82-86 (2000).
58 Chen, R. The Uniqueness of the Eigenvalue Assemblage for
Optimum Differential Equations, Chinese Journal of Engi-
neering Mathematics 20, 121-124(2003).
59 Critchfield, C. L. Scalar potentials in the Dirac equation.
Journal of Mathematical Physics 17, 261-266 (1976).
60 Lapidus, I. R. Relativistic one-dimensional hydrogen atom.
American Journal of Physics 51, 1036-1038 (1983).
61 Spector, H. N. & Lee, J. Relativistic one-dimensional hydro-
gen atom. American Journal of Physics 53, 248-251 (1985).
62 Grosse, H. New solitons connected to the Dirac equation.
Physics reports 134, 297-304 (1986).
63 Calkin, M., Kiang, D. & Nogami, Y. Proper treatment of the
delta function potential in the one-dimensional Dirac equa-
tion. American Journal of Physics 55, 737-739 (1987).
64 Coutinho, F. A. B., Nogami, Y. & Toyama, F. General aspect-
s of the bound-state solutions of the one-dimensional Dirac
equation. American Journal of Physics 56, 904-907 (1988).
65 Calkin, M., Kiang, D. & Nogami, Y. Nonlocal separable po-
tential in the one-dimensional Dirac equation. Physical Re-
view C 38, 1076 (1988).
66 Dom´ınguez-Adame, F. & Gonz´alez, M. Solvable linear po-
tentials in the Dirac equation. Europhysics Letters 13, 193
(1990).
67 Anderson, A. Intertwining of exactly solvable Dirac equations
with one-dimensional potentials. Physical Review A 43, 4602
(1991).
68 Nogami, Y. & Toyama, F. Transparent potential for the one-
dimensional Dirac equation. Physical Review A 45, 5258
(1992).
69 Nogami, Y. & Toyama, F. Supersymmetry aspects of the
Dirac equation in one dimension with a Lorentz scalar po-
tential. Physical Review A 47, 1708 (1993).
70 Alonso, V., De Vincenzo, S. & Mondino, L. On the bound-
ary conditions for the Dirac equation. European Journal of
Physics 18, 315 (1997).
71 Nogami, Y. & Toyama, F. Reflectionless potentials for the
one-dimensional Dirac equation: Pseudoscalar potentials.
Physical Review A 57, 93 (1998).
72 Bocquet, M. Some spectral properties of the one-dimensional
disordered Dirac equation. Nuclear Physics B 546, 621-646
(1999).
73 Gu, X. Y., Ma, Z. Q. & Dong, S. H. Exact solutions to the
Dirac equation for a Coulomb potential in D+ 1 dimension-
s. International Journal of Modern Physics E 11, 335-346
(2002).
74 Hiller, J. R. Solution of the one-dimensional Dirac equation
with a linear scalar potential. American Journal of Physics
70, 522-524 (2002).
75 Nieto, L., Pecheritsin, A. & Samsonov, B. F. Intertwining
technique for the one-dimensional stationary Dirac equation.
Annals of Physics 305, 151-189 (2003).
76 Jiang, Y., Dong, S. H., Antillon, A. & Lozada-Cassou, M.
Low momentum scattering of the Dirac particlewith an asym-
metric cusp potential. The European Physical Journal C-
Particles and Fields 45, 525-528 (2006).
77 Ma, Z. Q., Dong, S. H. & Wang, L. Y. Levinson theorem for
the Dirac equation in one dimension. Physical Review A 74,
012712 (2006).
78 Gonz´alez-D´ıaz, L. & Villalba, V. M. Resonances in the one-
dimensional Dirac equation in the presence of a point inter-
action and a constant electric field. Physics Letters A 352,
202-205 (2006).
79 Hossain, M. S. & Faruque, S. Influence of a generalized uncer-
tainty principle on the energy spectrum of (1+ 1)-dimensional
Dirac equation with linear potential. Physica Scripta 78,
035006 (2008).
80 Peres, N. Scattering in one-dimensional heterostructures de-
scribed by the Dirac equation. Journal of Physics: Con-
densed Matter 21, 095501 (2009).
81 Panella, O., Biondini, S. & Arda, A. New exact solution of
the one-dimensional Dirac equation for the Woods–Saxon po-
tential within the effective mass case. Journal of Physics A:
Mathematical and Theoretical 43, 325302 (2010).
82 Gerritsma, R. et al. Quantum simulation of the Dirac equa-
tion. Nature 463, 68-71 (2010).
83 Allahverdiev, B. P. & Tuna, H. One-dimensional q-Dirac e-
quation. Mathematical methods in the applied sciences 40,
7287-7306 (2017).
84 Silva, T. L., Taillebois, E. R., Gomes, R., Walborn, S. P. &
Avelar, A. T. Optical simulation of the free Dirac equation.
Physical Review A 99, 022332 (2019).
85 Dongfang, X. D. Relativistic Equation Failure for LIGO Sig-
nals. Mathematics & Nature 1, 202103 (2021).
86 Dongfang, X. D. Dongfang Com Quantum Equations for
LIGO Signal. Mathematics & Nature 1, 202106 (2021).
87 Dongfang, X. D. Com Quantum Proof of LIGO Binary Merg-
ers Failure. Mathematics & Nature 1, 202107 (2021).