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Article
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Mathematics and Physics
Neutron State Solution of Dongfang Modified Dirac Equation
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
The challenging solution of the Dirac equation of the Coulomb field satisfying exact boundary con-
ditions is further studied. If the Dirac equation is effective, then the intrinsic angle quantum number
determined by the exact solution of the equation must be introduced to modify the Dirac equation to
make it self-consistent. The solution of the modified Coulomb field Dirac equation satisfying the ex-
act boundary conditions leads to a variety of breakthrough conclusions that overturns the traditional
physical thinking. 1) The modified Dirac equation of Coulomb field has a neutron state solution cor-
resp onding to the neutron structure, and the binding energy of the neutron has a certain value, while
the calculation result of the intrinsic radius written in the accurate boundary condition is equivalent to
the size of the atomic nucleus; 2) The energy eigenvalue formula of the modified Coulomb field Dirac
equation contains only radial quantum numbers and is independent of the intrinsic angular quantum
numbers, where the zero radial quantum number energy level corresponds to the neutron state, and
the nonzero radial quantum number energy level corresponds to the atomic state, and the accuracy
of each atomic state energy level is equivalent to the Bohr energy level, while the Dirac energy level
formula as the expected solution no longer exists; 3) The intrinsic angular quantum number of the
mo dified Coulomb field Dirac equation indirectly negates the Dirac algebra theory that constructs
the Dirac angular quantum number beyond the mathematical calculation rules; 4) The neutron state
wave function component of the modified Coulomb Dirac equation is the terminated Yukawa p oten-
tial function, which reflects the physics dilemma that the wave function is wrongly described as a
p otential function to establish a Yukawa pseudo-scientific theory that can also be infinitely developed
and admired by physicists around the world, exposing the false prosperity of modern physics. It is
concluded that the Dirac equation of the Coulomb field defined by Dirac algebra is not self-consistent,
and the exact boundary condition solution of the modified self-consistent Dirac equation of the an-
gular quantum number regression intrinsic physical quantity negates the Dirac electronic theory of
fabricating the energy level formula of the fine structure of the hydrogen atom spectrum, and the
microscopic quantum theory urgently needs to find a more reasonable wave equation that describes
the fine spectral structure of the hydrogen atom.
Keywords: Unitary principle; Dirac equation; inevitable solution; quantum neutron radius.
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
The Loren transformation in the sense of relativity
[1-3]
is not a set of spatiotemporal parameter transfor-
mations with complete concepts. The spatiotemporal
transformation of the complete concept in the sense of
relativity shows that the constant speed of light is on-
ly a mathematical definition applicable to the isolated
reference frame. However, the theory of relativity based
on the assumption that the speed of light is constant
seems to have achieved irreplaceable and surprising re-
sults in many aspects. In particular, a large number of
experimental reports claim to confirm various inferences
of relativity. This forces us to try to break through the
traditional thinking mode of either or, and to test the
logical self consistency of each major developmental the-
ory of relativity and the authenticity of those influential
experimental reports in a more in-depth and detailed
way. Theoretically, there is no essential difference be-
tween relativity and Newton’s theory. If relativity is a
scientific theory, the conclusion of relativity of all prob-
lems should be the same as that of Newtonian mechanics
when the relative speed is far less than the speed of light.
The relativistic Dirac equation
[4, 5]
is the product of
the combination of special relativity and quantum me-
chanics with morbid equation of quantum numbers
[6]
.
Then, how can a theory with serious defects be combined
with another theory with serious defects to achieve a his-
torical breakthrough in physics? In fact, the traditional
)Citation: Dongfang, X. D. Neutron State Solution of Dongfang Modified Dirac Equation. Mathematics & Nature 2, 202207 (2022).
*This article is a rewritten version of the article entitled “Correction to Solution of Dirac Equation” published on the Internet by the
author’s signature Rui Chen in his early years.
202207-2 X. D. Dongfang Neutron State Solution of Dongfang Modified Dirac Equation
solution of the evolution Dirac equation for describing
the hydrogen atom leads to various morbid problem-
s, such as the virtual energy and the divergence of S-
state wave function
[7]
. The latter obviously leads to the
counterfactual conclusion of the collapse of the universe,
which are unacceptable. But the Dirac equation pieced
together the most expected energy level formula of the
fine structure of the hydrogen atom in that era. This is
one of the elusive and strange phenomena of quantum
mechanics. Constructing equations that do not conform
to mathematical and physical logic may get the expect-
ed energy level formula. In this case, we always try to
find the beneficial factors in the distorted logic so as to
correct the distorted logic. Replacing the rough bound-
ary condition without considering the size of the atomic
nucleus with the exact boundary condition with the size
of the atomic nucleus, the challenge solution obtained
by re solving the Dirac equation is self consistent, but
the expected energy level formula can not be obtained.
At the same time, it also brings more mathematical and
physical problems to be solved. Perhaps only physicists
who study the elementary proof of Fermat’s final theo-
rem can find the endless plausible and complex problems
concealed in quantum mechanics. Not every physics pa-
per or theory can reasonably describe the laws of nature.
Only those physics papers or theories that can inspire
people to break through the confinement of tradition-
al incorrect ideas are worthy of in-depth study. Based
on this idea, we further study the challenging solution of
the Dirac equation of hydrogen like atom satisfying exac-
t boundary conditions. The challenge solution may not
be popular because it fails to piece together the expect-
ed formula of fine energy level structure of the hydrogen
atom, but it can break through the confinement of tra-
ditional incorrect physical ideas better than almost all
the conclusions about the Dirac equation in history.
There is another strange phenomenon in quantum me-
chanics. Some wave equations established from different
angles are not exactly the same, although they have en-
ergy level formulas with the same accuracy but different
expressions. This is not in line with the Dongfang uni-
tary principle that natural science theories must follow.
Whether testing the logical self consistency of the old
theory or establishing a new self consistency theory, the
unitary principle is very useful. The specific contents
of this universal principle are: There is a definite trans-
formation relationship between different metrics describ-
ing 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 under dif-
ferent metrics is transformed into one metric, the result
must be the same as the inherent form under this met-
ric, 1=1, meaning the transformation is unitary
[3, 8, 9]
.
Its concise expression is as follows: The result of the
transformation of the mathematical form of the natural
law in different metrics to the same metric is unique.
The unitary principle is the most effective principle to
test the self consistency of theoretical and experimental
reports. Conforming to the unitary principle is a neces-
sary condition for the self consistency of theoretical logic.
