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Article
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Mathematics and Physics
Dongfang Challenge Solution of Dirac Hydrogen Equation
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
In quantum mechanics, the Schr¨odinger 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
co ordinate 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.
Keywords: Dirac equation; boundary condition; energy eigenvalues; uniqueness of solution.
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
In history, it was not meeting expectations to use
the Klein-Gordon equation
[1, 2]
to describe the hydrogen
atom because its eigenvalues of the quantum energy is
incompletely agrees accurately with the experimentally
observed hydrogen spectra. Considering the relativis-
tic effect, Dirac introduced his relativistic wave equation
for the single electron
[3]
in 1928. Darwin
[4]
and Gordon
[5]
first obtain the exact solution of the Dirac equation with
a Coulomb potential. Biedenharm
[6]
, Wong, Yeh
[7, 8]
and
Su
[9]
etc. also constructed the different second order
Dirac equation and obtain the different form of solu-
tions. Nenciu
[10]
, Kalus and W¨ust
[11]
investigated the
different construction methods of self-adjoint extension
of the Dirac operators with coulomb potential, and it
is also showed that the distinguished self-adjoint exten-
sions given by the two methods are identical. It should
be noted that, there is no essential difference in vari-
ous treatment methods of the Dirac equation in modern
mathematical physics textbooks
[12]
. Formally, the Dirac
equation combines quantum mechanics and relativity to
describe the spin and magnetic moment of electrons in
a completely natural way. Especially, the distinguished
Dirac formula of energy levels in a Coulomb field can
explain the fine-structure of the hydrogen atom. This is
considered to be one of the important indicates to the
achievements of the Diac theory.
The Dirac equation is generally considered to be suc-
cessful in many ways. However, the results of the sys-
tematic examination of Dirac’s electronic theory and its
various development theories by using Dongfang’s uni-
tary principle show that all kinds of extremely exag-
gerated beautification descriptions of Dirac’s equation
are basically not true, and there are many false calcula-
tions and distorted calculations purely for the purpose
of piecing together the expected conclusions
[13-17]
. Al-
though physicists have been unable to face a large num-
)Citation: Dongfang, X. D. Dongfang Challenge Solution of Dirac Hydrogen Equation. Mathematics & Nature 2, 202206 (2022).
*This article revises the author’s article entitled New Exact Solution of Dirac-Coulomb Equation with Exact Boundary Condition signed
Ruida Chen published in Int J Theor Phys in 2008 to express the author’s clear scientific position.
202206-2 X. D. Dongfang Dongfang Challenge Solution of Dirac Hydrogen Equation
ber of major quantum mechanical problems that will
overturn traditional cognition, such as the morbid equa-
tions of quantum numbers
[18]
, the operator equations
of angular motion laws
[19]
, the macroscopic com quan-
tum equations
[20-22]
, and the Yukawa nuclear force me-
son theory
[23]
distorting mathematics
[24, 25]
, and so on.
Perhaps because of the complex mathematical process
in form, including the introduction to Dirac’s achieve-
ments cited above, physicists always have a preconceived
understanding of Dirac’s equation, and few have tested
the authenticity of various conclusions through further
in-depth mathematical calculations
[26, 27]
. Just say that
the so-called exact solution of the Dirac equation for
the hydrogen atom, which is used to fit the energy lev-
el formula of fine structure, has enough mathematical
problems to need clear answers.
In order to facilitate the distinction, the recognized
solution of Dirac equation respected in the textbook is
called the traditional solution of the Dirac equation. In
fact, it belongs to one of the formal solutions of Dirac
equation that do not conform to the mathematical op-
eration rules
[14]
, which means that there are other com-
pletely different formal solutions, but it has not been
found by theoretical physics scholars, the Dirac equa-
tion itself has many mathematical difficulties that are
more difficult to be noticed by theoretical physics schol-
ars and can not be solved at present. Dirac equation
combines quantum mechanics with special relativity
[28]
,
and the difficulty of negative kinetic energy appeared at
the beginning. The negative kinetic energy solution was
originally a veto solution, but it was interpreted as that
the interaction may lead to the transition to the negative
energy state, making the matter in this framework im-
possible to be stable, so a veto solution was turned into
a new progress. This logic of mo dern physics is difficult
to reverse. However, it seems that only Dirac equation
can be used to fit the so-called fine energy level structure
formula of hydrogen atom, so Dirac equation has always
been very attractive. There is a practical problem: if
the explanation of the abnormal inference of quantum
mechanics deviates too far from the mathematical logic
of quantum mechanics itself, human understanding of it
will b e very unclear, so quantum mechanics will appear
very mysterious, so that more people will stagnate in
the stage of popular science description, which will in-
evitably lead to dogmatism. As we all know, there is still
no clear solution to the localization problem and Klein
paradox in quantum electrodynamics. Therefore, when
describing the achievements of the Dirac equation, we’d
better not go too far from Dirac’s point of view: “. . . a
book on the new physics, if not purely descriptive of
experimental work, must be essentially mathematical.”
This means that the logical test of theoretical physics
from the perspective of mathematics is not only reliable,
but also necessary.
When I promote the optimization differential
theory
[29, 30]
, I hope to find practical examples of gen-
eralized optimization differential equations, and my at-
tention is attracted to the second order Dirac equation.
