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Article
.
Mathematics
Dongfang Special Entangled Solution of Schr¨odinger
Hydrogen Equation
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Abstract: The textbook solutions of Schr¨odinger equations are usually represented as
the combination of multiple special function symbols, resulting in a large number of miss-
ing exact solutions not being discovered. Taking the Schr¨odinger wave function of a
hydrogen atom as an example, regressing to the analytical expression that can be ex-
tended and programmed for testing, two types of sp ecial entangled wave functions that
have not been described by established theories have been discovered. Specifically, when
the magnetic quantum number is 0, the traditional Schr¨odinger wave function degener-
ates into a binary function about radial variables r and angle θ without angles ϕ. The
conclusion of this reasoning process, which seems very rigorous in the traditional sense, is
actually extremely untrue. The Schr¨odinger equation for hydrogen atoms has two types of
sp ecial entangled solutions with magnetic quantum numbers of 0, and the sine and cosine
functions of angle ϕ are included in it, implying that there are other unknown wave func-
tions that satisfy the Schr¨odinger equation and even affect the energy eigenvalue formula.
The three-dimensional function image of the special entangled solution of the hydrogen
atom Schr¨odinger equation is further drawn. The results show intuitively and clearly that
there is no one-to-one correspondence between the zero or extreme point of the modulus
function of the ternary function and that of the square of the modulus function of the
ternary function. It is concluded that the definition of the square of the wave function
mo dulus as the probability density function lacks causality, and it is an urgent problem
to derive the probability density function according to the basic principle.
Keywords: Hydrogen atom, Schr¨odinger equation, special entangled spherical har-
monic function, Schr¨odinger wave function, special entangled wave function, probability
density function.
MSC(2020) Subject Classification: 35J05, 43A90, 33E10
Contents
1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2
2 The traditional analytical solution of Schr¨odinger hydrogen equation · · · · · · · · 2
3 Zero magnetic quantum number special entangled wave function of hydrogen · · 4
4 Three types of special wave functions and probability density diagrams · · · · · · 8
5 Integrated special entanglement Schr¨odinger wave function of hydrogen · · · · · · 10
6 Complete set of special entangled Schr¨odinger wave functions for hydrogen · · · · 14
7 The uncertainty of normalized Schr¨odinger wave functions · · · · · · · · · · · · · · · 15
8 Comments · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 17
Citation: Dongfang, X. D. Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation. Mathematics & Nature
202303 (2023).
2 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
1 Introduction
The mathematical processes of quantum mechanics seem perfect enough. However, new re-
search has found that the set of known eigensolutions of certain partial differential equations,
such as spherical harmonic partial differential equations, is not the complete solution set of the
equation, but one of the subsets. Early partial differential equation theory tended to obtain ex-
act solutions to equations, while early quantum mechanics tended to obtain energy eigenvalues
through exact solutions to wave equations, leading to the neglect of the integrity of the solution
set of partial differential equations in both mathematics and physics. Of course, now we urgently
need to understand the complete solution sets of various wave equations in order to present the
quantum mechanical landscape of truly perfect mathematical processes.
The local exact solution of the Schr¨odinger equation for hydrogen atoms contains energy eigen-
value that are astonishingly consistent with the Bohr hydrogen atom energy formula. This un-
expected gain promoted the birth of quantum mechanics and its rapid promotion and chain
development, which in turn gave rise to a large number of financially successful large-scale quan-
tum experimental projects. Experimental projects fed back beautification projects, forcing strict
ideological unity. The argument that the Schr¨odinger equation for hydrogen atoms perfectly re-
veals the law of low-speed microscopic motion is therefore, deeply ingrained in people’s minds.
The Bohr hydrogen atom theory based on the assumption of angular momentum quantization ex-
plains the spectral structure of hydrogen atoms. The Schr¨odinger equation itself is a hypothesis,
but it constructs a theoretical framework for describing microscopic phenomena with partial dif-
ferential equations satisfied by wave functions of being not truly clear in meaning, thus becoming
the fundamental equation of quantum mechanics.
The spherical harmonic partial differential equation obtained by decomposing the Schr¨odinger
equation in spherical coordinates is completely consistent with the angular momentum square
operator equation, which hides various forms of missing solutions. Therefore, as a second-
order partial differential equation, the traditional exact solution of the time-dependent stationary
Schr¨odinger equation is only a subset of the equation’s intrinsic solution set. The Schr¨odinger
equation has other sets of intrinsic solutions that satisfy boundary conditions. The discovery of a
large number of missing exact solutions to wave equations will inject new vitality into quantum
mechanics.
To briefly illustrate the phenomenon of missing solutions to those wave equations, this paper
first introduces two types of zero magnetic quantum number special entangled solutions for the
Schrodinger equation of hydrogen atoms that have not been described by established theories. The
complete set of special entangled function general solution containing local meanings of known
solutions is given, indicating that the boundedness and normalization conditions of the wave
function are not sufficient to determine the special solution. Although the energy eigenvalues of
hydrogen atoms are not currently affected by special entanglement solutions, except for the ground
state case, the three-dimensional contour images clearly indicate that the spatial distribution of
the square of the Schr¨odinger wave function modulus defined as a probability density function
does not correspond one-to-one with the spatial distribution of the wave function modulus, and
the experimental observability of the statistical significance of the wave function is broken.
2 The traditional analytical solution of Schr¨odinger hydrogen equation
Usually, Z represents the nuclear charge of a hydrogen like atom; ~ represents the reduced
Planck constant defined by the so-called Planck constant h, α represents the fine-structure con-
stant, and c represents the speed of light in the vacuum. Marking A = Zα, the potential energy of
a hydrogen like atomic system in a spherical coordinate system can be expressed in the following
Mathematics & Nature Vol. 3 (2023) 3
simple form,
U (r) = −
A~c
r
(2.1)
The charge e of electrons is included in the fine-structure constant.
It should be pointed out that theoretically, the so-called fine structure constant α or Planck
constant h is not a true constant. α and h should be referred to as fine structure parameters
and Planck parameters, respectively. It is strongly recommended that experimental physicists
accurately determine the distribution table of fine structural parameters α or Planck parameter
h, as this is a significant scientific experiment that subverts traditional limiting thinking. But
for hydrogen atoms, the product αh is a constant. Therefore, expression (2.1) is not affected by
future experimental conclusions.
There are too many conjectures or hypotheses about the principles of quantum mechanics.
A correct conjecture or assumption necessarily implies a strict causal relationship, which may
not actually be complex, but may not be noticed. The experimental data is often corrected as
needed, and there are actually many different or even completely opposite explanations for the
conclusions. Therefore, rigorous logic is the preferred choice for scientific theory. In order to
maintain logical rigor and conciseness, we summarize the characteristic solution theory of the
Schr¨odinger equation for hydrogen like atoms as the following definite solution problem.