Although the theory that conforms to the unitary prin-
ciple may not be the true description of natural laws or
social laws, the theory that violates the unitary princi-
ple must be incorrect. If a theory used to describe the
laws of nature does not conform to the unitary princi-
ple, it must contain principled mathematical calculation
errors or philosophical errors. Nuclear meson theory
[10]
,
relativity
[11-13]
based on the pseudo proposition of con-
stant speed of light, quantum mechanics
[14]
concealing
morbid equations of quantum umbers, the thermody-
namic theory with completely wrong basic equations
[15]
and Maxwell electromagnetic theory with wrong equa-
tion solution
[16-18]
do not accord with the unitary prin-
ciple, so they must be treated
[19, 20]
. Hendrik Schon
experiment
[21, 22]
and LIGO’s spiral binary gravitation-
al wave
[23]
cannot pass the test of unitary principle
[24-26]
.
Now, there is a unified scientific method for exposing
the truth that famous scientists or scientific groups fab-
ricate experimental data or confuse different experimen-
tal data and publish the so-called major experimental
reports. Just looking at the challenging problems such
as the morbid equation of quantum numbers hidden in
the Schr¨odinger equation, which is considered to be per-
fect, we know that quantum mechanics has not been
really understood. The Bonn statistical interpretation
of the wave function of quantum mechanics, which has
always been mysterious, is abstract and vague. If the
probability of electron particles appearing in the speci-
fied region in the Coulomb field is written independent-
ly, it will be found that the so-called probability density
function does not satisfy the Schr¨odinger equation or
the Dirac equation. God can’t roll the dice
[27, 28]
. That’s
right. Therefore, the concept of orbital density is pro-
posed here, but these may not be the ultimate answer.
From this corner, theoretical physics is still in its infan-
cy, far from entering the view where its heyday can be
predicted.
In fact, even assuming that the theory of relativity
and quantum mechanics are correct, studying whether
the construction of the Dirac equation conforms to the
unitary principle will lead to many important problem-
s that have not been discovered in the past
[29, 30]
. So
we return to the mathematics of the Dirac equation and
gradually discuss the logic and inference of the Dirac
equation that is rarely known. This paper introduces
the advanced research results of the challenging solution
of the Dirac equation for the hydrogen-like atom satis-
fying the exact boundary conditions. The intrinsic an-
gular quantum number determined by the equation is
introduced to replace Dirac’s definition of angular quan-
tum number to modify the Dirac equation of Coulom-
b field, and the precise boundary condition containing
the size of atomic nucleus is used to replace the rough
boundary condition without the size of atomic nucleus,
Mathematics & Nature (2022) Vol. 2 No. 1 202207-3
and then the modified Dirac hydrogen equation is solved
accurately. The results show that the modified locally
self-consistent Dirac equation of the hydrogen-like atom
has the lowest energy state solution corresponding to
the neutron state. However, the accuracy of other ener-
gy levels corresponding to the energy level formula as-
sociated with the exact solution of the modified Dirac
equation is equivalent to that of the Bohr energy level
formula. This is not the expected result of theoretical
physics workers. To distort the logic of mathematics in
order to obtain the desired results will not have lasting
vitality. Any new subversive conclusion of the challeng-
ing solution of the Dirac equation for the hydrogen atom
needs to be reprocessed. The neutron state solution of
the modified Dirac equation of the Coulomb field is self-
consistent; The nuclear radius used to write accurate
boundary conditions belongs to the intrinsic parameter
of the modified Dirac equation, which is equivalent to
the radius of neutrons, so the binding energy of neutron-
s seems to have an exact calculation result; The angular
quantum number is the intrinsic parameter of the mod-
ified Dirac equation, which is determined by the exact
solution of the equation, which means that it is illogical
to construct the angular quantum number through the
user-defined Dirac algebra. So, is it possible that the
neutron state of the modified Dirac equation that meet-
s the exact boundary conditions is just another formal
solution that seemingly conforms to the mathematical
and physical logic? Is it another coincidence that the
neutron state solution which seems to describe the neu-
tron structure well
[31, 32]
?
2 Dongfang modified Dirac equation for
hydrogen-like atoms
In Dirac electron theory, dealing the relativistic Dirac
equation for hydrogen like atoms is finally reduced to
solving the first-order radial differential equations about
the wave functions of the upper and lower components
cα · ˆp+
1 0
0 −1
mc
2
−
Ze
2
4πε
0
r
ψ
1
ψ
2
=E
ψ
1
ψ
2
(1)
where α is the Dirac matrix, ˆp = −i~∇, ~ = h/2π with
the plank constant h, c the velocity of light in a vacuum,
and mthe rest mass of electron. α · ˜p is defined by the
Dirac algebra
α · ˜p=
0 −i
i 0
−i~
∂
∂r
−
i~
r
+
i~
r
1 0
0 −1
⌢
κ
(2)
Where the angular quantum number κ =
±1, ±2, ±3, ··· , which is constructed by Dirac al-
gebra independent of mathematics, actually belongs to
artificial angular quantum number The main reason is
that Dirac algebra does not conform to the basic rules
of mathematical operation. This part will be discussed
separately as a special topic. Function transformation
is usually introduced for two-component wave function
ψ
1
ψ
2
=
1
r
F
G
(3)
In Dirac theory, the boundary conditions satisfied by the
wave function
lim
r→0
1
r
F
G
=
0
0
, lim
r→∞
1
r
F
G
= 0
1
r
F (0 < r < ∞)
G (0 < r < ∞)
̸=
±∞
±∞
] (4)
has been interpreted as
lim
r→0
F
G
=
0
0
, lim
r→∞
F
G
= 0
F (0 < r < ∞)
G (0 < r < ∞)
̸=
±∞
±∞
This interpretation makes it difficult to detect the logi-
cal contradiction of the divergence of S-state Dirac wave
function at the coordinate origin
[7]
, and the divergence of
wave function means the abnormal inference of the col-
lapse of the universe, which is one of the fatal problems
of Dirac hydrogen atom theory.
Now let us focus on the angular quantum number in
the Dirac equation. One of the characteristics of quan-
tum mechanics is that the angular quantum number is
the eigensolution of the eigenequation of the angular mo-
mentum operator acting on the angular wave function,
which exists in the wave equation. For example, solv-
ing the Schrodinger equation or the Klein Gordon equa-
tion naturally obtains the eigenvalue of angular momen-
tum. However, Dirac theory does not write the angular
wave equation corresponding to the angular momentum
operator, but constructs the angular quantum number
κ = ±1, ±2, ±3, ··· through the self-defined Dirac al-
gebra independent of mathematics. The so-called Dirac
algebra does not conform to the basic mathematical op-
eration rules. It must be reiterated that the important
symbol of the success of quantum mechanics is that the
quantized angular momentum and quantized energy of
the bound state system are naturally derived from the
exact solution of the wave equation satisfying the bound-
ary conditions. It is futile and a waste to comment too
much on Dirac algebra now. According to the unitary
principle, we can inversely test whether the conclusion
κ = ±1, ±2, ±3, ··· of Dirac algebra is an inevitable
corollary.