When solving the real second order Dirac hydrogen equa-
tion, it was unexpectedly found that the result of deter-
mining the exact solution of the equation with boundary
conditions did not completely meet the boundary condi-
tions. This cannot be accepted by mathematical logic,
especially the distortion of the description of the uni-
verse structure constitutes a serious physical logic para-
dox. Now we focus on the divergence of the S-state wave
function of the original Dirac equation of the hydrogen
atom traditional solution and the problem of the vir-
tual energy in the Dirac energy level formula when the
nuclear charge number Z > 137
[31]
. It is one of the
mathematical difficulties implied in Dirac theory. From
a mathematical point of view, the divergence of the wave
function as the traditional solution of Dirac equation at
the coordinate origin means that the traditional solution
of the equation does not meet the definite solution con-
ditions, which shows that the logic of Dirac traditional
solution is not self consistent. From a physical point
of view, the wave function of hydrogen atom represents
the electron probability density, and the wave function
as the traditional solution of Dirac equation diverges at
the coordinate origin, which means that the atom col-
lapses into neutrons or neutron-like, indicating that the
Dirac traditional solution is contrary to the fact of the
cosmic structure. The most concise is that the appear-
ance of virtual energy in Dirac energy level formula is a
direct veto solution. If the solution of a wave equation
hides some mathematical or physical logical difficulties
or even errors, it must mean some new and importan-
t events, including new mathematical problems to be
found and solved.
As we all know, the most attractive part of quan-
tum mechanics is to find the solution of the Schr¨odinger
equation
[32, 33]
of the hydrogen atom satisfying the natu-
ral boundary conditions, and the Bohr energy level for-
mula is naturally obtained
[34]
. This seems to be the most
difficult mystery of mathematical physics. Although we
can write wave equations that are far from known form-
s (which will be introduced one by one later) and ob-
tain different forms of energy level formulas with the
same accuracy, at present, we can not find the reason
why various wave equations of quantum mechanics hide
similar energy level formulas. According to the result-
s of all aspects of analysis, just talking about the logic
of mathematics, the traditional solution of Dirac equa-
tion implies irreconcilable logical contradictions, which
is attributed to the fact that the definite solution condi-
tions of the equation directly misappropriate the rough
boundary conditions of Schr¨odinger equation without
considering the size of an atomic nucleus. Therefore,
considering the fact that the nucleus has a certain size,
the exact boundary conditions of the hydrogen atom can
be written. It is necessary to generalize the analytical
theory of optimization differential equations in order to
Mathematics & Nature (2022) Vol. 2 No. 1 202206-3
re solve the Schr¨odinger equation with exact boundary
conditions. However, there seems to be no difficulty in
re solving the Dirac equation with exact boundary con-
ditions, and we get the bounded wave function in the
whole space, and the corresponding energy level formu-
la does not appear the virtual energy when the nuclear
charge numberZ > 137. However, the energy level for-
mula determined by the exact boundary has the same
accuracy as the Bohr energy formula, and there is no
so-called fine structure.
Therefore, Dirac equation, as the core principle of rel-
ativistic quantum mechanics
[35-41]
, faces many difficult
choices. Or hunting for ingenious explanations to di-
lute the contradiction between mathematical and phys-
ical logic, and choosing rough boundary to solve the e-
quation to obtain the expected traditional solution; Or
follow the logic rules of mathematics and physics and
choose the exact boundary to solve the equation and
accept the regular solution that does not meet the ex-
pectation; Or find a new wave equation to establish a
better theory. In fact, Dirac equation hides more logi-
cal difficulties than we know or even can imagine. The
exact boundary condition solution may not be the final
answer, but it can be accepted by some physicists. The
mathematical inference of exact boundary conditions is
at least better than the cover up of logical difficulties by
qualitative explanations. All solutions of the Dirac equa-
tion of the hydrogen atom under precise boundary con-
ditions, regardless of the relationship with experimental
observations, pose a challenge to the relativistic quan-
tum mechanics, which is called the challenging solution
of the Dirac equation for the hydrogen atom.
2 Rough boundary condition and diver-
gence of Dirac function
In 1928, Dirac constructed the wave equation of quan-
tum system in line with the meaning of special relativity,
called Dirac equation, which opened the era of relativis-
tic quantum mechanics. This section only introduces
the main conclusions of the Dirac theory of hydrogen
like atoms, including the Dirac energy level formula and
Dirac wave function, to illustrate the contradiction prob-
lems such as divergence and virtual energy implied in the
traditional solution of the Dirac equation determined by
rough boundary. For the detailed process of solving the
Dirac equation of hydrogen like atom to obtain the Dirac
energy level formula and Dirac wave function, you can
read the relevant chapters of various quantum mechanics
textbooks
[42-44]
. The dirac equation is used to describe
hydrogen like atom. In order to obtain the expected
energy level formula, the problem is finally transformed
into solving the Dirac radial equation,
cˆp
r
0 i
−i 0
−
~cκ
r
0 1
1 0
+ mc
2
1 0
0 −1
R =
E +
Ze
2
4πε
0
r
R
Boundary conditions play a decisive role in solv-
ing wave equations
[45-52]
. In quantum mechanics, the
Schr¨odinger equation is used to describe the hydrogen
atomic system, and the size of an atomic nucleus is usu-
ally not considered in the determination of boundary
conditions. The boundary conditions of the atom like
the Schr¨odinger equation with nuclear charge number Z
are written in the following rough form
lim
r→0
ψ = 0
ψ (0 < r < ∞) ̸= ±∞
lim
r→∞
ψ = 0
(1)
where R is the radial wave function. Solving the
Schr¨odinger equation by using this rough boundary con-
dition, the Bohr formula of the energy levels is natural-
ly obtained. It is a landmark work to open the era of
quantum mechanics and is considered to be perfect in
mathematics. Relativistic quantum mechanics still uses
the above rough boundary conditions to solve the Dirac
equation, and obtains the Dirac energy level formula of
the energy level in the Coulomb field, which is consistent
with the expectation
[42-44]
E =
mc
2
1 +
Z
2
α
2
(
n
r
+
√
κ
2
−Z
2
α
2
)
2
(2)
where n
r
= 0, 1, 2, ···, α is the fine structure constant,
while κ = ±1, ±2, ±3, ···, is an artificial constan-
t constructed by the so-called angular momentum cou-
pling theory that bypasses the basic mathematical oper-
ation rules. In fact, it does not accord with the physical
meaning. Taking a few special cases as implicit axiom-
s and defining a set of calculation methods to achieve
the expected purpose is a common writing technique
in morden physics. Its interference with mathematics
will inevitably bring irreconcilable logical contradiction-
s. Saying alone the Dirac ground state or Dirac S state
of the hydrogen atom, because n = 0, κ = ±1, when
z > 137, the energy of the system become an imagi-
nary number. This is a purely mathematical problem.