Problem: Assuming the reduced mass of a hydrogen like atom is µ, the exact solution of the
Schr¨odinger equation that satisfies the boundedness and normalization conditions of the wave
function constitutes the definite solution problem
−
~
2
2µ
∇
2
ψ −
A~c
r
ψ = Eψ
ψ (r → ∞) = 0, ψ (0 6 r, 0 6 θ 6 π, 0 6 φ 6 2π) ̸= ±∞
ψ (r, θ + 2π, φ + 2π) = ψ (r, θ, φ)
∞
r=0
π
θ=0
2π
φ=0
ψ (r, θ, φ) ψ
∗
(r, θ, φ) r
2
sin θdφ
dθ
dr = 1
(2.2)
Where
∇
2
ψ =
1
r
2
∂
∂r
r
2
∂ψ
∂r
+
1
r
2
∂
2
ψ
∂θ
2
+
cos θ
sin θ
∂ψ
∂θ
+
1
sin
2
θ
∂
2
ψ
∂ϕ
2
(2.3)
The definite solution problem is a concise and accurate concrete expression of the quantum
mechanical principles of Schr¨odinger hydrogen atoms. The Schr¨odinger equation is usually de-
composed into radial ordinary differential equations and angular differential equations using the
method of separating variables. The angular differential equation is a spherical harmonic partial
differential equation. The solution of the radial ordinary differential equation is represented by
the Laguerre function, and the solution of the spherical harmonic partial differential equation
is represented by the associated Legendre function. The textbook solution to the Schr¨odinger
equation for a hydrogen atom commonly uses abbreviations for these functions. Abbreviations
may seem concise, but in reality, due to being too abstract, the question of whether the given
solution is a complete set solution is completely ignored. The analytical form of the regression
wave function makes the structure of the exact solution to the Schr¨odinger equation for hydrogen
atoms clear, which not only facilitates the verification of the solution, but also helps to discover
missing solutions. Here, a lemma is used to represent the textbook solution of the Schr¨odinger
equation for hydrogen atoms.
Lemma 1 (Traditional solution of Schr¨odinger equation): If the integer m, natural number l,
4 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
and positive integer n satisfy the relationships |m| 6 l and l < n, then the wave function
ψ
⟨m⟩
n,l
(r, θ, φ) =
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×
a
⟨m⟩
n,l
cos mφ + a
⟨−m⟩
n,l
sin mφ
sin
m
θ
×
∞
j=0
j
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2j
θ
l−m=0,2,4,···
;
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×
b
⟨m⟩
n,l
cos mφ + b
⟨−m⟩
n,l
sin mφ
sin
m
θ
×
∞
j=0
j
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2j+1
θ
l−m=1,3,5,···
(2.4)
with two undetermined coefficients a
⟨m⟩
n,l
and a
⟨−m⟩
n,l
or b
⟨m⟩
n,l
and b
⟨−m⟩
n,l
satisfies the stationary
Schr¨odinger equation
∇
2
ψ +
2Aµc
~r
ψ +
2µE
~
2
ψ = 0 (2.5)
i.e.
1
r
2
∂
∂r
r
2
∂ψ
∂r
+
1
r
2
∂
2
ψ
∂θ
2
+
cos θ
sin θ
∂ψ
∂θ
+
1
sin
2
θ
∂
2
ψ
∂ϕ
2
+
2Aµc
~r
+
2µE
~
2
ψ = 0 (2.6)
the energy eigenvalue in the equation is
E = −
µA
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
The wave function satisfies the natural period condition ψ (r, θ + 2π, ϕ + 2π) = ψ (r, θ, ϕ) and the
bounded condition.
The integers m, l and n in Lemma 1 are respectively referred to as magnetic quantum numbers,
orbital angular momentum quantum numbers, and total quantum numbers. Lemma 1 is a com-
prehensive form of the traditional exact solution to the Schr¨odinger equation for hydrogen atoms,
and no further proof is needed. Unlike traditional specific representations, the uncertainty of the
undetermined coefficients in the wave function (2.4) is manifested. To determine the specific form
of the wave function, at least another definite solution condition is required in addition to the nor-
malization condition. The traditional representation of normalization coefficients given through
a special descriptive order is not mathematically meaningful and cannot actually normalize the
wave function. This is only a narrow case, and in the broad case, the complete solution set of the
Schr¨odinger equation has enough undetermined coefficients to be undetermined.
3 Zero magnetic quantum number special entangled wave function of hy-
drogen
In Lemma 1, let the magnetic quantum number m = 0 obtain a special wave function in the
traditional sense, which is a binary function of the radial variable r and the angular variable θ,
meaning that the wave function of the zero magnetic quantum number is indep endent of the angle
ϕ. This mathematical inference seems strict, but the conclusion does not conform to the facts.
Mathematics & Nature Vol. 3 (2023) 5
The traditional Legendre sp ecial bounded wave function for zero magnetic quantum numbers
of hydrogen atoms satisfies the Schr¨odinger equation, but it is not the only wave function for
zero magnetic quantum numbers of hydrogen atoms. We present two special wave functions with
zero magnetic quantum numbers containing the angle ϕ, both of which satisfy the Schr¨odinger
equation for hydrogen atoms.
3.1 The traditional special direct product wave function of hydrogen
When the magnetic quantum number m = 0, the traditional exact solution (2.4) of the
Schr¨odinger equation degenerates into a binary function, known as the Legendre special bounded
wave function of hydrogen atoms.
Lemma 2 (Legendre special bounded wave function): If the natural number l and the positive
integer n satisfy the condition l < n, then the Legendre special bounded wave function
ψ
⟨0⟩
n,l
(
r, θ, φ
) =
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×a
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2j
θ
l=0,2,4,···
;
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×b
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2j+1
θ
l=1,3,5,···
(3.1)
with an undetermined coefficient a
n,l
or b
n,l
satisfies the stationary Schr¨odinger equation
1
r
2
∂
∂r
r
2
∂ψ
∂r
+
1
r
2
∂
2
ψ
∂θ
2
+
cos θ
sin θ
∂ψ
∂θ
+
1
sin
2
θ
∂
2
ψ
∂ϕ
2
+
2Aµc
~r
+
2µE
~
2
ψ = 0 (3.2)
the energy eigenvalue in the equation is
E = −
µA
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
The traditional wave function of the Schr¨odinger equation for the zero magnetic quantum number
of hydrogen atoms does not include an angle ϕ. Table 3.1 lists the traditional special solutions
of the Schr¨odinger equation for hydrogen atoms calculated based on (3.1) when n = 6 and l =
0, 1, · · · , 5, respectively. It can be verified that all wave functions in the table satisfy the stationary
Schr¨odinger equation for hydrogen atoms as described in the lemma.