If the Dirac equation holds, first assume that the an-
gular quantum number is an intrinsic parameter C of the
Dirac equation, solve the Dirac equation and keep the
calculation consistent from beginning to end, then the
inevitable result of C can be obtained. This means that
C should b e obtained naturally in the process of solving
202207-4 X. D. Dongfang Neutron State Solution of Dongfang Modified Dirac Equation
the Dirac equation. Whether the calculation result of C
meets the expectation of Dirac algebra or not will make
the problem suddenly clear. Equation (2) is reduced to
α · ˆp=
0 −i
i 0
−i~
∂
∂r
−
i~
r
+
i~C
r
1 0
0 −1
(5)
The potential energy of hydrogen like atom can be writ-
ten as follows
V (r) = −
Ze
2
4πε
0
r
= −
Zα~c
r
(6)
Substitute equation (5) into equation (1) to obtain the
angular quantum number C as the intrinsic parameter
equation, which is called Dongfang modified Dirac equa-
tion
c
0 −i
i 0
−i~
∂
∂r
−
i~
r
+
i~C
r
1 0
0 −1
+
1 0
0 −1
mc
2
−
Zα~c
r
ψ
1
ψ
2
=E
ψ
1
ψ
2
(7)
Although it is a historic breakthrough to revise the an-
gular quantum number defined by Dirac to an intrinsic
parameter, this revision contains an unclear causal rela-
tionship. The name of the equation (7) is only for distin-
guishing and attracting attention. It does not mean the
end of the problem. Whether and how to construct a
more reasonable equation to replace the equation is the
key. The Dongfang modified Dirac equation (7) is local-
ly self-consistent after introducing the intrinsic angular
quantum number C determined by the exact solution of
the equation. Expressed by the result of a matrix prod-
uct, equation (7) is transformed into a first-order radial
equation system of hydrogen-like atoms composed of two
first-order differential equations
dψ
2
dr
+
C + 1
r
ψ
2
−
mc
2
− E
~c
−
Zα
r
ψ
1
= 0
dψ
1
dr
−
C − 1
r
ψ
1
−
mc
2
+ E
~c
+
Zα
r
ψ
2
= 0
(8)
From equation (2) to equation (8), readers are advised
to take up the pen and deduce it again, and then de-
duce Dirac theory again, instead of just remembering
and interpreting the traditional theory. Over the past
100 years, generations of theoretical physicists have on-
ly continued their high-intensity memory, but failed to
find the mathematical and physical logic difficulties hid-
den in Dirac theory, which is more confusing than those
enough puzzles left by the Dirac equation. Even for sim-
ple mathematics, the understanding of relevant mathe-
matical logic in the process of personal derivation is often
completely different from the unforgettable memory, not
to mention finding and solving many less obvious math-
ematical and physical problems hidden in Dirac theory.
The Dirac electron theory has always adopted the
two-component wave function transformation of form
(3). In this way, the original two-component wave func-
tion should be returned after the accurate solution is ob-
tained, so as to it is found that the solution (also called
the Dirac wave function) of the Dirac equation satisfying
rough boundary conditions is divergence at the coordi-
nate origin, that is, the conclusion is inconsistent with
the natural boundary conditions
[7]
. The transformation
formula (3) of the upp er and lower component wave func-
tions is substituted into equation (8) to obtain the the in-
duction equation of the modified first-order radial Dirac
equation for hydrogen-like atoms
dG
dr
+
C
r
G +
Zα
r
F −
mc
2
− E
~c
F = 0
dF
dr
−
C
r
F −
Zα
r
G −
mc
2
+ E
~c
G = 0
(9)
Thus, the treatment of the modified Dirac equation for
hydrogen-like atoms is left to a purely mathematical
problem, and inference can always be given various phys-
ical meanings.
3 Statistical interpretation of orbital wave
function and exact boundary conditions
What kind of physical quantity is the wave function-
s? Quantum mechanics has no definite answer. This
has created the brilliance of Born’s statistical interpre-
tation, which holds that the wave function corresponds
to the probability of electrons appearing in the space
around the nucleus
[33, 34]
. The Born statistical interpre-
tation of wave function is generally accepted, forcing the
concept of orbit of electron motion in the atom to be qui-
etly denied, which is another misunderstanding hidden
in quantum mechanics.
The Born statistical interpretation of the wave func-
tion means that electrons appear and disappear around
the nucleus like ghosts, but the probability of electrons
appearing at any position can be determined, which is
calculated from the wave function. This does not confor-
m to the unitary principle. The counter example in the
experiment is that the trajectory of the electron beam
in the cathode ray tube can be displayed by the fluores-
cence effect. According to the unitary principle, if the
statistical interpretation of the wave function is the on-
ly correct, the trajectory of the electron beam can be
determined by the wave function of the electron beam
in the cathode ray tube. Actually not! The physical
theory must satisfy the unitary principle. Otherwise it
must be changed. Of course, we can also study the mo-
tion of electrons from other angles to further investigate
the real physical meaning of wave function. For exam-
ple, the probability of the electron appearing around the
atomic nucleus can be directly calculated by the motion
Mathematics & Nature (2022) Vol. 2 No. 1 202207-5
law of the electron governed by the Coulomb force. So
is its functional form or some agreed deformation form
the solution of the wave equation in quantum mechanics,
which will prove that the Born statistical interpretation
of the wave function is only correct?
It is noted that the b oundary condition (4), which is
the definite solution condition of the quantum mechani-
cal wave equation, obviously does not take into account
the size of the nucleus, which is equivalent to the tacit
consent that electrons can appear in the central neigh-
borhood of the nucleus. For the hydrogen atom, it mean-
s that the electron can almost coincide with the proton,
which has no scientific basis and do es not accord with
the fact. The boundary condition (4) thus written is
a rough boundary condition. The modified boundary
condition should be able to solve the problem of S-state
Dirac wave function divergence
[35]
.
d
Figure 1 Schematic drawing for the electron’s probability den-
sity and motion orbits. Each group of ellipse curves of different
colours that express the orbit of the electron corresponds to the
certain special energy states. The density of curves with the speed
weight of moving electron is proportional to the magnitude of the
probability density of the electron.
Even if the Born statistical interpretation of the wave
function is considered to be the only correct one, it does
not contradict the concept of the classical orbit of par-
ticles. Considering an artificial satellite turning around
the earth, for example, its probability density can be
translated as the density of the orbit number (the orbit
number per unit volume). In fact, classical mechanic-
s theory and quantum mechanics theory are just two
metrologies for describing the order of nature. The con-
cept of the classical orbit will not suddenly disappear
because of quantum mechanics theory. According to the
unitary principle, using the probability amplitudes for
the position of the particle to determine the orbit of the
particle is necessarily equivalent to using the density of
the orbit number to determine the probability of finding
the particle. FIG.1 shows symbolically the responding
relation between the wave function’s probability distri-
bution and the motion orbit of an electron outside the
nucleus of an atom. For the same energy level, an elec-
tron may have the different circle or the ellipse orbits.