Of course, on this issue, morden physics has a set of
logic of causal inversion, which connects the maximum
number of nuclear charges with the speed of light. The
specific reason is that no particle with mass, including
electrons, can reach or exceed the speed of light, and it
gives the reader a formal logic of non inevitable causal-
ity: the atomic number of the heaviest element in the
universe must therefore be less than 137. Therefore,
the aforementioned logical contradiction was complete-
ly covered up by this illusion. Later, few people found
implicit mathematical difficulties of Dirac equation from
the perspective of mathematical and physical logic, and
202206-4 X. D. Dongfang Dongfang Challenge Solution of Dirac Hydrogen Equation
could not fundamentally solve these difficulties. Part of
physics has been being in such a logical cycle of causal
inversion.
Eloquence, which deviates far from the essence of
mathematics and physics, is often widely accepted as
the symbol of the wisdom of mankind, and has a pro-
found impact on the thinking of later physics scholars.
However, an explanation of cause and effect inversion
can only cover up a single logical contradiction, and the
theory presenting a certain logical contradiction must
imply many logical contradictions. Now let’s discuss the
implied divergence of Dirac wave function of hydrogen-
like atom. The two-component Dirac wave function of a
hydrogen-like atom is
R (r) =
e
−ar
n
r
ν=0
b
ν
(ar)
√
κ
2
−Z
2
α
2
+ν−1
e
−ar
n
r
ν=0
d
ν
(ar)
√
κ
2
−Z
2
α
2
+ν−1
(3)
where a =
√
m
2
c
4
− E
2
~c and the coefficients of the
polynomial satisfy the system of the following recurrence
relations
mc
2
+ E
mc
2
− E
b
ν−1
+ d
ν−1
− Zαb
ν
−
κ +
κ
2
− Z
2
α
2
+ ν
d
ν
= 0
b
ν−1
+
mc
2
− E
mc
2
+ E
d
ν−1
+
κ −
κ
2
− Z
2
α
2
− ν
b
ν
+ Zαd
ν
= 0
(4)
This is a mixed recurrence system of equations that cannot be transformed into a single series (polynomial) coefficient
recurrence relationship. However, the Dirac wave function for the S state of the hydrogen-like atom is divergent as
κ = ±1 (S state) . It can be seen that whatever the radial quantum number n
r
takes any value, the first term of the
two-component Dirac wave function (3) is the same. We have,
lim
r→0
|R| = lim
r→0
e
−ar
b
0
(ar)
√
1−Z
2
α
2
−1
+
n
r
ν=1
b
ν
(ar)
√
1−Z
2
α
2
+ν−1
e
−ar
d
0
(ar)
√
1−Z
2
α
2
−1
+
n
r
ν=1
d
ν
(ar)
√
1−Z
2
α
2
+ν−1
=
∞
∞
(5)
It can be seen that the traditional solution of the Dirac equation does not meet the definite solution conditions of
the equation. This is a typical logical contradiction in a pure mathematical sense, which will not disappear because
of the defining relationship between the nuclear charge limit and the speed of light limit.
For the time being, we don’t discuss those problems that the abnormal solutions of some equations in modern
physics often turn into the problem of how strange inferences are generally accepted. When looking from a physical
point of view, the divergence of the Dirac wave function for S state implies that the probability density of the electron
around the nucleus rapidly increases as it close to the atomic nucleus. The probability density corresponding to the
wave function of two components is defined as
ρ (r, t) = R
∗
(r, t) R (r, t) (6)
According to (3), we have
R
∗
(r) = e
−ar
n
r
ν=0
b
ν
(ar)
ν+
√
κ
2
−Z
2
α
2
−1
n
r
ν=0
d
ν
(ar)
ν+
√
κ
2
−Z
2
α
2
−1
(7)
In this case the radial probability density for the electron of the so called relativistic hydrogen atom is as follows
ρ =
e
−ar
n
r
ν=0
b
ν
(ar)
ν+
√
κ
2
−Z
2
α
2
−1
2
+
e
−ar
n
r
ν=0
d
ν
(ar)
ν+
√
κ
2
−Z
2
α
2
−1
2
(8)
For S-state which implies κ = ±1, as r → 0, the above formula becomes
lim ρ
r→0
= lim
r→0
e
−ar
b
0
(ar)
1−
√
1−Z
2
α
2
+ ···
2
+
e
−ar
d
0
(ar)
1−
√
1−Z
2
α
2
+ ···
2
= ∞ (9)
Mathematics & Nature (2022) Vol. 2 No. 1 202206-5
What this result predicts should be that the hydro-
gen and hydrogen-like atom in the ground state must
rapidly collapse to the neutron-like. However the fac-
t is not thusness. That is to say, the original solution
of the Dirac equation for the hydrogen and hydrogen-
like atom neither agrees with the mathematical prin-
ciple nor agrees with the physical signification. Unex-
pectedly, such divergence was defined as so-called “mild
divergence”
[53-55]
so that hardly might one open out it-
s actual meaning, and the correct deduction has been
buried. We know that the Klein-Gordon for the meson
without spin has the same divergence, but the Klein-
Gordon divergence in the Coulomb field can be elimi-
nated by the suitable mathematical method. This part
will be introduced in detail after expounding the relevant
new mathematical theorems.