Table 3.1 Partial Legendre special solutions for zero magnetic quantum numbers in the
Schr¨odinger equation for hydrogen atoms
ψ
⟨0⟩
6,0
= a
6,0
e
−
cAµr
6~
1 −
c
5
A
5
µ
5
r
5
174960~
5
+
c
4
A
4
µ
4
r
4
1944~
4
−
5c
3
A
3
µ
3
r
3
324~
3
+
5c
2
A
2
µ
2
r
2
27~
2
−
5cAµr
6~
ψ
⟨0⟩
6,1
= b
6,1
e
−
cAµr
6~
r + 6r
c
4
A
4
µ
4
r
4
408240~
4
−
c
3
A
3
µ
3
r
3
4860~
3
+
c
2
A
2
µ
2
r
2
180~
2
−
cAµr
18~
cos θ
6 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
ψ
⟨0⟩
6,2
= a
6,2
e
−
cAµr
6~
r
2
+ 120r
2
−
c
3
A
3
µ
3
r
3
1088640~
3
+
c
2
A
2
µ
2
r
2
15120~
2
−
cAµr
720~
1 − 3cos
2
θ
ψ
⟨0⟩
6,3
= b
6,3
e
−
cAµr
6~
r
3
+ 5040r
3
c
2
A
2
µ
2
r
2
3265920~
2
−
cAµr
60480~
cos θ −
5cos
3
θ
3
ψ
⟨0⟩
6,4
= a
6,4
e
−
cAµr
6~
r
4
−
cAµr
5
30~
1 − 10cos
2
θ +
35cos
4
θ
3
ψ
⟨0⟩
6,5
= b
6,5
e
−
cAµr
6~
r
5
cos θ −
14cos
3
θ
3
+
21cos
5
θ
5
3.2 The first type of special entangled wave function for hydrogen
Replacing cos θ with sin ϕ sin θ in the traditional wave function (3.1) of the Schr¨odinger equation
for zero magnetic quantum numbers of hydrogen atoms yields a new series of wave functions that
satisfy the Schr¨odinger equation, known as the first kind of special entangled wave function for
hydrogen atoms.
Theorem 1 (The first kind of special bounded entangled wave function): If the natural number
l and the positive integer n satisfy the condition l < n, then the first kind of special bounded
entangled wave function
ξ
⟨0⟩
n,l
(r, θ, φ) =
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×c
⟨0⟩
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2j
l=0,2,4,···
;
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×d
⟨0⟩
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2j+1
l=1,3,5,···
(3.3)
with an undetermined coefficient c
n,l
or d
n,l
satisfies the stationary Schr¨odinger equation
1
r
2
∂
∂r
r
2
∂ξ
∂r
+
1
r
2
∂
2
ξ
∂θ
2
+
cos θ
sin θ
∂ξ
∂θ
+
1
sin
2
θ
∂
2
ξ
∂ϕ
2
+
2Aµc
~r
+
2µE
~
2
ξ = 0 (3.4)
the energy eigenvalue in the equation is
E = −
µA
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
The proof of Theorem 1 is divided into two parts: the radial wave function satisfies the radial
Schr¨odinger equation and the angular wave function satisfies the spherical harmonic partial dif-
ferential equation. The radial wave function is a well-known weighted Laguerre function and does
not require further proof. The proof that the angular entanglement function satisfies the spherical
harmonic equation is just a simple but lengthy calculation, as seen in the article entitled special
entanglement spherical harmonic function
[1]
, which is omitted here. Table 3.2 lists the first kind of
special entanglement solutions for the Schr¨odinger equation of hydrogen atoms calculated based
on (3.3), taking n − l = 0, 1, · · · , 10, respectively. It can be verified that all wave functions in the
Mathematics & Nature Vol. 3 (2023) 7
table satisfy the stationary Schr¨odinger equation for hydrogen atoms as stated in the theorem.
Table 3.2 Partial first type special entanglement solutions for zero magnetic quantum numbers
in the Schr¨odinger equation for hydrogen atoms
ξ
⟨0⟩
6,0
= c
6,0
e
−
cAµr
6~
1 −
c
5
A
5
µ
5
r
5
174960~
5
+
c
4
A
4
µ
4
r
4
1944~
4
−
5c
3
A
3
µ
3
r
3
324~
3
+
5c
2
A
2
µ
2
r
2
27~
2
−
5cAµr
6~
ξ
⟨0⟩
6,1
= d
6,1
e
−
cAµr
6~
r + 6r
c
4
A
4
µ
4
r
4
408240~
4
−
c
3
A
3
µ
3
r
3
4860~
3
+
c
2
A
2
µ
2
r
2
180~
2
−
cAµr
18~
sin φ sin θ
ξ
⟨0⟩
6,2
= c
6,2
e
−
cAµr
6~
r
2
+ 120r
2
−
c
3
A
3
µ
3
r
3
1088640~
3
+
c
2
A
2
µ
2
r
2
15120~
2
−
cAµr
720~
1 − 3sin
2
φsin
2
θ
ξ
⟨0⟩
6,3
= d
6,3
e
−
cAµr
6~
r
3
+ 5040r
3
c
2
A
2
µ
2
r
2
3265920~
2
−
cAµr
60480~
sin φ sin θ −
5sin
3
φsin
3
θ
3
ξ
⟨0⟩
6,4
= c
6,4
e
−
cAµr
6~
r
4
−
cAµr
5
30~
1 − 10sin
2
φsin
2
θ +
35sin
4
φsin
4
θ
3
ξ
⟨0⟩
6,5
= d
6,5
e
−
cAµr
6~
r
5
sin φ sin θ −
14sin
3
φsin
3
θ
3
+
21sin
5
φsin
5
θ
5
3.3 The second type of special entangled wave function for hydrogen
In the traditional wave function (3.1) of the zero magnetic quantum number Schr¨odinger equa-
tion for hydrogen atoms, cos θ is replaced with cos ϕ sin θ to obtain a new series of wave functions
that also satisfy the Schr¨odinger equation. The second kind of special entangled wave function
called hydrogen atoms.
Theorem 2 (The second kind of special bounded entangled wave function) If the natural
number l and the positive integer n satisfy the condition l < n, then the second kind of special
bounded entangled wave function
ζ
⟨0⟩
n,l
(r, θ, φ) =
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×f
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2j
l=0, 2, 4, ···
;
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×g
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2j+1
l=1, 3, 5, ···
(3.5)
with an undetermined coefficient f
n,l
or g
n,l
satisfies the stationary Schr¨odinger equation
1
r
2
∂
∂r
r
2
∂ζ
∂r
+
1
r
2
∂
2
ζ
∂θ
2
+
cos θ
sin θ
∂ζ
∂θ
+
1
sin
2
θ
∂
2
ζ
∂ϕ
2
+
2Aµc
~r
+
2µE
~
2
ζ = 0 (3.6)
the energy eigenvalue in the equation is
E = −
µA
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
8 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
The proof process of Theorem 2 is the same as that of Theorem 1, which is omitted here. Table
3.3 lists the second kind of special entangled solutions of the Schr¨odinger equation of hydrogen
atom calculated by taking n − l = 0, 1, · · · , 10 in (3.5) respectively. It can be verified that all wave
functions in the table satisfy the stationary Schr¨odinger equation of the hydrogen atom described
in the theorem.
