MATHEMATICS & NATURE
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Random Vol. 3 Sn: 202304
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⃝ Mathematics & Nature–Free Media of Eternal Truth, China, 2021 https://orcid.org/0000-0002-3644-5170
.
Article
.
Mathematics
Dongfang Special Entangled Schr¨odinger Wave Function of
Hydrogen
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Abstract: 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 Schr¨odinger 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
describ ed 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.
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 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 1
2 The special complete set entangled wave function of hydrogen · · · · · · · · · · · · 2
3 Problem of existence and uniqueness of probability density function· · · · · · · · · 7
4 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 8
1 Introduction
The recently discovered special entangled spherical harmonic functions that are independent of
the commonly described spherical harmonic functions
[1]
indicate that the large numbers of exact
solutions to certain partial differential equations are not described by established mathematical
theories, indicating that various wave equations hide many missing solutions. Here, we briefly
introduce two types of special entangled solutions with zero magnetic quantum numbers
[2]
that
have not been described in quantum mechanics. Their linear combination with traditional exact
solutions forms a class of local complete set general solutions for the Schr¨odinger hydrogen equa-
tion. The results indicate that, in addition to the normalization condition, the specific form of the
Citation: Dongfang, X. D. Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen. Mathematics & Nature 3,
202304 (2023).
2 X. D. Dongfang Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen
exact solution to the Schr¨odinger hydrogen equation increases with the increase of energy levels,
and the natural period condition, natural boundary condition, and normalization condition are
not sufficient to form a definite solution problem with wave equations such as the Schr¨odinger
hydrogen equation. However, the uncertainty of the special entangled wave function has no effect
on the energy eigenvalue formula of the Schr¨odinger hydrogen equation. The three-dimensional
contour lines of a special entangled wave function in the form of a real variable function are drawn
by assigning undetermined coefficients, and the results intuitively and clearly demonstrate that
there is no one-to-one correspondence between the wave function modulus and the wave function
modulus square due to multivariate reasons. Therefore, the quantum mechanical definition that
the probability density of particle appearance is the square of the wave function modulus is not
supported by mathematical principles, and this conclusion also provides sufficient counterevidence
for the uncertainty of any even power of the wave function.
2 The special complete set entangled wave function of hydrogen
The spiritual essence of quantum mechanics
[3, 4]
is that the non-completely exact solution of
the wave equation that satisfies the definite solution condition leads to the quantized energy
eigenvalues
[5-7]
, especially the energy eigenvalues of the Schr¨odinger equations
[8, 9]
. for hydrogen
atoms, which conform to the Bohr hydrogen atom quantized energy level formula
[10, 11]
. This
energy level formula has been tested by experimental observations and becomes the standard
for whether the wave equation is accepted, and determines the direction of the development of
quantum mechanics
[12, 13]
. So, are there or how many different partial differential equations with
exactly the same eigenvalue under the same definite solution conditions? This type of question
breaks a single mindset, and the answer does not support the claim that quantum forces are
mathematically precise and perfect.
Examine the completeness and rationality of exact solutions to wave equations. Let’s first re-
view the textbo ok exact solutions of the Schr¨odinger equation for hydrogen atoms, and then focus
on the special entangled solutions of zero magnetic quantum numbers and their mathematical
conclusions that have been overlooked. For the sake of simplicity and clarity, the Schr¨odinger
wave equation theory for hydrogen atoms is summarized here as a lemma or theorem for definite
solution problems and problem answers. This should develop into a fixed format for describing
wave equation theory, in order to reduce the lengthy discourse that often amplifies misleading
functions.