Using a cut-off procedure for the wave function that
is similar to the case of considering an extended nucleus
to blench the divergence should be independent of the
exact solution for the Dirac equation for the hydrogen-
like atom, otherwise it may oppresses the correct wave
equation and it real solution. For the traditional solu-
tion of the Dirac equation for the hydrogen-like atom,
why coming forth the mathematical difficulty such as
the expression (5) and (9) and the virtual energies is
that the size of the nucleus of the hydrogen-like atom
is not considered in the rough boundary condition (1),
and the nuclear are regarded as the point in geomet-
rical meaning. The point in geometrical meaning falls
short of the actual case of the atomic nucleus. In fact,
the necessary of the normalizable wave function of the
hydrogen atom has been discussed home and widely in
some modern physics textbook
[42-44]
. Now one should
consider the actual size of the atomic nucleus to rewrite
the boundary condition then find the eigensolution of
the Dirac equation for the hydrogen and hydrogen-like
atom.
3 Exact boundary condition and challenge
solution of Dirac equation
If the wave equation theory of quantum mechanics it-
self will not be challenged, then boundary conditions for
various wave equations of quantum mechanics can be
written out based on the structure of the physical mod-
el and distributing character of the physical quantity.
There are two basic facts, one is that the atomic nucle-
us has definite size, we suppose its equivalent radius or
barrier width is δ, and another is that the electron does
not enter the inside of the atomic nucleus, and does not
collide to and rub with the atomic nucleus. In this way,
any wave equation that describes the atom has the same
exact boundary condition
R (r = δ) ̸= ±∞,
R (r → ∞) = 0,
−∞ < R (δ < r < ∞) < ∞
(10)
One can use this exact condition to solve the Schr¨odinger
equation for the hydrogen atom so as to recover the Bohr
formula of the energy levels. It is well known that the
Schr¨odinger equation of the hydrogen-like atom has no
virtual energy solution, and there is no mathematical
contradiction that the solution of the equation does not
meet the conditions of a definite solution. Then, in order
to solve the problem that the virtual energy of Dirac’s
solution under rough boundary conditions and the di-
vergence of the solution of the equation do not meet
the boundary condition, what is the result of finding
the solution of Dirac’s equation under accurate bound-
ary conditions? Although formally one would obtain the
satisfying formula that is as exact as the distinguished
Dirac formula when considering the spin-orbit coupling
in the Dirac equation, and further solutions can also be
made from the point of view that the boundary condi-
tions are related to the self adjoint of the operator, so as
to give a solution to avoid the divergence of Dirac func-
tion. We hope to jump out of all kinds of logical thinking
of non inevitable causality and reprocess Dirac equation
with accurate boundary conditions from the perspective
of pure mathematics
[56]
. This is because there are simi-
lar problems that need to develop relevant mathematical
theories.
It must be reiterated, as described in the text after
equation (2), the Dirac theory of hydrogen atom uses
an unconventional operation method to determine the
angular quantum number κ = ±1, ±2, ···, so as to ob-
tain the radial Dirac equation of the hydrogen atom or
hydrogen-like atom
cˆp
r
0 i
−i 0
−
~cκ
r
0 1
1 0
+ mc
2
1 0
0 −1
R =
E +
e
2
4πε
0
r
R (11)
Where the radial momentum operator is controversial-
ly defined as ˆp
r
= −i~ (∂/∂r + 1/r), whether it is rea-
sonable or not has been discussed in detail and clearly
concluded
[19]
. This means that it is necessary to under-
stand the solution of the regular radial Dirac equation
derived from the radial momentum operator that con-
forms to the laws of mathematical operation rather than
man-made definition. This part is left to the reader to
solve. Following the radial momentum operator defined
by the Dirac theory, this paper focuses on the solution
of the recognized radial Dirac equation under accurate
boundary conditions. It usually introduces a mathemat-
ical transformation
R =
1
r
F (r)
1
r
G (r)
(12)
202206-6 X. D. Dongfang Dongfang Challenge Solution of Dirac Hydrogen Equation
and translate the radial Dirac equation for the hydrogen
atom
[42-44, 57]
into the following form
E − mc
2
~c
+
α
r
F +
κ
r
+
d
dr
G = 0
E + mc
2
~c
+
α
r
G +
κ
r
−
d
dr
F = 0
(13)
Considering the exact boundary condition (10), intro-
duce the transform
ξ = r − δ (ξ > 0) (14)
the boundary condition (10) can be overwritten as fol-
lows
R (ξ → 0) ̸= ±∞
R (ξ → ∞) = 0
−∞ < R (0 < ξ < ∞) < ∞
(15)
then r = ξ + δ, substituting it into (13), one obtains
E − mc
2
~c
+
α
ξ + δ
F +
κ
ξ + δ
+
d
dξ
G = 0
E + mc
2
~c
+
α
ξ + δ
G +
κ
ξ + δ
−
d
dξ
F = 0
(16)
For this kind of variable coefficient differential equation,
the undetermined form of the exact solution of the e-
quation is usually written by using the asymptotic solu-
tion of the differential equation, and then the equation
is solved. The asymptotic solution comes from the solu-
tion of the asymptotic differential equation with ξ → ∞.
When ξ → ∞, the asymptotic form of equation (16) is
dG
dξ
−
mc
2
− E
~c
F ≈ 0
dF
dξ
−
mc
2
+ E
~c
G ≈ 0
From this, two second-order asymptotic second-order d-
ifferential equations are obtained
dF
2
dξ
2
−
m
2
c
4
− E
2
~
2
c
2
F ≈ 0
dG
2
dξ
2
−
m
2
c
4
− E
2
~
2
c
2
G ≈ 0
Their asymptotic solutions are the same,
F ≈ e
−aξ
, G ≈ e
−aξ
Where a =
√
m
2
c
4
− E
2
~c. the exact solution form of
equation (16) takes its asymptotic solution as the weight
function, and let the specific form of the exact solution
be
F = e
−aξ
f (ξ) , G = e
−aξ
g (ξ) (17)
By transforming the first-order differential equations in-
to second-order differential equations, it can be proved
that the asymptotic solution method of differential equa-
tions with variable coefficients is reasonable. Substitut-
ing (17) into equation (16), one then obtains
E − mc
2
~c
(ξ + δ) + α
f + (ξ + δ)
dg
dξ
+ [κ − a (ξ + δ)] g = 0
E + mc
2
~c
(ξ + δ) + α
g − (ξ + δ)
df
dξ
+ [κ + (aξ + δ)] f = 0
(18)
the eigensolutions of equations (18) correspond to quantum energy are two interrupted series, which the number of
terms is determined by the eigenvalues.