Table 3.3 Part of the second kind of special entangled solutions of the zero magnetic quantum
number of the Schr¨odinger equation for the hydrogen atom
ζ
⟨0⟩
6,0
= f
6,0
e
−
cAµr
6~
1 −
c
5
A
5
µ
5
r
5
174960~
5
+
c
4
A
4
µ
4
r
4
1944~
4
−
5c
3
A
3
µ
3
r
3
324~
3
+
5c
2
A
2
µ
2
r
2
27~
2
−
5cAµr
6~
ζ
⟨0⟩
6,1
= g
6,1
e
−
cAµr
6~
r + 6r
c
4
A
4
µ
4
r
4
408240~
4
−
c
3
A
3
µ
3
r
3
4860~
3
+
c
2
A
2
µ
2
r
2
180~
2
−
cAµr
18~
sin θ cos φ
ζ
⟨0⟩
6,2
= f
6,2
e
−
cAµr
6~
r
2
+ 120r
2
−
c
3
A
3
µ
3
r
3
1088640~
3
+
c
2
A
2
µ
2
r
2
15120~
2
−
cAµr
720~
1 − 3sin
2
θcos
2
φ
ζ
⟨0⟩
6,3
= g
6,3
e
−
cAµr
6~
r
3
+ 5040r
3
c
2
A
2
µ
2
r
2
3265920~
2
−
cAµr
60480~
sin θ cos φ −
5sin
3
θcos
3
φ
3
ζ
⟨0⟩
6,4
= f
6,4
e
−
cAµr
6~
r
4
−
cAµr
5
30~
1 − 10sin
2
θcos
2
φ +
35sin
4
θcos
4
φ
3
ζ
⟨0⟩
6,5
= g
6,5
e
−
cAµr
6~
r
5
sin θ cos φ −
14sin
3
θcos
3
φ
3
+
21sin
5
θcos
5
φ
5
4 Three types of special wave functions and probability density diagrams
The dogmatic idea of quantum mechanics is that the Bohr hydrogen atom energy formula is an
inevitable result of the Schr¨odinger equation satisfying the definite solution condition, which is
considered an important symbol of the success of quantum mechanics. Therefore, the application
of partial differential equations to quantum theory seems to have achieved new breakthroughs.
However, partial differential equations conceal numerous mathematical principles that subvert
scientific theories and have been overlooked. The application of partial differential equations
in physics and other scientific theories may not necessarily conform to mathematical and scien-
tific significance, and may even violate mathematical and scientific common sense. The concise
conclusion is that mathematical and scientific theories seriously lack a correct understanding of
partial differential equations. In the future, we will gradually introduce and explain in detail
the answers to relevant questions. Here we first focus on the intrinsic physical meaning of the
wave function. There is an inevitable causality that if the square of the modulus function of
the wave function represents the probability density, then for a specific physical model, such as
the hydrogen atom model, the probability density function should be derived by using the basic
principles of mathematics and physics, and then it is proved that the probability density function
satisfies the Schr¨odinger equation. In fact, if the probability density function with clear meaning
is derived, it does not satisfy the Schr¨odinger equation. This is an unsolved mystery of quantum
mechanics.
Born proposed a statistical interpretation of the wave function and was quickly accepted. The
product of a pair of conjugate wave functions is defined as the probability density of particles
appearing in space. This definition is unproven and unprovable. It may come from the extension
of the characteristics of univariate oscillatory real functions. The modulus of one-dimensional
wave function can represent the relative distribution of the quantity described by the oscillating
function, but the calculus operation of the modulus of the wave function is inconvenient. From
the absolute value of the one-variable oscillating real function to the square of the one-variable
Mathematics & Nature Vol. 3 (2023) 9
oscillating real function, the extreme point and the zero point are unchanged, and have the
characteristics of conformal transformation. Extending to the complex variable function, the
product of a pair of conjugate univariate complex variable functions is the square of the modular
function. From the modular function of the complex variable function to the square of the modular
function, it also has the characteristics of conformal transformation, and the extreme point and
zero point are unchanged. The unsolved problem implied in the logic that the abstract meaning is
greater than the concrete meaning is that from the absolute value of the unary real function to the
even power of the unary real function, or from the modulus function of the unary complex variable
function to the even p ower of the modulus function of the unary complex variable function, there
are the characteristics of conformal transformation. How to prove that the probability density is
the product of the modulus function of the wave function rather than some other even power of
the modulus function ?
However, for multivariate wave functions, from the absolute value of the function to the square
of the function, or from the modulus function of the complex variable function to the square of
the modulus function, there is no feature of conformal transformation, and the extreme points
and zero points cannot remain unchanged. The special wave function of zero magnetic quantum
number is a real function, which is very convenient to clarify the essence of the problem. Table
4.1 draws the contour lines of the modulus function and the square of the modulus function
of the special wave function of three zero magnetic quantum numbers. The range is θ ∈ [0, π],
ϕ ∈ [0, 2π], r ∈ [0, 2~/Aµc] in the case of l = 0, r ∈ [0, 25~/Aµc] in the case of l > 0. The blank areas
in the figure indicate that the function values are relatively large and insignificant. It is found
that only the modulus function of the radial one-dimensional wave function with orbital angle
quantum number l = 0 is similar to the square of the modulus function, while the square of the
three-dimensional wave function with l > 0 from the modulus function to the modulus function
is distorted. This proves that defining the probability density as the square of the modulus
function cannot prove whether it is scientific, and the definition itself is completely negated by
the multivariate wave function.
Table 4.1 Three kinds of special bounded wave function mo dulus and three-dimensional contour
of probability density for n = 6
l
ψ
⟨0⟩
6,l
ψ
⟨0⟩
6,l
2
ξ
⟨0⟩
6,l
ξ
⟨0⟩
6,l
2
ζ
⟨0⟩
6,l
ζ
⟨0⟩
6,l
2
0
1
2
10 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
l
ψ
⟨0⟩
6,l
ψ
⟨0⟩
6,l
2
ξ
⟨0⟩
6,l
ξ
⟨0⟩
6,l
2
ζ
⟨0⟩
6,l
ζ
⟨0⟩
6,l
2
3
4
5
5 Integrated special entanglement Schr¨odinger wave function of hydrogen
Any multivariate function will inevitably satisfy many partial differential equations. As a
solution to the Schr ¨o dinger equation of the second-order partial differential equation of a multi-
variate function, the square of the wave function’s modulus is used to define probability density,
an extremely abstract physical quantity. This definition does not have any causal relationship,
but rather a construction that yields to specific expectations. It is necessary to determine the
specific form of the wave function, but quantum mechanics cannot provide the initial conditions
for second-order partial differential equations, and it is not enough to only assign bounded con-
ditions to the wave equation. Therefore, the normalization condition is proposed. Proposing
normalization conditions is quite creative. However, the normalization condition only applies to
positively definite bounded functions and not to unbounded and oscillatory functions. Taking
the square of the wave function modulus that satisfies the bounded condition of the function
yields a positive definite function, which is the reason defining the probability density function.
This definition, as we now know, does not satisfy the uniqueness theorem. The serious problem
is that the bounded and normalized conditions of the wave function cannot actually determine
the specific form of the wave function. The specific forms of wave functions given by established
theories are not logical inferences that conform to the existence and uniqueness theorems, but
rather cunning conclusions that force specific computational orders. Given the powerful logical
power of proof by contradiction, here we discuss the integration of special entangled Schr¨odinger
wave functions to sp ecifically prove the above statement.