Its orbit plane is varying continuously because of the
electromagnetic disturbance. The farther the electron is
away from the nuclear, the smaller the probability of the
electron crossing a given surface appearing to be. The
electrons absorb and radiate the energy to produce the
orbit transition. On the surface of the atomic nucleus,
the probability of electron appearance must be a non-
zero finite value and cannot go to infinity. From this
macroscopic picture of statistical interpretation on the
wave function, it is not difficult to find that the quantum
mechanics should not give the meaning of God play dice.
The initial-boundary value conditions play an impor-
tant role for determining the logical solution of a wave
equation from its general solutions. In order to over-
come the divergence of relativistic wave functions at the
origin
[35]
, we should consider the quantum radius of the
atomic nucleus. As one of the reliable treatments, we
assume that equation (5d) holds only for r > δ. Insid-
e δ, the potential has to be modified from a Coulomb
Ze
2
4πε
0
δ potential to one corresponding to a spread-
out charge distribution. Therefore, to do a completely
correct calculation, one solves the Dirac equation sepa-
rately outside of and inside of δ, with two different po-
tentials, and then matches, at r = δ, the outside solu-
tion (i.e., the original Dirac-Coulomb one) to the inside
solution. The inside solution is the finite constant; its
effect is to modify the energy-level formula by a small
correction that takes into account the finite radius of the
nuclear. This idea just supports introducing the exact
boundary condition to the wave equations for hydrogen-
like atom.
lim
r→δ
ψ
1
ψ
2
̸=
±∞
±∞
, lim
r→∞
ψ
1
ψ
2
= 0 (10)
where δ > 0 is the quantum radius of the atomic nucleus.
Around 1996, I replaced the rough boundary without
considering the size of the atomic nucleus with the ex-
act boundary condition written in the size of the atom-
ic nucleus, and re solved the Schr¨odinger equation of
the hydrogen atom, but the energy level formula as the
eigenvalue of the equation did not change. It can be in-
ferred that the energy level formula corresponding to the
solution of the Klein Gordon equation satisfying exact
boundary conditions is also invariant, including the solu-
tion of the real number wave equation
[9]
which gives the
meaning of a steady state again. To solve Schr¨odinger
equation and the Klein Gordon equation with exact
boundary conditions, it is necessary to generalize the
202207-6 X. D. Dongfang Neutron State Solution of Dongfang Modified Dirac Equation
relevant theorems of optimization differential equation-
s. However, in the past many years, efforts to publish
relevant papers in mainstream journals have failed.
The challenge solution of the hydrogen atom Dirac
equation meeting the exact boundary condition
[7]
elimi-
nates the divergence and virtual energy of the hydrogen
atom Dirac wave function, but the corresponding ener-
gy level formula is ignored by physicists because it does
not meet expectations. The age of interest causes people
who study scientific theories to care less about the cor-
rectness of theories. Theoretical physics has always been
indulged in the impetuous atmosphere of fabricating ex-
pected results with the wrong calculations and has de-
veloped to the extreme stage of fabricating observation
data to conjecture imaginary results and then making
wild speculation to achieve unprecedented success. Sci-
entific journals generally refuse to disclose famous scien-
tific theories with false calculations and serious logical
contradictions, and famous scientific experiment reports
with untrue data. The responsibility of scientific jour-
nals is more committed to maintaining the reputation of
famous scientists and constantly developing their theo-
ries. The main reason is that scientific journals are wide-
ly controlled by the makers and their successors of false
theories and untrue experimental reports. For relativis-
tic quantum mechanics, abandoning the precise bound-
ary conditions and desalinating the fatal logic problems
hidden in the traditional solution of Dirac equation to
maintain Dirac theory will not only keep the quantum
theory stuck in the wrong ideological framework forever,
but also seriously hinder the revelation of the mathe-
matical essence of quantum mechanics and thus hinder
the unification of macroscopic and microscopic quantum
theories. However, in any era, there is always a need for
someone to adhere to the truth.
4 The neutron state solution of the modi-
fied Dirac equation
Under the same solution conditions, is it possible that
the exact solution of a differential equation or system of
differential equations may be different due to different
methods of solving the equations, thus violating the uni-
tary principle? If the answer is yes, the solution of the
induced equation of the modified Dirac equation (9) sat-
isfying the exact boundary condition (10) is a local self
consistent solution. The local self consistent definition
leads to new problems, such as whether the global self
consistent solution of the Dirac equation exists or even
what form the global self consistent Dirac equation is,
which are left to the reader to study.
Now let’s look for the exact solution of the induced e-
quation of the modified Dirac equation (9) that satisfies
the exact boundary condition (10). For the convenience
of writing and calculation, a new variable ξ = r − δ is
introduced, where ξ > 0, and the exact boundary condi-
tion (10) becomes
lim
ξ→0
F
G
̸=
±∞
±∞
, lim
ξ→∞
F
G
= 0 (11)
Equation group (9) is equivalent to the following differ-
ential equations
dG
dξ
+
C
ξ + δ
G +
A
ξ + δ
F −
mc
2
− E
~c
F = 0
dF
dξ
−
C
ξ + δ
F −
A
ξ + δ
G −
mc
2
+ E
~c
G = 0
(12)
For the convenience of later calculation, we write the