In order to find the general series solutions for equations (18), it is assumed that the formal solutions are
f (ξ) =
∞
ν=0
b
ν
ξ
σ + ν
, g (ξ) =
∞
ν=0
d
ν
ξ
σ +ν
(19)
Substituting into equations (18), one obtains the linear system of recurrence relations
∞
ν=0
E − mc
2
~c
b
ν−1
+
E − mc
2
~c
δb
ν
+ αb
ν
+ Kd
ν
+ (σ + ν) d
ν
+ δ (σ + ν + 1) d
ν+1
− ad
ν−1
− δad
ν
ξ
σ +ν
= 0
∞
ν=0
E + mc
2
~c
d
ν−1
+
E + mc
2
~c
δd
ν
+ αd
ν
+ Kb
ν
− (σ + ν) b
ν
− δ (σ + ν + 1) b
ν+1
+ ab
ν−1
+ δab
ν
ξ
σ +ν
= 0
(20)
hence the coefficient of the power series satisfy the following system of recurrence relations
E − mc
2
~c
b
ν−1
+
E − mc
2
~c
δ + α
b
ν
− ad
ν−1
+ δ (σ + ν + 1) d
ν+1
+ (κ + σ + ν − δa) d
ν
= 0
E + mc
2
~c
d
ν−1
+
E + mc
2
~c
δ + α
d
ν
+ ab
ν−1
− δ (σ + ν + 1) b
ν+1
+ (κ − σ − ν + δa) b
ν
= 0
(21)
Mathematics & Nature (2022) Vol. 2 No. 1 202206-7
Corresponding to ν = −1 the indicial equations are given that δσb
0
= 0 and δσd
0
= 0. Because δ ̸= 0, b
0
̸= 0 and
d
0
̸= 0, one obtains
σ = 0 (22)
so that the wave functions satisfy the boundary condition at r → δ namely ξ → 0, the above equations reduce to
E − mc
2
~c
b
ν−1
+
E − mc
2
~c
δ + α
b
ν
− ad
ν−1
+ δ (ν + 1) d
ν+1
+ (κ + ν − δa) d
ν
= 0
E + mc
2
~c
d
ν−1
+
E + mc
2
~c
δ + α
d
ν
+ ab
ν−1
− δ (ν + 1) b
ν+1
+ (κ − ν + δa) b
ν
= 0
(23)
Respectively evaluate for ν = 0, 1, 2, ··· , n
r
, b
n
r
+1
= d
n
r
+1
= 0, make use of that b
−2
= d
−2
= 0 and b
−1
= d
−1
= 0,
equations (23) give
E − mc
2
~c
δ + α
b
0
+ (κ − δa) d
0
+ δd
1
= 0
(κ + δa) b
0
+
E + mc
2
~c
δ + α
d
0
− δb
1
= 0
E − mc
2
~c
b
0
+
E − mc
2
~c
δ + α
b
1
− ad
0
+ (κ + 1 − δa) d
1
+ 2δd
2
= 0
ab
0
+ (κ − 1 + δa) b
1
+
E + mc
2
~c
d
0
+
E + mc
2
~c
δ + α
d
1
− 2δb
2
= 0
.
.
.
E − mc
2
~c
b
n
r
−1
+
E − mc
2
~c
δ + α
b
n
r
− ad
n
r
−1
+ (κ + n
r
− δa) d
n
r
= 0
ab
n
r
−1
+ (κ − n
r
+ δa) b
n
r
+
E + mc
2
~c
d
n
r
−1
+
E + mc
2
~c
δ + α
d
n
r
= 0
E − mc
2
~c
b
n
r
− ad
n
r
= 0
ab
n
r
+
E + mc
2
~c
d
n
r
= 0
(24)
The last two formulas are linearly dependent. Note that a =
√
m
2
c
4
− E
2
~c and use
E + mc
2
~c to multiply the
third formula from bottom and use
√
m
2
c
4
− E
2
~c to multiply the fourth formula from bottom, and then add the
two new formulas, it is given as follows
α
E + mc
2
+ (κ − n
r
)
m
2
c
4
− E
2
b
n
r
+
(κ + n
r
)
E + mc
2
+ α
m
2
c
4
− E
2
d
n
r
= 0 (25)
Substituting for the second formula from the bottom, one will obtain a new formula of the energy levels for the
hydrogen atom
E =
mc
2
1 +
α
2
n
2
r
, (n
r
= 0, 1, 2, 3, ···) (26)
It is different from the Dirac formula of the energy levels for the hydrogen atom. This result is the inevitable
deduction of the Dirac equation with the exact boundary condition for the hydrogen atom. With the exact boundary
condition (10) and the new formula of the energy levels (26), all of the corresponding wave functions satisfy the
boundary conditions and there is not any virtual energy.