The linear combination of all special solutions satisfying the linear differential equation is
the general solution of the linear differential equation. The general solutions of linear ordinary
differential equations obtained by different methods are equivalent. The general solutions of a
linear partial differential equation obtained by different methods are not necessarily equivalent,
but are various local general solutions. The linear combination of all local general solutions is the
complete set general solution of the linear partial differential equation. However, there is no basic
principle that can explain how many local general solutions a linear partial differential equation
has. The linear combination of multiple local general solutions of a linear partial differential
equation may be only a subset of the complete set general solution. We summarize Lemma 2,
Mathematics & Nature Vol. 3 (2023) 11
Theorem 1 and Theorem 2 to obtain the integrated special entanglement solution of the zero
magnetic quantum number of the hydrogen atom Schr¨odinger equation.
Theorem 3 (Integrated specially bounded entangled wave function): If any natural number l
and positive integer n satisfy the condition l < n, then the integrated specially bounded entangled
wave function
Υ
⟨0⟩
n,l
(r, θ, φ) =
e
−
Aµcr
n~
r
l
+
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
l+ν
×
a
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2j
θ
+c
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2j
+f
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2j
l=0, 2, 4, ···
;
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×
b
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2j+1
θ
+d
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2j+1
+g
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2j+1
l=1, 3, 5, ···
(5.1)
with three undetermined coefficients a
n,l
, c
n,l
and f
n,l
( or b
n,l
, d
n,l
and g
n,l
) satisfies the stationary
Schr¨odinger equation
1
r
2
∂
∂r
r
2
∂Υ
∂r
+
1
r
2
∂
2
Υ
∂θ
2
+
cos θ
sin θ
∂Υ
∂θ
+
1
sin
2
θ
∂
2
Υ
∂ϕ
2
+
2Aµc
~r
+
2µE
~
2
Υ = 0 (5.2)
the energy eigenvalue in the equation is
E = −
µA
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
When the total quantum numb er n = 6, the orbital quantum numbers l = 0 , 1, 2, 3, 4, 5 are
substituted into (5.1), or the wave functions with the same orbital quantum number in Table 3.1,
Table 3.2 and Table 3.3 are added directly, and all the integrated special entangled wave functions
with the total quantum number n = 6 and the magnetic quantum number zero are obtained. The
results are listed in 5.1. It can be checked that these integrated special entangled wave functions
satisfy the Schr¨odinger equation.
The integrated entangled special entangled wave function with zero magnetic quantum number
has three undetermined constants. The complete wave function of this special case cannot be
determined only by the normalization condition. In order to obtain a complete special entangled
wave function, two definite conditions must be supplemented. In the past, the undetermined
12 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
coefficients of three kinds of special wave functions were determined by normalization conditions,
and then linear combination was carried out. In fact, the integrated wave function proposed by
this reversed logic still has three undetermined coefficients, and the specific form of the wave
function cannot be determined by the normalization condition alone. The reversal of logic is
extremely confusing. The integrated entangled special entangled wave function with zero magnetic
quantum number shows that the Schr¨odinger wave function of hydrogen atom is uncertain in the
established quantum mechanics framework.
Table 5.1 Partially integrated special entanglement solutions of zero magnetic quantum
number for Schr¨odinger equation of hydrogen atom
Υ
⟨0⟩
6,0
= α
6,0
e
−
cAµr
6~
1 −
c
5
A
5
µ
5
r
5
174960~
5
+
c
4
A
4
µ
4
r
4
1944~
4
−
5c
3
A
3
µ
3
r
3
324~
3
+
5c
2
A
2
µ
2
r
2
27~
2
−
5cAµr
6~
Υ
⟨0⟩
6,1
=
e
−
cAµr
6~
r + 6r
c
4
A
4
µ
4
r
4
408240~
4
−
c
3
A
3
µ
3
r
3
4860~
3
+
c
2
A
2
µ
2
r
2
180~
2
−
cAµr
18~
× {b
6,1
cos θ + d
6,1
sin φ sin θ + g
6,1
sin θ cos φ}
Υ
⟨0⟩
6,2
=
e
−
cAµr
6~
r
2
+ 120r
2
−
c
3
A
3
µ
3
r
3
1088640~
3
+
c
2
A
2
µ
2
r
2
15120~
2
−
cAµr
720~
×
a
6,2
1 − 3cos
2
θ
+ c
6,2
1 − 3sin
2
φsin
2
θ
+ f
6,2
1 − 3sin
2
θcos
2
φ
Υ
⟨0⟩
6,3
=
e
−
cAµr
6~
r
3
+ 5040r
3
c
2
A
2
µ
2
r
2
3265920~
2
−
cAµr
60480~
×
b
6,3
cos θ −
5
3
cos
3
θ
+ d
6,3
sin φ sin θ −
5
3
sin
3
φsin
3
θ
+g
6,3
sin θ cos φ −
5
3
sin
3
θcos
3
φ
Υ
⟨0⟩
6,4
= e
−
cAµr
6~
r
4
−
cAµr
5
30~
a
6,4
1 − 10cos
2
θ +
35
3
cos
4
θ
+c
6,4
1 − 10sin
2
φsin
2
θ +
35
3
sin
4
φsin
4
θ
+f
6,4
1 − 10sin
2
θcos
2
φ +
35
3
sin
4
θcos
4
φ
Υ
⟨0⟩
6,5
= e
−
cAµr
6~
r
5
b
6,5
cos θ −
14
3
cos
3
θ +
21
5
cos
5
θ
+d
6,5
sin φ sin θ −
14
3
sin
3
φsin
3
θ +
21
5
sin
5
φsin
5
θ
+g
6
,
5
sin θ cos φ −
14
3
sin
3
θcos
3
φ +
21
5
sin
5
θcos
5
φ
The ultimate goal of wave equation theory is not to obtain energy eigenvalues. The fundamental
purpose of solving the wave equation is to obtain the specific wave function that meets the definite
solution conditions and give the wave function real meaning. To determine the specific integrated
wave function of the Schr¨odinger hydrogen equation in the special case of zero magnetic quantum
number, it is critical to supplement two appropriate definite conditions, but this is not easy. Table
5.2 draws the three-dimensional contour of the integrated special wave function of several different
combinations of undetermined coefficients in the range of θ ∈ [0, π], ϕ ∈ [0, 2π], r ∈ [0, 2~/(cAµ)] in
the case of l = 0 , and r ∈ [0, 25~/(cAµ)] in the case of l > 0. The blank difference indicates that
Mathematics & Nature Vol. 3 (2023) 13
the function value is relatively large. The three-dimensional contour intuitively shows that the
distribution of the special integrated wave functions of different combinations of undetermined
coefficients is different for the zero magnetic quantum number determined by any total quantum
number n and orbital angular momentum quantum number l. Therefore, the urgent problem to
be solved in the wave equation theory of quantum mechanics is to propose suitable and sufficient
conditions for the definite solution. In the absence of sufficient and reasonable definite conditions,
the specific form of the wave function cannot be determined, and the development results of the
wave equation theory of quantum mechanics imply greater uncertainty.