differential equation in the following form
(ξ + δ)
dG
dξ
+ CG + AF −
mc
2
− E
~c
(ξ + δ) F = 0
(ξ + δ)
dF
dξ
− CF − AG −
mc
2
+ E
~c
(ξ + δ) G = 0
(13)
The undetermined quantum number C and the undeter-
mined nuclear equivalent radius δ in the equation are
intrinsic parameters. Let the analytical solution of equa-
tion (13) be
F = e
−
√
m
2
c
4
−E
2
~c
ξ
∞
ν=0
b
ν
ξ
σ +ν
G = e
−
√
m
2
c
4
−E
2
~c
ξ
∞
ν=0
d
ν
ξ
σ + ν
(14)
Their first derivatives are
dF
dξ
=e
−
√
m
2
c
4
−E
2
~c
ξ
∞
ν=0
(σ+ν +1) b
ν+1
−
√
m
2
c
4
−E
2
~c
b
ν
ξ
σ+ν
dG
dξ
=e
−
√
m
2
c
4
−E
2
~c
ξ
∞
ν=0
(σ+ν +1) d
ν+1
−
√
m
2
c
4
−E
2
~c
d
ν
ξ
σ+ν
(15)
Mathematics & Nature (2022) Vol. 2 No. 1 202207-7
Substituting (14) and (15) into equation (13) gives a recursive relationship group
(σ+ν +2) δd
ν+2
+
σ+ν +1−
√
m
2
c
4
−E
2
~c
δ+C
d
ν+1
+
A−
mc
2
−E
~c
δ
b
ν+1
−
√
m
2
c
4
−E
2
~c
d
ν
−
mc
2
−E
~c
b
ν
= 0
(σ+ν +2) δb
ν+2
−
A+
mc
2
+E
~c
δ
d
ν+1
+
σ+ν +1−
√
m
2
c
4
−E
2
~c
δ−C
b
ν+1
−
mc
2
+E
~c
d
ν
−
√
m
2
c
4
−E
2
~c
b
ν
=0
(16)
From δ ̸= 0, b
0
̸= 0, d
0
̸= 0, b
−1
= b
−2
= ··· = 0, d
−1
= d
−2
= ··· = 0, let ν = −2, σ = 0 is obtained. The above
recursive relationship is simplified to
(ν +2) δd
ν+2
+
ν +1−
√
m
2
c
4
−E
2
~c
δ+C
d
ν+1
+
A−
mc
2
−E
~c
δ
b
ν+1
−
√
m
2
c
4
−E
2
~c
d
ν
−
mc
2
−E
~c
b
ν
=0
(ν +2) δb
ν+2
−
A+
mc
2
+E
~c
δ
d
ν+1
+
ν +1−
√
m
2
c
4
−E
2
~c
δ−C
b
ν+1
−
mc
2
+E
~c
d
ν
−
√
m
2
c
4
−E
2
~c
b
ν
=0
(17)
The boundary condition (11) requires the series interruption in the radial wave function (14) to be polynomial. Let
the highest power of the polynomial be n, so the general solution of Dirac radial equations (8) for hydrogen like
atoms is
R =
ψ
1
ψ
2
=
1
r
F
1
r
G
=
e
−
√
m
2
c
4
−E
2
~c
(r−δ)
1
r
n
ν=0
b
ν
(r − δ)
ν
e
−
√
m
2
c
4
−E
2
~c
(r−δ)
1
r
n
ν=0
d
ν
(r − δ)
ν
, (n = 0, 1, 2, ···) (18)
The expansion form of recursive relation group (16) is as follows
δd
1
+
−δ
√
m
2
c
4
− E
2
~c
+ C
d
0
+
A −
mc
2
− E
~c
δ
b
0
= 0
δb
1
−
A +
mc
2
+ E
~c
δ
d
0
+
−δ
√
m
2
c
4
− E
2
~c
− C
b
0
= 0
2δd
2
+
1 − δ
√
m
2
c
4
− E
2
~c
+ C
d
1
+
A −
mc
2
− E
~c
δ
b
1
−
√
m
2
c
4
− E
2
~c
d
0
−
mc
2
− E
~c
b
0
= 0
2δb
2
−
A +
mc
2
+ E
~c
δ
d
1
+
1 − δ
√
m
2
c
4
− E
2
~c
− C
b
1
−
mc
2
+ E
~c
d
0
−
√
m
2
c
4
− E
2
~c
b
0
= 0
.
.
.
(ν + 2) δd
ν +2
+
ν + 1 − δ
√
m
2
c
4
− E
2
~c
+ C
d
ν +1
+
A −
mc
2
− E
~c
δ
b
ν +1
−
√
m
2
c
4
− E
2
~c
d
ν
−
mc
2
− E
~c
b
ν
= 0
(ν + 2) δb
ν +2
−
A +
mc
2
+ E
~c
δ
d
ν +1
+
ν + 1 − δ
√
m
2
c
4
− E
2
~c
− C
b
ν +1
−
mc
2
+ E
~c
d
ν
−
√
m
2
c
4
− E
2
~c
b
ν
= 0
.
.
.
nδd
n
+
n − 1 − δ
√
m
2
c
4
− E
2
~c
+ C
d
n−1
+
A −
mc
2
− E
~c
δ
b
n−1
−
√
m
2
c
4
− E
2
~c
d
n−2
−
mc
2
− E
~c
b
n−2
= 0
nδb
n
−
A +
mc
2
+ E
~c
δ
d
n−1
+
n − 1 − δ
√
m
2
c
4
− E
2
~c
− C
b
n−1
−
mc
2
+ E
~c
d
n−2
−
√
m
2
c
4
− E
2
~c
b
n−2
= 0
n − δ
√
m
2
c
4
− E
2
~c
+ C
d
n
+
A −
mc
2
− E
~c
δ
b
n
−
√
m
2
c
4
− E
2
~c
d
n−1
−
mc
2
− E
~c
b
n−1
= 0
−
A +
mc
2
+ E
~c
δ
d
n
+
n − δ
√
m
2
c
4
− E
2
~c
− C
b
n
−
mc
2
+ E
~c
d
n−1
−
√
m
2
c
4
− E
2
~c
b
n−1
= 0
−
√
m
2
c
4
− E
2
~c
d
n
−
mc
2
− E
~c
b
n
= 0
−
mc
2
+ E
~c
d
n
−
√
m
2
c
4
− E
2
~c
b
n
= 0
(19)
202207-8 X. D. Dongfang Neutron State Solution of Dongfang Modified Dirac Equation
Multiply the penultimate formula by
mc
2
+ E
~c and the penultimate formula by −
√
m
2
c
4
− E
2
~c to get
n−δ
√
m
2
c
4
−E
2
~c
+C
mc
2
+E
~c
d
n
+
A−
mc
2
−E
~c
δ
mc
2
+E
~c
b
n
−
mc
2
+E
√
m
2
c
4
−E
2
~
2
c
2
d
n−1
−
m
2
c
4
−E
2
~
2
c
2
b
n−1
=0
A+
mc
2
+E
~c
δ
√
m
2
c
4
−E
2
~c
d
n
−
n−δ
√
m
2
c
4
−E
2
~c
−C
√
m
2
c
4
−E
2
~c
b
n
+
mc
2
+E
√
m
2
c
4
−E
2
~
2
c
2
d
n−1
+
m
2
c
4
−E
2
~
2
c
2
b
n−1
=0
The left and right sides of the two recursive relations are added correspondingly, and finally a simplified equation is
obtained
(n + C)
mc
2
+ E
~c
+ A
√
m
2
c
4
− E
2
~c
d
n
+
A
mc
2
+ E
~c
− (n − C)
√
m
2
c
4
− E
2
~c
b
n
= 0 (20)
The condition that the homogeneous system of equation-
s composed of the above equation and the last equation
of the recursive relationship group (19) has a nontrivial
solution is
(n + C)
mc
2
+E
~c
+ A
√
m
2
c
4
−E
2
~c
A
mc
2
+E
~c
− (n − C)
√
m
2
c
4
−E
2
~c
−
mc
2
+E
~c
−
√
m
2
c
4
−E
2
~c
= 0
The result is
√
m
2
c
4
− E
2
− AE/n = 0. Because A > 0,
n > 0, this equation shows thatE > 0. The energy
eigenvalues containing only radial quantum numbers are
obtained
E =
mc
2
1 +
A
2
n
2
=
mc
2
1 +
Z
2
α
2
n
2
(n = 0, 1, 2, 3, ···) (21)
Excluding the so-called static energy, this energy lev-
el formula is equivalent to the energy eigenvalue of the
Schrodinger equation, that is, the Bohr energy level for-
mula, which is indep endent of the angular quantum num-
ber C, and there is no fine structure prediction. However,
the solution of the Dirac equation of hydrogen atom sat-
isfying the exact b oundary is locally self consistent, while
the so-called spectral fine structure prediction of the tra-
ditional solution is inseparable from the collapse of the
atom, which leads to the collapse of the universe, hiding
fatal mathematical difficulties and disastrous physical
conclusions. The Dirac theory of the so-called energy
level formula of hydrogen atom fine structure takes in-
to account one thing and loses the other, and violates
the unitary principle. Any theory should at least ensure
logical self consistency, and should not aim at patching
up the expected conclusions. When the radial quantum
number n = 0 , the self-consistent energy level formula
(21) of the induced equation (9) of the modified Dirac
equation becomes
E
0
= 0 (22)
This is a unique conclusion that the traditional solution
of the Dirac equation for the hydrogen atom has never
appeared. It belongs to the unique solution of the mod-
ified Dirac equation (7), that is, (8). When Z = 1, the
lowest energy state corresponding to E
0
= 0 is the neu-
tron state, and the values of the minimum self-consistent
angle quantum number C
0
and the radius δ of the nucle-
us are determinable.