According to (11), (14), (16), (19), (23), the whole wave function with the exact boundary condition is as follows
R =
e
−aξ
ξ+δ
n
r
ν=0
b
ν
ξ
ν
e
−aξ
ξ+δ
n
r
ν=0
d
ν
ξ
ν
=
e
−
√
m
2
c
4
−E
2
~c
(r−δ)
1
r
n
r
ν=0
b
ν
(r − δ)
ν
e
−
√
m
2
c
4
−E
2
~c
(r−δ)
1
r
n
r
ν=0
d
ν
(r − δ)
ν
, (n
r
= 0, 1, 2, ···) (27)
202206-8 X. D. Dongfang Dongfang Challenge Solution of Dirac Hydrogen Equation
the coefficients of the corresponding polynomial are determined by the system of recurrence relations (23). All
appearance, at the boundary of the hydrogen atom
lim
r→δ
R = lim
ξ→0
R =
Constant
Constant
lim
r→∞
R = lim
ξ→∞
R =
0
0
(28)
Make use of the definition (6), the probability density of the electron appearing outside the nucleus of the hydrogen
atom takes the form
ρ =
e
−aξ
ξ + δ
n
r
ν=0
b
ν
ξ
ν
2
+
e
−aξ
ξ + δ
n
r
ν=0
d
ν
ξ
ν
2
Namely,
ρ =
e
−
√
m
2
c
4
−E
2
~c
(r−δ)
1
r
n
r
ν=0
b
ν
(r − δ)
ν
2
+
e
−
√
m
2
c
4
−E
2
~c
(r−δ)
1
r
n
r
ν=0
d
ν
(r − δ)
ν
2
(29)
homoplastically, one has
lim
r→δ
ρ = lim
ξ→0
ρ = Constant
lim
r→∞
ρ = lim
ξ→∞
ρ = 0
(30)
Obviously, the exact solution of the Dirac equation of the
hydrogen atom under exact boundary conditions does
not show obvious mathematical contradiction. Howev-
er, the energy level formula thus obtained is no different
from Bohr energy level formula. This is obviously not
the result expected by quantum mechanics. The exact
solution of the Dirac equation of the hydrogen atom un-
der rough boundary conditions is only formal. Although
it pieced together the exact energy formula of the hydro-
gen atom that meets the expectation, there are enough
irreconcilable mathematical and physical logical contra-
dictions. In order to obtain the desired formula, it is
not a choice that scientists should make at the expense
of mathematical rules and objective logic. Therefore,
the exact solution of the Dirac equation of the hydrogen
atom under the exact boundary poses a severe challenge
to relativistic quantum mechanics, which is called the
challenging solution of the Dirac equation. Consider-
ing the situation how much the problems hidden in the
Dirac equation that will be introduced step by step later
are beyond our guess, it can be inferred that the Dirac
equation may not be the best choice to describe the fine
structure of the hydrogen atom. If readers think that
this conclusion damages the reputation of celebrities and
produces resistance or even disgust, it will not be con-
ducive to the progress of scientific theory. You can find
a better wave equation to describe the fine structure of
the hydrogen atom and obtain the ideal energy level for-
mula. Although in the author’s opinion, it may still not
be the best solution, it is always better than the theory
full of contradictions.
4 Conclusions and comments
Famous theories about natural phenomena as well as
incredible fairy tales that are wildly under the aura of
science can always more effectively stimulate the enthu-
siasm of those who pursue truth to test those logic and
inferences. In this paper we expatiated that the diver-
gences of the Dirac function and the virtual energy of
the Dirac formula of the energy levels for the hydrogen
and hydrogen-like atom are due to the traditional rough
boundary condition. By using the exact boundary con-
dition one can obtain a new solution of the Dirac equa-
tion in the Coulomb field. The new solution without
any mathematical difficulty gives a new formula of the
energy levels which is different from the distinguished
Dirac formula. One can find that in the new solution
the values constructed by Dirac have been written in
the radius of the atomic nucleus and they are indepen-
dent of the new formula for the energy levels. Only when
looking from the point of view of the formula of the en-
ergy levels, the new formula is not as exact as the Dirac
formula. However, considering the spin-orbit coupling
or some new potential parameters, one will obtain the
more exact formula.
It generally believes that the Dirac equation has been
successful in many important predictions
[58]
. The Dirac
equation has an infinite number of negative energy solu-
tions, which are considered to lead to the discovery of the
existence of positrons and interpreted as an unexpected
relativistic interaction between the translational motion
of electrons and spin, resulting in violent oscillations of
particles at very high frequencies and a distance of about
one Compton wavelength. This leads to the concept of
Dirac electron. The further description is that as long
as the electron wave function contains positive energy
and negative energy components, and zeitbergen will oc-
cur. Both sets of states need to establish an arbitrary
electronic state. As we all know, the wonderful part of
Mathematics & Nature (2022) Vol. 2 No. 1 202206-9
quantum mechanics is to naturally obtain the energy lev-
el formula of bound states by solving the wave equation
with boundary conditions. All other follow-up products
are just academic whitewash. Entanglement of quantum
states leads to the birth of quantum computer? So what
is the real meaning of wave function? Let’s look forward
to the wonderful ending of the story. Since the exact so-
lution of the Dirac equation that meets the expectation
can only be obtained under the rough boundary condi-
tions that do not meet the atomic structure, the Dirac
exact energy level formula should be treated again. This
paper reveals the hidden mathematical and physical dif-
ficulties in the Dirac equation of Coulomb potential, and
gives the exact solution of the Dirac equation of the hy-
drogen atom under the exact boundary conditions. The
purpose is to accurately present the inevitable mathe-
matical logic of different wave equations. At least the
divergence can be eliminated by using the exact bound-
ary conditions and abandoning the rough boundary con-
ditions, although this may not be the ultimate answer.
The solution of the Dirac equation, which ostensibly e-
liminates the mathematical contradiction, can not give
the formula of fine structure energy level of the hydro-
gen atom. Failure to meet the expectation is the driving
force to find the ultimate answer.