Tab. 5.2 Three-dimensional contour lines of partially integrated special wave functions for
Schr¨odinger equation of hydrogen atom
(a
n,l
, c
n,l
, f
n,l
) = (b
n,l
, d
n,l
, g
n,l
)
Υ
⟨0⟩
n,l
(1, 1, 0) (0, 1, 1) (1, 0, 1) (1, 1, 1) (1, 5, 10) (10, 5, 1)
Υ
⟨0⟩
6,0
Υ
⟨0⟩
6,1
Υ
⟨0⟩
6,2
Υ
⟨0⟩
6,3
Υ
⟨0⟩
6,4
Υ
⟨0⟩
6,5
14 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
6 Complete set of special entangled Schr¨odinger wave functions for hydrogen
The integrated special entangled wave function of the Schr¨odinger equation for hydrogen atoms
with zero magnetic quantum numbers listed earlier belongs to the general form of the sub solution
set of the Schr¨odinger equation. The subset of solutions cannot be summed up for the total
quantum number n, but it can be summed up for all orbital angular quantum numbers l that
satisfy the condition l < n. The resulting wave function belongs to the special wave function
of the complete set with local significance. The complete set special wave function has 3n − 2
undetermined coefficients, and relying solely on the normalization condition can obviously only
determine the ground state wave function with n = 1. However, with the increase of the main
quantum number, the required solution condition increases nearly threefold. This decisive factor
has been overlooked in the past, leading to a biased development of the theory. The complete set
of the Schr¨odinger equation for hydrogen atoms and the expression for the special wave function
are as follows.
Theorem 4 ( Complete set special bounded entangled wave function ) Let any natural number l
and positive integer n satisfy the condition l < n, then the complete set special b ounded entangled
wave function
Υ
⟨0⟩
n
(r, θ, φ) =
n−1
l=0
Υ
⟨0⟩
n,l
(r, θ, φ)
=
n−1
l=0
e
−
Aµcr
n~
r
l
+
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
l+ν
×
a
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2j
θ
+c
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2j
+f
n,l
∞
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2j
l=0, 2, 4, ···
+
e
−
Aµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2Aµc
n~
ν
r
ν
×
b
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2j+1
θ
+d
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2j+1
+g
n,l
∞
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2j+1
l=1, 3, 5, ···
(6.1)
with 3n − 2 independent undetermined coefficients a
n,l
, c
n,l
and f
n,l
( or b
n,l
, d
n,l
and g
n,l
) satisfies
the stationary Schr¨odinger equation
1
Υ
∂
∂r
r
2
∂Υ
∂r
+
2Aµc
~
r +
2µE
~
2
r
2
+
1
Υ
∂
2
Υ
∂θ
2
+
cos θ
sin θ
∂Υ
∂θ
+
1
sin
2
θ
∂
2
Υ
∂ϕ
2
= 0 (6.2)
Mathematics & Nature Vol. 3 (2023) 15
the energy eigenvalue in the equation is
E = −
µA
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
In the special case of only zero magnetic quantum number, there are 3n undetermined coefficients
in the special general solution form of the complete set of the Schr¨odinger equation of hydrogen
atom. However, the three terms of l = 0 are combined into one item, and the undetermined
coefficient is equivalent to a
n,0
+ c
n,0
+ f
n,0
= β
n,0
. In fact, there are 3n − 2 independent unde-
termined constants. In addition to the normalization conditions, 3n − 3 definite conditions are
needed to determine the specific form of the wave function. The principal quantum number n can
be a non-negative arbitrary integer, which means that the specific wave function cannot be deter-
mined according to the special general solution of the complete set of the Schr¨odinger equation
of hydrogen atom. The normalized wave function of hydrogen atom given by quantum mechanics
is an illusion rather than an inevitable inference, and its process does not conform to mathe-
matical principles. Using the wave function of uncertain specific form to obtain the determined
energy eigenvalue, or using the theory of uncertainty of credibility to deduce the conclusion of
expectation, there is a coincidence and therefore it is extremely mysterious.
There are some implicit conditions in modern physics that determine the development of theory.
Some of these conditions are not based on the promotion of theoretical perfection, but only to
avoid the orientation of inevitable causality. Extreme expectations that do not conform to the
facts are often described as the inevitable laws of nature. Using the square of the wave function
modulus to define the probability density of particles appearing in space is only an example.
The problem of the solution set of the Schr¨odinger equation, which belongs to the second-
order partial differential equation, has not been correctly handled in modern physics. The basic
principle of the solution of the differential equation is that the linear combination of all the
solutions satisfying the linear differential constitutes the general solution of the linear differential
equation, and the specific solution of the linear differential equation is a special function that
can satisfy the definite solution condition. Partial differential equations lead to different kinds
of local general solutions because of different methods. The linear system of these local general
solutions constitutes a wider range of general solutions of partial differential equations. The local
general solution is a sub-solution set. The simpler the coefficients of linear partial differential
equations are, the more the sub solution sets are. The Laplace equation is the simplest linear
partial differential equation. At present, it seems that it is impossible to give a conclusion on
how many sub-solution sets it has. It is too early to give a conclusion about how many sub-
solution sets Schr¨odinger equation has. The special entanglement solution set of the Schr¨odinger
equation with zero magnetic quantum numbers is sufficient to demonstrate that the credibility of
established scientific theories described by partial differential equations is very low.