5 Radius and angular quantum number of
the neutron
The Dirac equation is the product of the combination
of energy dynamic relationship of special relativity and
quantum mechanics. Although it can be proved theoret-
ically that relativity is not a correct theory, if modern
physics with many mathematical errors
[7]
is regarded as
an empirical theory, it is understandable that its logic
has problems and the conclusion is consistent with the
observed results. It is inferred that the Dirac equation
may be locally effective, which means that there are at
least some unknown factors in line with natural phenom-
ena. We need to calculate the radius and self consistent
angular quantum number C corresponding to the atom-
ic nucleus, so as to obtain the local self consistent com-
plete solution of the Dirac equation satisfying the exact
boundary conditions. Readers can also directly find the
neutron state solution of Dirac equation (13), take
F = b
0
e
−
√
m
2
c
4
−E
2
0
~c
ξ
G = d
0
e
−
√
m
2
c
4
−E
2
0
~c
ξ
(23)
Substitute into equation (9) to obtain the neutron s-
tate recursive relation group for b
0
, d
0
, δ and C
0
, which
can also be obtained by making making d
0
̸= 0 and
Mathematics & Nature (2022) Vol. 2 No. 1 202207-9
d
1
= d
2
= ··· = 0 in the recursive relation group (19).
−δ
m
2
c
4
−E
2
0
~c
+C
0
d
0
+
A−
mc
2
−E
0
~c
δ
b
0
=0
A+
mc
2
+E
0
~c
δ
d
0
+
δ
m
2
c
4
−E
2
0
~c
+C
0
b
0
=0
−
m
2
c
4
−E
2
0
~c
d
0
−
mc
2
−E
0
~c
b
0
=0
−
mc
2
+E
0
~c
d
0
−
m
2
c
4
−E
2
0
~c
b
0
=0
(24)
This homogeneous system of equations has a unique so-
lution, but its solution method is unusual.
The latter two equations of equation set (24) are lin-
early related, and only the third equation is taken in the
following calculation. The necessary and sufficient con-
dition for the homogeneous equations composed of the
first two equations to have nontrivial solutions is
−δ
√
m
2
c
4
−E
2
0
~c
+ C
0
A −
mc
2
−E
0
~c
δ
−
A + δ
mc
2
+E
0
~c
−δ
√
m
2
c
4
−E
2
0
~c
− C
0
= 0
The formula for calculating δ is thereby obtained
δ =
~c
C
2
0
− A
2
2AE
0
(25)
The necessary and sufficient condition for a homoge-
neous system of equations composed of the first equation
and the third equation to have nontrivial solutions is
−δ
√
m
2
c
4
−E
2
0
~c
+ C
0
A − δ
mc
2
−E
0
~c
−
√
m
2
c
4
−E
2
0
~c
−
mc
2
−E
0
~c
= 0
The formula for calculating C
0
or E
0
is thereby obtained
C
0
= A
mc
2
+ E
0
mc
2
− E
0
> 0
E
0
=
C
2
0
− A
2
mc
2
C
2
0
+ A
2
(26)
The necessary and sufficient condition for the system of
homogeneous equations composed of the second equa-
tion and the third equation to have nontrivial solutions
is
−
A + δ
mc
2
+E
0
~c
−δ
√
m
2
c
4
−E
2
0
~c
− C
0
−
√
m
2
c
4
−E
2
0
~c
−
mc
2
−E
0
~c
= 0
Thus, another calculation formula of C
0
or E
0
is ob-
tained
C
0
= A
mc
2
− E
0
mc
2
+ E
0
> 0
E
0
=
A
2
− C
2
0
mc
2
C
2
0
+ A
2
(27)
The only solution satisfying equations (26) and (27) si-
multaneously is
C
0
= A = Zα, E
0
= 0 (28)
The energy eigenvalue is the same as the above infer-
ence (22). According to the above results, equation (25)
for calculating P has the significance of limit calculation.
Substituting E in (26) and (27) into (25) respectively,
two results of nuclear radius are obtained
δ = ±A
~c
mc
2
(29)
But the radius of the nucleus can’t be negative. Discard
the negative root and keep the positive root, so
δ = A
~
mc
=
Ze
2
4πε
0
mc
2
(30)
Finally, by substituting the above results and ξ = r − δ
into equation (23), the zero energy state wave function
is obtained
ψ
1
ψ
2
=
b
0
r
e
−
mc
~
(r−δ)
−
b
0
r
e
−
mc
~
(r−δ)
=
B
0
r
e
−
mc
~
r
−
B
0
r
e
−
mc
~
r
(31)
Where B
0
is the constant to be normalized. How should
we explain that the two-component wave functions of
the neutron state of the Dongfang modified Dirac equa-
tion of the hydrogen-like atom are opposite to each oth-
er? Now we find that the so-called Yukawa potential
function is the zero energy state wave function hidden
in the modified Dirac hydrogen equation, which surpris-
ingly derives the nuclear meson theory
[36-39]
. How to
evaluate the relationship between the two or how to re
understand Yukawa potential function
[10, 40]
, requires a
bit of real mathematics and physics. Readers may wish
to prove by themselves that Yukawa potential function
is the product of completely distorting the principles of
quantum mechanics and brutally dismembering the rela-
tivistic momentum energy equation
[41]
, while the meson-
s that do not exist in the atomic nucleus are vividly
described as the medium of nucleon interaction, which
makes the theory continue to grow. This far fetched sub-
jective logic is unprecedented, and it is also a microcosm
of the development process of modern physics.