No matter which wave equation is used to describe
the quantum system of the bound statge, their eigenval-
ues set and eigensolutions set must be in agreement with
the uniqueness, and the solution must accord with the
conditions for the exact solution but not any approx-
imate solution such as cut off potential. In principle,
we cannot immolate the mathematical rule to obtain a
formula only for agreeing with the experimentally ob-
served hydrogen spectra. Actually the divergence of the
original solution for the Dirac equation in the Coulom-
b field is nonexistent. The Dirac equation is more and
more widely applied for various models. One should note
that using different mathematical methods and different
boundary condition to solve the differential equation will
obtain different results
[59, 60]
. The rough boundary con-
dition for the Dirac equation with the Coulomb poten-
tial brings on so mathematical contradictions. We need
to revise all incorrect mathematical methods and search
the correct mathematical methods to find the correct
and exact solution of the various wave equations. One
can find that the classical solution of the plane trans-
verse electromagnetic mode of the Maxwell equation is
also incorrect
[61, 62]
, because of the incorrect mathemati-
cal method for solving the wave equation. For quantum
mechanics, only when the solution method of the wave
equation conforms to the law of mathematical opera-
tion and the exact solution does not violate the natural
law, the corresponding energy level formula is meaning-
ful. Further research should explore the exacter formula
of the energy levels for the hydrogen and hydrogen-like
atom by using the exact boundary condition that ac-
cords with the structure of the atoms. Here it should
be pointed out that the Schr¨odinger equation and the
Klein-Gordon equation in the Coulomb field with the
exact boundary condition have the same corresponding
formula of the energy levels.
Usually we can try to maintain famous theories. How-
ever, when irreconcilable mathematical irreconcilable
contradictions and physical and logical contradiction-
s implied in the famous theory are found, and bet-
ter treatment methods are presented, we should make
some changes. The author’s research does not mean
a complete denial of the achievements of his predeces-
sors. Without the brilliant footprints of predecessors,
the later researchers can not make any breakthrough
in mathematics and physics. Correcting the mistakes
of predecessors is an important link in the development
of science. From the perspective of mathematics, cor-
rectly dealing with logical contradictions implied in the
Dirac equation will bring not only the development of
mathematics, but also the profound reform of quantum
mechanics, and even the great reform of the whole the-
oretical physics
[14-17, 19, 20]
. Isn’t it a blessing that sci-
entific theory is facing a great opportunity for profound
change?
1 Gordon, W. Der comptoneffekt nach der Schr¨odingerschen
theorie. Zeitschrift f¨ur Physik 40, 117-133 (1926).
2 Klein, O. Quantentheorie und f¨unfdimensionale Relativ-
it¨atstheorie. Zeitschrift f¨ur Physik 37, 895-906 (1926).
3 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).
4 Darwin, C. G. The wave equations of the electron. Proceed-
ings of the Royal Society of London. Series A, Containing
Papers of a Mathematical and Physical Character 118, 654-
680 (1928).
5 Gordon, W. Die energieniveaus des wasserstoffatoms nach
der diracschen quantentheorie des elektrons. Zeitschrift f¨ur
Physik 48, 11-14 (1928).
6 Biedenharn, L. C. Remarks on the relativistic Kepler prob-
lem. Physical Review 126, 845 (1962).
7 Wong, M. & Yeh, H.-Y. Simplified solution of the Dirac e-
quation with a Coulomb potential. Physical Review D 25,
3396 (1982).
8 Wong, M. & Yeh, H.-Y. Exact solution of the Dirac-Coulomb
equation and its application to bound-state problems. II. In-
teraction with radiation. Physical Review A 27, 2305 (1983).
9 Su, J.-Y. Simplified solutions of the Dirac-Coulomb equation.
Physical Review A 32, 3251 (1985).
10 Nenciu, G. Self-adjointness and invariance of the essential
spectrum for Dirac operators defined as quadratic form-
s. Communications in Mathematical Physics 48, 235-247
(1976).
11 Klaus, M. & W¨us, R. Characterization and uniqueness of dis-
202206-10 X. D. Dongfang Dongfang Challenge Solution of Dirac Hydrogen Equation
tinguished self-adjoint extensions of Dirac operators. Com-
munications in Mathematical Physics 64, 171-176 (1979).
12 B. Thaller, The Dirac Equation , Springer, New York, 1992.
Preface.
13 Dongfang, X. D. The End of Klein-Gordon Equation for
Coulomb Field. Mathematics & Nature 2, 202201 (2022).
14 Dongfang, X. D. The End of Teratogenic Simplified Dirac Hy-
drogen Equations. Mathematics & Nature 2, 202202 (2022).
15 Dongfang, X. D. Dongfang Solution of Induced Second Order
Dirac Equations. Mathematics & Nature 2, 202203 (2022).
16 Dongfang, X. D. The End of Isomeric Second Order Dirac Hy-
drogen Equations. Mathematics & Nature 2, 202204 (2022).
17 Dongfang, X. D. The End of Expectation for True Second Or-
der Dirac Equation. Mathematics & Nature 2, 202205 (2022).
18 Dongfang, X. D. The Morbid Equation of Quantum Numbers.
Mathematics & Nature 1, 202102 (2021).
19 Dongfang, X. D. Dongfang Angular Motion Law and Opera-
tor Equations. Mathematics & Nature 1, 202105 (2021).
20 Dongfang, X. D. Relativistic Equation Failure for LIGO Sig-
nals. Mathematics & Nature 1, 202103 (2021).
21 Dongfang, X. D. Dongfang Com Quantum Equations for
LIGO Signal. Mathematics & Nature 1, 202106 (2021).
22 Dongfang, X. D. Com Quantum Proof of LIGO Binary Merg-
ers Failure. Mathematics & Nature 1, 202107 (2021).
23 Yukawa, H. On the Interaction of Elementary Particles I.
Proc. Phys. Math. Soc. Jpn. 17, 48 (1935).
24 Dongfang, X. D. Nuclear Force Constants Mapped by Yukawa
Potential. Mathematics & Nature 1, 202109 (2021).
25 Dongfang, X. D. The End of Yukawa Meson Theory of Nu-
clear Forces. Mathematics & Nature 1, 202110(2021).
26 Dongfang, X. D. Dongfang Modified Equations of Molecular
Dynamics. Mathematics & Nature 1, 202104 (2021).
27 Dongfang, X. D. Dongfang Modified Equations of Electro-
magnetic Wave. Mathematics & Nature 1, 202108 (2021).