7 The uncertainty of normalized Schr¨odinger wave functions
The above Equation (6.1) gives the zero magnetic quantum number general solution of the
Schr¨odinger equation for hydrogen atom or hydrogen-like atom. Using the normalization condi-
tion, only the ground state solution can be determined, so as to obtain the ground state wave
function, and any excited state wave function can not be determined. Normalization conditions of
the special bounded entangled wave function of the Schr¨odinger complete set of hydrogen atoms
are usually written as follows,
∞
0
π
0
2π
0
Υ
⟨0⟩
n
(r, θ, ϕ) dϕ
sin θdθ
r
2
dr =
∞
0
π
0
2π
0
5
l=0
Υ
⟨0⟩
n,l
(r, θ, ϕ) dϕ
sin θdθ
r
2
dr = 1 (7.1)
16 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
The linear combination of the six integrated special wave functions in Table 5.1 is the calculation
result of n = 6 in (5.1). In the form, this gives the complete set entangled wave function of the
Schr¨odinger theory of the hydrogen atom with 16 undetermined coefficients and zero magnetic
quantum numbers, which is represented by the symbol Υ
⟨0⟩
6
=
5
l=0
Υ
⟨0⟩
6,l
. The result is,
Υ
⟨0⟩
6
=
5
l=0
Υ
⟨0⟩
6,l
=
α
6,0
e
−
cAµr
6~
1 −
c
5
A
5
µ
5
r
5
174960~
5
+
c
4
A
4
µ
4
r
4
1944~
4
−
5c
3
A
3
µ
3
r
3
324~
3
+
5c
2
A
2
µ
2
r
2
27~
2
−
5cAµr
6~
+
e
−
cAµr
6~
r + 6r
c
4
A
4
µ
4
r
4
408240~
4
−
c
3
A
3
µ
3
r
3
4860~
3
+
c
2
A
2
µ
2
r
2
180~
2
−
cAµr
18~
× { b
6,1
cos θ + d
6,1
sin φ sin θ + g
6,1
sin θ cos φ}
+
e
−
cAµr
6~
r
2
+ 120r
2
−
c
3
A
3
µ
3
r
3
1088640~
3
+
c
2
A
2
µ
2
r
2
15120~
2
−
cAµr
720~
×
a
6,2
1 − 3cos
2
θ
+ c
6,2
1 − 3sin
2
φsin
2
θ
+ f
6,2
1 − 3sin
2
θcos
2
φ
+
e
−
cAµr
6~
r
3
+ 5040r
3
c
2
A
2
µ
2
r
2
3265920~
2
−
cAµr
60480~
×
b
6,3
cos θ −
5
3
cos
3
θ
+ d
6,3
sin φ sin θ −
5
3
sin
3
φsin
3
θ
+g
6,3
sin θ cos φ −
5
3
sin
3
θcos
3
φ
(7.2)
+ e
−
cAµr
6~
r
4
−
cAµr
5
30~
a
6,4
1 − 10cos
2
θ +
35
3
cos
4
θ
+c
6,4
1 − 10sin
2
φsin
2
θ +
35
3
sin
4
φsin
4
θ
+f
6,4
1 − 10sin
2
θcos
2
φ +
35
3
sin
4
θcos
4
φ
+ e
−
cAµr
6~
r
5
b
6,5
cos θ −
14
3
cos
3
θ +
21
5
cos
5
θ
+d
6,5
sin φ sin θ −
14
3
sin
3
φsin
3
θ +
21
5
sin
5
φsin
5
θ
+g
6,5
sin θ cos φ −
14
3
sin
3
θcos
3
φ +
21
5
sin
5
θcos
5
φ
This integrated special entangled wave function is a bounded function, but it has 16 undeter-
mined coefficients and its specific form cannot be determined.
The logic of established quantum mechanics theory is to force a calculation sequence to nor-
malize the function terms of each independent coefficient, determine each independent coefficient,
and then integrate them, adding a new undetermined coefficient before each term, and focusing
on describing the discrete wave functions of the independent coefficients. The calculation process
does not conform to the basic principle of determining specific solutions from general solutions of
differential equations. In fact, it was not possible to obtain a complete integrated wave function.
However, the very cunning description hides serious logical issues.
If the wave function (7.2) is substituted into the so-called normalization condition (7.1), the
Mathematics & Nature Vol. 3 (2023) 17
calculation result is a quadratic equation with 16 undetermined coefficients,
1 =
∞
0
π
0
2π
0
Υ
⟨0⟩
6
dφ
sin θdθ
r
2
dr =
∞
0
π
0
2π
0
5
l=0
Υ
⟨0⟩
6,l
dφ
sin θdθ
r
2
dr
=
216π~
3
35c
13
A
13
µ
13
35c
10
A
10
µ
10
a
2
6,0
+ 29160c
6
A
6
µ
6
~
4
a
2
6,2
+ 39504568320c
2
A
2
µ
2
~
8
a
2
6,4
+108c
8
A
8
µ
8
~
2
b
2
6,1
+ 1360800c
4
A
4
µ
4
~
6
b
2
6,3
+ 12799480135680~
10
b
2
6,5
+35c
10
A
10
µ
10
c
2
6,0
+ 29160c
6
A
6
µ
6
~
4
c
2
6,2
+ 29628426240c
2
A
2
µ
2
~
8
a
6,4
c
6,4
+39504568320c
2
A
2
µ
2
~
8
c
2
6,4
+ 108c
8
A
8
µ
8
~
2
d
2
6,1
+ 1360800c
4
A
4
µ
4
~
6
d
2
6,3
+12799480135680~
10
d
2
6,5
+ 70c
10
A
10
µ
10
c
6,0
f
6,0
+ 35c
10
A
10
µ
10
f
2
6,0
+70c
10
A
10
µ
10
a
6,0
(c
6,0
+ f
6,0
) − 29160c
6
A
6
µ
6
~
4
c
6,2
f
6,2
+ 29160c
6
A
6
µ
6
~
4
f
2
6,2
−29160c
6
A
6
µ
6
~
4
a
6,2
(c
6,2
+ f
6,2
) + 29628426240c
2
A
2
µ
2
~
8
a
6,4
f
6,4
+29628426240c
2
A
2
µ
2
~
8
c
6,4
f
6,4
+ 39504568320c
2
A
2
µ
2
~
8
f
2
6,4
+108c
8
A
8
µ
8
~
2
g
2
6,1
+ 1360800c
4
A
4
µ
4
~
6
g
2
6,3
+ 12799480135680~
10
g
2
6,5
(7.3)
Obviously, when n = 6, 15 definite conditions are needed to determine the total 16 undetermined
coefficients of the special entanglement wave function of the Schr¨odinger equation, so as to de-
termine the specific special entanglement solution of the Schr¨odinger equation corresponding to
the n = 6 energy level.
In the field of quantum mechanics, it is impossible to provide the other 15 definite solution
conditions for determining the specific form of the wave function beyond the traditional normal-
ization conditions. This is only a sp ecial case where the magnetic quantum number is zero. The
Schr¨odinger wave function of hydrogen atoms can have many undetermined coefficients. Per-
haps the uncertainty principle of quantum mechanics can have a more basic explanation, but the
inherent uncertainty of Schr¨odinger wave functions in quantum mechanics cannot be solved. Nor-
malization, as the only condition for determining bounded wave functions, was misunderstood,
but historically it has been technically described perfectly. Human intelligence can always write
theories by reversing right and wrong, just like applying normalization conditions to local general
solutions of differential equations and then linearly combining them to give pseudo complete wave
functions. However, the irreversible conclusion is that the normalization condition cannot be used
to determine the non-ground state solution of the Schr¨odinger equation.
8 Comments
The traditional solution of the hydrogen atom Schr¨odinger equation expressed by the product
of the weighted Laguerre function and the traditional spherical harmonic function is incomplete.
The traditional spherical harmonic function is a direct product function. It has been proved
that there are two kinds of special entangled spherical harmonic functions even if the magnetic
quantum number is zero. The linear combination of three kinds of special spherical harmonic
functions constitutes the complete set special entangled function solution of the hydrogen atom
Schr¨odinger equation as a second-order partial differential equation, which is the Schr¨odinger
complete set special entangled wave function of the hydrogen atom. The undetermined coefficients
increase with the increase of the main quantum numb er, and there are a total of undetermined
coefficients which cannot be determined by normalization conditions. There is no definite excited
state solution for the rigorous mathematical definite solution problem of the Schr¨odinger equation
of the hydrogen atom. The existence of entangled wave function proves that the mathematical
18 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
deduction of Schr¨odinger equation theory in quantum mechanics is incomplete.