For hydrogen atom, the angular quantum number
C
0
= A = α of the neutron state is a very small val-
ue, which has not appeared in quantum theory before.
It is generally believed that quantum mechanics breaks
through the classical theory, which is a flashy bias. Quan-
tum mechanics is just a different description. According
to classical theory, the angular quantum number C
0
= α
of hydrogen atom corresponds to the angular momentum
202207-10 X. D. Dongfang Neutron State Solution of Dongfang Modified Dirac Equation
of electrons moving around protons at the speed of light.
But because of Lorentz transformation, relativity denies
that the relative speed of matter reaches or exceeds the
speed of light, including the relative propagation speed
of light. We can find the physical model that the theory
of the limit of the speed of light of the matter does not
accord with the unitary principle within the framework
of relativity. Of course, this deviates from the theme of
this paper. In fact, C
0
= A = α has a reasonable ex-
planation within the framework of quantum theory, and
this problem is left to the readers for the time being.
The angular quantum number C, which is the self con-
sistent solution of the proxy equation of the modified
Dirac equation, is consistent with the logic determined
by the exact solution of the Schrodinger equation, and is
different from the angular quantum number constructed
by Dirac algebra. It inversely shows that the logically
distorted Dirac algebra does not accord with the basic
rules of mathematics and the inference does not accord
with the physical meaning.
The radius of the hydrogen-like atomic neutron state
is c δ = Ze
2
4πε
0
mc
2
, and it seems to correspond to
neutron-like quantum radius. The special eigenvalue
E
0
= 0 denotes the unique energy state of the neutron-
like. For Z = 1, the formula (13) reads the neutron
binding energy
∆E = E
∞
− E
0
= mc
2
(32)
where E
∞
is the energy of the hydrogen atom corre-
sponding to n
r
= ∞ in the formula (13), and E
0
is the
energy of a neutron. Physics is a theory that describes
natural phenomena in mathematical language. The so-
lution of the equation, the method of solving the equa-
tion and the popular science interpretation of inference
must comply with the unitary principle. (30) and (32)
form the self consistent solution of the radial Dirac e-
quation with a Coulomb potential. The neutron binding
energy implies that the neutron can be broken up by
a photon of the energy m
e
c
2
, or perhaps an electron
and a proton could combine into a neutron and emit
a photon of the energy m
e
c
2
at the same time. Usual-
ly, δ = e
2
4πε
0
mc
2
(=2.8117940285 fm) is regarded as
the classical electron radius. The above analysis shows
that it should be the quantum neutron radius. This is
about triplication of the neutron radius, which the re-
cent value is rep orted to be 0.8418 fm, the early results
are 0.805(12)
[42]
, 0.861(26)fm
[43]
, 0.862(13)fm
[44]
, 0.8768
fm, 0.88014 fm.
[45]
, 0.89014 fm and 0.895±0.018 fm.
[46]
and so on. These data are actually calculated by Lamb
shift
[47-49]
.
6 Test of neutron state wave function for
modified Dirac equation
Subversive conclusions about famous traditional the-
ories are always doubted, even though mathematical in-
ferences can be easily tested by repeated or non repeat-
ed calculations in different orders. The neutron state
wave function (31) satisfying the exact boundary condi-
tion (10) is the true solution of the modified Dirac equa-
tion. The reader can verify that the neutron state wave
function (31) satisfies the modified Dirac equation (8) of
hydrogen-like atoms. Substitute E = 0 and C = A = Zα
into equation (8) to obtain
dψ
2
dr
+
Zα + 1
r
ψ
2
−
mc
2
~c
−
Zα
r
ψ
1
= 0
dψ
1
dr
−
Zα − 1
r
ψ
1
−
mc
2
~c
+
Zα
r
ψ
2
= 0
(33)
Substitute the wave function (31) into the above two
equations successively, and the results
dψ
2
dr
+
Zα + 1
r
ψ
2
−
mc
2
~c
−
Zα
r
ψ
1
=
d
dr
−B
0
1
r
e
−
mc
~
r
+
Zα + 1
r
−B
0
1
r
e
−
mc
~
r
−
mc
2
~c
−
Zα
r
B
0
1
r
e
−
mc
~
r
=
e
−
mcr
~
B
0
r
2
+
mce
−
mcr
~
B
0
~r
+
Zα + 1
r
−B
0
1
r
e
−
mc
~
r
−
mc
2
~c
−
Zα
r
B
0
1
r
e
−
mc
~
r
= B
0
e
−
mcr
~
1
r
2
+
mc
~r
−
Zα + 1
r
2
−
mc
~
−
Zα
r
1
r
= 0
(34)
Mathematics & Nature (2022) Vol. 2 No. 1 202207-11
and
dψ
1
dr
−
Zα − 1
r
ψ
1
−
mc
2
~c
+
Zα
r
ψ
2
=
d
dr
B
0
1
r
e
−
mc
~
r
−
Zα − 1
r
B
0
1
r
e
−
mc
~
r
−
mc
2
~c
+
Zα
r
−B
0
1
r
e
−
mc
~
r
=
−
e
−
cmr
~
B
0
r
2
−
ce
−
cmr
~
mB
0
r~
−
Zα − 1
r
B
0
1
r
e
−
mc
~
r
−
mc
2
~c
+
Zα
r
−B
0
1
r
e
−
mc
~
r
= B
0
e
−
cmr
~
−
1
r
2
−
mc
~r
−
Zα − 1
r
2
+
mc
2
~c
+
Zα
r
1
r
= 0
(35)
The test result of neutron state wave function is valid.
The introduction of the intrinsic angular quantum num-
ber C to replace the Dirac electron theory and the angu-
lar quantum number κ = ±1 ±2 ±3 ··· defined by the
unique mathematical operation rules are consistent with
the logic that the quantized angular momentum is the
intrinsic solution of the angular equation of the wave e-
quation. The Dirac equation for the hydrogen-like atom
modified by the intrinsic angular quantum number is self-
consistent, while the angular quantum number construct-
ed by Dirac does not conform to the logic and should be
abandoned. The exact boundary condition including nu-
clear size is the actual boundary condition of the wave
equation.