28 Dongfang, X. D. On the Relativity of the Speed of Light.
Mathematics & Nature, 1, 202101 (2021).
29 Chen, R. The Optimum Differential Equations. Chinese
Journal of Engineering Mathematics 17, 82-86 (2000).
30 Chen, R. The Uniqueness of the Eigenvalue Assemblage for
Optimum Differential Equations, Chinese Journal of Engi-
neering Mathematics 20, 121-124(2003).
31 Chen, R. New exact solution of Dirac-Coulomb equation with
exact boundary condition. International Journal of Theoret-
ical Physics 47, 881-890 (2008).
32 Schr¨odinger, E. An undulatory theory of the mechanics of
atoms and molecules. Physical review 28, 1049 (1926)..
33 Schr¨odinger, E. Quantisierung als eigenwertproblem. An-
nalen der physik 385, 437-490 (1926).
34 Bohr, N. On the constitution of atoms and molecules. Par
III. Systems containing several nuclei. Phil. Mag. 6, p. 875
(1913).
35 Villalba, V. M. Bound states of the hydrogen atom in the p-
resence of a magnetic monopole field and an Aharonov-Bohm
potential. Physics Letters A 193, 218-222 (1994).
36 Dominguez-Adame, F. & Rodriguez, A. A one-dimensional
relativistic screened Coulomb potential. Physics Letters A
198, 275-278 (1995).
37 Horbatsch, M. & Shapoval, D. Variational solution of the
two-center Dirac-Coulomb problem in the free-particle repre-
sentation. Physics Letters A 209, 63-68 (1995).
38 Gu, Y. Repulsive bound states produced by momentum cutoff
in Coulomb systems. Physics Letters A 295, 81-86 (2002).
39 Dong, S.-H. & Lozada-Cassou, M. Scattering of the Dirac par-
ticle by a Coulomb plus scalar potential in two dimensions.
Physics Letters A 330, 168-172 (2004).
40 Marques, G. d. A., Bezerra, V. B. & Fernandes, S. G. Exact
solution of the Dirac equation for a Coulomb and scalar po-
tentials in the gravitational field of a cosmic string. Physics
Letters A 341, 39-47 (2005).
41 Ritchie, B. Relativistic bound states. Physics Letters A 352,
239-241 (2006).
42 Dirac, P. A. M. The principles of Quantum Mechanics,
Clarendon Press, Oxford, 1958, pp268-272
43 Greiner, W. Relativistic Quantum Mechanics: Wave Equa-
tions, Springer, 2000, 3rd Edition, pp225-260.
44 Schiff, L. I. Quantum Mechanics. M cG raw-H ill, New York
(1968).
45 Ciftci, H., Hall, R. L. & Saad, N. Iterative solutions to the
Dirac equation. Physical Review A 72, 022101 (2005).
46 Nakatsuji, H. & Nakashima, H. Analytically solving the rel-
ativistic Dirac-Coulomb equation for atoms and molecules.
Physical review letters 95, 050407 (2005).
47 Gavrilov, S. P. & Gitman, D. M. Vacuum instability in ex-
ternal fields. Physical Review D 53, 7162 (1996).
48 Vakarchuk, I. On Dirac theory in the space with deformed
Heisenberg algebra: exact solutions. Journal of Physics A:
Mathematical and General 38, 7567 (2005).
49 Villalba, V. M. in Journal of Physics: Conference Series.
016 (IOP Publishing).
50 Torrente-Lujan, E. Baryon asymmetry at the weak phase
transition in the presence of arbitrary CP violation. Physics
of Atomic Nuclei 69, 147-158 (2006).
51 Berkdemir, C., Berkdemir, A. & Sever, R. Systematical ap-
proach to the exact solution of the Dirac equation for a
deformed form of the Woods–Saxon potential. Journal of
Physics A: Mathematical and General 39, 13455 (2006).
52 Morsink, S. M. & Mann, R. B. Black hole radiation of Dirac
particles in 1+ 1 dimensions. Classical and Quantum Gravity
8, 2257 (1991).
53 Taub, A. A special method for solving the dirac equations.
Reviews of Modern Physics 21, 388 (1949).
54 Yennie, D., Ravenhall, D. G. & Wilson, R. Phase-shift cal-
culation of high-energy electron scattering. Physical Review
95, 500 (1954).
55 Bethe, H. & Salpeter, E. in Handbuch der Physik Vol. 35
(Springer-Verlag Berlin, 1957)..
56 Deck, R., Amar, J. G. & Fralick, G. Nuclear size corrections
to the energy levels of single-electron and-muon atoms. Jour-
nal of Physics B: Atomic, Molecular and Optical Physics 38,
2173 (2005).
57 Bjorken, J. D. & Drell, S. D. Relativistic quantum fields.
(McGraw-Hill, 1965).
58 Buchanan, M. Dirac gets the jitters. Nature Physics 1, 5-5
(2005).
59 Dong, S.-H. & Ma, Z.-Q. Exact solutions to the Dirac equa-
tion with a Coulomb p otential in 2+ 1 dimensions. Physics
Letters A 312, 78-83 (2003)..
60 Alhaidari, A. Solution of the Dirac equation with position-
dependent mass in the Coulomb field. Physics Letters A 322,
72-77 (2004)..
61 Chen, R. The Problem of Initial Value for the Plane Trans-
verse Electromagnetic Mode. Acta Phys. Sin-CH ED 49,
2518 (2000).
62 Chen, R., Li, X. J. Achievable Transverse Cylindrical Electro-
magnetic Mode, Progress In Electromagnetics Research Let-
ters 24, 59-68(2011).
Latest findings
Neutron State Solution of Dongfang Modified Dirac Equation
The challenging solution of the Dirac equation of the Coulomb field satisfying exact boundary conditions 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 exact 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 corresponding 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 modified 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 potential function, which reflects the physics dilemma that the wave function is wrongly described as a potential 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 angular 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.