The meaning of wave function touches the mathematical essence of quantum mechanics. The
statistical significance of the wave function satisfying Schr¨odinger is generally accepted. However,
in the past, based on the understanding of one-dimensional oscillatory functions such as harmonic
functions, the square of the modulus function of the wave function was used to represent the
probability density of the particle in space, which was completely negated by the multivariate
wave function. From the module of one-dimensional wave function to the square of the module
of one-dimensional wave function, it has the characteristics of conformal transformation, and the
zero point and the extreme point are unchanged. However, from the module of the multivariate
oscillation function to the square of the mo dule of the multivariate oscillation function, the shape
will be distorted. The probability density defined by the square of the modulus of the wave
function is only applicable to the univariate wave function but not to the multivariate wave
function. The solution of the Schr¨odinger equation of hydrogen atom is a linear combination of
multivariate functions. The mystery of quantum mechanics is actually derived from the lack of
basic mathematical knowledge and the excessive whitewash of deviating from the theme. The
undetermined coefficients of the wave function are determined by the normalization condition,
and the so-called probability density is expressed by the even power of the modulus of the wave
function. All of these can not be proved to be the inevitable causality in nature, but it is fully
proved that the results are uncertain and abnormal.
The conclusion of incomplete mathematical deduction is described as a universal law of nature,
and the credibility of its experimental test results is very low. The usual experimental test of
quantum mechanics is not intrinsically related to the probability density. Based on the physical
conclusions of mathematical reasoning, the prerequisite for experimental testing is that the math-
ematical reasoning process must be reliable. The conclusions of some false inferences, subjective
choices, and false calculations are often claimed to be confirmed by experiments. This is only
a different interpretation of the observed data, and many observed data have enough different
interpretations. The engineering of experimental verification of the expected conclusion must be
based on correct and complete logical inference. There are many mathematical and conceptual
problems to be solved in quantum mechanics. Although the development of quantum mechanics
has bypassed these decisive problems, more and more confusion has arisen, which is the funda-
mental reason why the mathematical nature of quantum mechanics is not known. Only by solving
enough basic problems that must be solved can quantum theory make a real breakthrough.
The Schr¨odinger equation theory of hydrogen atoms relies on mathematical reasoning. The
physical theory based on mathematical reasoning cannot be separated from the constraints of the
basic principles of mathematics and become an independent kingdom of singular logic. Perhaps
we should admit a fact that only mathematicians participate in the demonstration, test, criticism
and revision from beginning to end can we ensure the correctness, integrity and reliability of the
establishment and development of physical theory.
[1]
.
References
[1] Dongfang, X. D. Dongfang Special Entangled Spherical Harmonic Functions. Mathematics & Nature
3, 202302 (2023).
Notes: All breakthrough new conclusions that cannot be fully expressed using past methods are described
in the form of theorems, and these theorems and various new wave functions of the listed hydrogen atom
Schr¨odinger theory have passed the computational tests of Wolfram Mathematica. For lengthy equations,
formatting errors may occur during the process of writing Mathematica code. The equations need to be
split into several parts and processed separately, and then combined to perform operations.
Mathematics & Nature Vol. 3 (2023) 19
• PS1: Why do contemporary breakthrough papers choose to be published in self media
journals?
This groundbreaking paper based on Dongfang Special Entangled Spherical Harmonic Functions was
written within a week, aiming to reveal the serious confidence issues hidden in quantum mechanics and
promote significant changes in scientific theory.
Before writing this paper, Nature Physics had sent a solicitation letter to the author, mainly inquiring
ab out the author’s recent research status, the title and abstract of the paper the author plans to submit
to Nature Physics, and the estimated time required for the author to complete the research project.
However, based on the fact that breakthrough papers submitted to Nature Physics in the past were all
rejected, the author infers that Nature Physics is not genuinely soliciting submissions from the author,
but is tentatively attempting to steal the core scientific principles needed to create a new Newtonian
era and achieve the plan of revitalizing the country through science. The author did not reply to this
solicitation letter and deleted it as spam.
After careful consideration, the author decided to submit the new important paper to Nature Physics
for the last time, both to confirm his personal views on Nature and to respond to readers’ doubts
ab out the author’s choice of publication method. The article submission number is NPHYS-2024-06-
01859. In the submitted version, the author removed important experimental suggestions written after
the first equation. The reason for doing so is out of scientific belief and responsibility. The Nature
journal has an absolute advantage in enticing the world to submit major breakthrough research results
in a timely manner due to its enormous influence. Editors can efficiently assist domestic scientists in
plagiarizing breakthrough discoveries that have changed the scientific world. Although the Nature team
and its complex members, with their current level of scientific theory, do not have the ability to perfect
partial differential equation theory and discover the fundamental principles of unified macroscopic and
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fo cus to plagiarizing breakthrough experimental ideas, profound misunderstandings may occur, leading
science in a new wrong direction once again.
The author agreed to share this paper on Research Square when submitting it to Nature Physics. After
17 days of detailed dissection by numerous personnel, the editor of Nature Physics informed the author
that their decision was not to send the paper for external review. This is expected.
Readers should be aware of the reality of the scientific world. On the one hand, because scientific
theories play a crucial role in the field of ideology, scientific research in some countries is actually tied
to political tasks. On the one hand, due to the enormous benefits of scientific research, some individuals
who originally lacked the ability to conduct scientific research actively infiltrate the editorial and peer
review teams of renowned academic journals, often plagiarizing and selling breakthrough submissions to
maximize their personal interests. In the field of science, it usually takes a hundred or even hundreds of
years for a few pioneering researchers to emerge. If you are a pioneering researcher, you will inevitably
face numerous despicable opponents everywhere, so never expect your breakthrough research results to
b e accepted by reputable academic journals such as Nature. Instead, you should consider how to publicly
disclose your research results and effectively do anti-theft and anti murder work, ensuring that great
discoveries are not contaminated by ignorant and unethical editors or peer reviewers, and ensuring that
readers with independent research capabilities are timely informed of new discoveries and develop theories
correctly.
By reading numerous groundbreaking papers by authors, future pioneering researchers can also expe-
rience unique writing techniques that deter scientific thieves. However, there will always be just scientists
in the world, which is the fundamental guarantee for the continuous progress of human civilization.
20 X. D. Dongfang Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation
• PS2: Decision on Nature Physics manuscript NPHYS-2024-06-01859
Dear Professor Dongfang,
Thank you for submitting your manuscript entitled “Dongfang Special Entangled Solution of Schr¨odinger
Hydrogen Equation” to Nature Physics. Please accept our apologies for the delay in reaching a decision
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Latest findings
Dongfang Special Entangled Schrödinger Wave Function of Hydrogen
Based on the missed exact solution special case of the newly discovered spherical harmonic partial differential equation, the analytical expression of the spherical harmonic function is regressed, and the wave function of the zero magnetic quantum number that overturns the traditional understanding of the Schrödinger equation theory for hydrogen atoms is briefly introduced. At the same time, the answer to the question of the existence and uniqueness of the probability density function definition based on the statistical interpretation of Born in quantum mechanics is clarified. Whether it is the inherent causal relationship hidden in the basic definition or the completely accurate solution of the wave equation, new conclusions clearly indicate that scientific theories described using partial differential equations, especially quantum mechanics, are facing significant changes, and all related theories will inevitably be rewritten after improving the theory of partial differential equations.