7 Conclusions and comments
The Dirac electron theory is unreasonable to define
the angular quantum number through formal mathe-
matics. We know that the angular quantum number
of Schr¨odinger equation is the intrinsic solution of the
angular equation of the equation. From Schr¨odinger e-
quation to the Dirac equation, if the angular quantum
number changes from the eigensolution of the equation
to a definition, the theory does not conform to the uni-
tary principle. The intrinsic angular quantum number is
used to replace the angular quantum number defined by
Dirac electron theory to modify Dirac equation, which
solves the inconsistent difficulty hidden in the Dirac e-
quation. To solve the problem fundamentally, we need
to analyze the reason why Dirac electron theory defines
the angular quantum number, test the mathematical log-
ic of Dirac algebra, and prove the existence of an intrin-
sic angular quantum number. The self-consistency of
the Dirac equation modified by the Coulomb field and
the existence of the neutron state solution not only have
the demonstrative mathematical significance of proving
whether the wave equation constructed is self-consistent
and whether the exact solution of the equation is true,
but also have the demonstrative physical significance of
explaining the true and false relationship between the
various forms of solution of the wave equation and the
natural law with scientific logic.
Since Einstein’s assumption of constant speed of light
has been proved to be untenable, relativity based on the
assumption of constant speed of light is naturally incor-
rect, and the relationship between momentum and en-
ergy of relativity has no real scientific significance. Rel-
ativistic quantum mechanics based on the relationship
between relativistic momentum and energy should be
abandoned. The dirac equation is a representative equa-
tion of relativistic quantum mechanics. However, Dirac
theory of the hydrogen atom, which was considered to
be very successful in the past, contains fatal mathemat-
ical and physical difficulties such as the collapse of the
universe and virtual energy, so it does not belong to
scientific theory. Many negative results prove that rela-
tivistic quantum mechanics deviates from scientific logic.
However, we have been trying to find the factors that the
Dirac equation can retain. Therefore, we first consider
how to eliminate the divergence and virtual energy of
the original Dirac wave function, and then put forward
the exact boundary conditions written into the nuclear
radius, and give the challenging solution of the Dirac e-
quation for the hydrogen atom. This is done in order to
avoid possible bias and draw unreasonable conclusion-
s. The solution of the Schrodinger equation and the
solution Klein Gordon equation of the hydrogen atom
satisfying the exact boundary condition (10) and satis-
fying rough boundary condition (4) corresponds to the
same energy level formula
1
. However, the challenging
solution of the Dirac equation of the hydrogen atom sat-
isfying the exact b oundary condition (10) corresponds
to a completely different energy level formula from the
traditional solution satisfying the rough boundary condi-
tion (4). The energy level formula corresponding to the
exact boundary condition solution is equivalent to the
energy level formula of the Schrodinger equation, and
the intention of the Dirac equation is to find an energy
level formula that is more in line with the expectation
than the Klein Gordon equation. This does not conform
to the unitary principle.
The existence problem of the solution of the linear
recursive relation system caused by the challenge so-
lution of the Dirac equation requires the introduction
of the intrinsic angular undetermined quantum num-
ber C instead of the artificial angular quantum number
κ = ±1 ±2 ··· to modify the Dirac hydrogen equation.
202207-12 X. D. Dongfang Neutron State Solution of Dongfang Modified Dirac Equation
The energy parameter E, undetermined angular quan-
tum number C and the nuclear radius δ constitute three
intrinsic parameters of Dongfang modified Dirac hydro-
gen equation. The radius and minimum angular quan-
tum number of the hydrogen like nuclei are obtained by
solving the lowest energy state solution. The lowest en-
ergy state solution of the hydrogen atom model seems
to exactly describe the structure of hydrogen-like neu-
trons. This is the best result of conservative treatment
of Dirac electron theory. The result of non-conservative
treatment will be to establish a new accurate wave equa-
tion. From the perspective of energy, the result of remov-
ing the so-called static energy from relativistic energy is
within the same accuracy range as Newtonian mechan-
ical energy. This shows that the establishment of the
Dirac equation and the Klein Gordon equation based on
the relationship between relativistic momentum and en-
ergy does not seem so outrageous, but belongs to an addi-
tional operation showing new ideas. Therefore, qualita-
tively speaking, relativistic Klein Gordon equation and
the Dirac equation should not have so-called fine energy
level structure expectation, otherwise it would violate
the unitary principle. Quantitatively, Dirac energy level
formula is a patchwork formula that distorts mathemat-
ics. The divergence of S-state Dirac wave function hid-
den behind it and the Dirac energy level formula that
cover up the virtual energy confirms this inference of
the unitary principle. Here we should initially realize
the powerful logical power of the unitary principle.
Dirac theory has been developed to quantum field
theory
[50]
. Many contents of modern physics are the
product of blindly promoting wrong theories. The
essence of quantum mechanics has always been hidden,
but quantum mechanics has been developing bravely,
and even the so-called quantum communication, quan-
tum computer, quantum food and other concepts have
emerged. Of course, the modified Dirac equation is not
the ultimate answer. The modified Dirac equation with-
out the virtual energy solution of Dirac electron theory
and the divergent S-state wave function is locally self
consistent, which is much more reasonable than the log-
ic of Dirac electron theory. We are always used to try
to defend famous theories that do not match the name.
The starting point of mo difying the Dirac hydrogen e-
quation is still to maintain the widely accepted theo-
ry. However, the solution of the modified Dirac hydro-
gen equation that meets the exact boundary conditions
cannot fit the fine structure energy level formula, which
exposes the systematic defects of Dirac electron theory.
There are enough problems in relativity, quantum me-
chanics and relativistic quantum mechanics that need
to be clarified step by step. In the final analysis, the
problems of relativistic quantum mechanics belong to
relativistic problems. The speed of light is the limit of
Lorentz transformation, but the so-called superluminal
phenomenon
[51-53]
may not be the best solution to the
problem. Mathematical problems caused by the Dirac
equation are worthy of further study. To find the exact
wave equation
[54-56]
containing the energy level formula
of fine structure, whether we can jump out of the log-
ical cycle of relativity or not, we must ensure that the
mathematical rules are not destroyed.
The Dirac equation is just a product of the combi-
nation of quantum mechanics and relativity. The for-
mer hides the morbid problem of quantum numbers that
needs to be solved urgently, while the latter is based
on the untenable assumption that the speed of light is
constant. There are enough mathematical loopholes in
quantum mechanics and relativity, which need to be radi-
cally corrected, and those parts that cannot be corrected
should be discarded. However, it is also one of the most
beautiful landscapes in physics to trace the famous physi-
cal theories that pieced together the expected conclusion-
s, uncover the magic veil that whitewashed the anoma-
lous logic, and finally destroy the magic illusion
[7, 57-60]
.
Perhaps the reader will soon be able to find the exact
wave equation to achieve the desired results. The neu-
tron state solution of the modified Dirac equation is an
ideal result that meets the requirements of mathematical
self-consistency. So, does the intrinsic angular quantum
number of the atomic ground state and excited state of
the modified Dirac equation exist? What are the intrin-
sic solutions of the ground and excited states of atoms?
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Latest findings
Ground State Solution of Dongfang Modified Dirac Equation
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 boundary 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 modified 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 Schrödinger equation. The two components 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 processing technology, it will help the development of mathematical theory to incorporate it into mathematics textbooks as a new mathematical model.