Problem (Schr¨odinger hydrogen atom): One of definite solution problems for the wave equa-
tion theory of hydrogen atoms with reduced mass µ, consisting of wave function bounded condi-
tions, natural period conditions, normalization conditions, and Schr¨odinger equation,
−
~
2
2µ
∇
2
ψ −
α~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
(2.1)
Where α is the fine structure constant, c is the speed of light in vacuum, ~ = h/2π is the reduced
Planck constant, E is the energy of the hydrogen atom, and ψ is the wave function. Operators
in spherical coordinate systems
∇
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
(2.2)
Mathematics & Nature Vol. 3 (2023) 3
The standard textbook theory uses the separation of variables method to solve the Schr¨odinger
equation mentioned above, and the results are usually represented as a combination of normalized
direct product bounded functions. This left a deep impression on us: the integral constant of
the bounded function solution of the Schr¨odinger equation seems to be completely determined
by the normalization condition. According to differential equation theory, the special solution of
a differential equation is a special case where the general solution satisfies the definite solution
condition. To solve the Schr¨odinger wave equation, which essentially belongs to second-order lin-
ear partial differential equations, the general solution form with undetermined integral constants
should still be given first. For the convenience of verification and new description, we now begin
to regress the analytical form of the special function, and summarize the general solution of the
traditional direct product function form of the Schr¨odinger equation for hydrogen atoms as the
following lemma.
Lemma (traditional solution of Schr¨odinger equation): For magnetic quantum number m, or-
bital angular momentum quantum number l, and total quantum number n that satisfy conditions
|m| 6 l and l < n, a bounded wave function
ψ
⟨m⟩
n,l
(r, θ, φ) =
e
−
αµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2αµc
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
−
αµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2αµc
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,···
(2.3)
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 natural periodic
conditionψ (r, θ + 2π, ϕ + 2π) = ψ (r, θ, ϕ). It is one of the exact solutions to the steady-state
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
+
2αµc
~r
+
2µE
~
2
ψ = 0 (2.4)
with an energy eigenvalue of
E = −
µα
2
c
2
2n
2
, (n = 1, 2, 3, · · ·) (2.5)
The standardized form of the traditional direct product function solution to the Schr¨odinger
equation for hydrogen atoms mentioned above also has multiple undetermined coefficients, which
cannot be determined using normalization conditions. But the standard textbo ok theory pro-
vides the normalization coefficient of the wave function. This is one of the hidden issues in the
Schr¨odinger theory of hydrogen atoms, which is difficult to pay attention to due to its technical
description. More importantly, due to serious flaws in the theory of partial differential equations,
4 X. D. Dongfang Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen
boundary problems of the same partial differential equation often have a large number of missing
solutions. There are two types of entangled function solutions for spherical harmonic partial dif-
ferential equations with zero magnetic quantum numbers that have not been discovered in history.
Any new spherical harmonic function will inevitably seriously affect the established Schr¨odinger
wave equation theory.
When the magnetic quantum number is m = 0, the traditional solution of the Schr¨odinger
equation degenerates into a binary function about r and θ, no longer containing the angle ϕ. This
seemingly rigorous mathematical inference is actually not true. Replacing the cosine function
cos θ in the direct product function (2.3) with sin ϕ sin θ and cos ϕ sin θ respectively, the two types
of special entangled wave functions obtained satisfy the Schr¨odinger equation for hydrogen atoms.
According to the theory of differential equations, the linear combination of the traditional solution
(2.3) and the two types of entangled wave functions obtained by substitution is the local general
solution of the Schr¨odinger equation, known as the special complete set entangled Schr¨odinger
wave function of hydrogen atoms. The lemma of traditional theory has been extended and
summarized as the following theorem.
Theorem (Schr¨odinger special entangled wave function): Let the magnetic quantum number
m = 0, for the orbital angular momentum quantum number l and total quantum number n that
satisfy the condition l < n, with no less than 3n−2 mutually independent undetermined coefficients
a
n,l
, c
n,l
, and f
n,l
(or b
n,l
, d
n,l
, and g
n,l
), the special complete set entangled wave function
ψ
⟨0⟩
n
(r, θ, φ) =
n−1
l=0
e
−
αµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2αµc
n~
ν
r
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) (2 k + 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
−
αµcr
n~
r
l
1 +
∞
ν=1
ν
η=1
(η − n + l)
η (η + 2l + 1)
2αµc
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) (2 k + l)
(2k + 1) 2k
(sin φ sin θ)
2j+1
+g
n,l
∞
j=0
j
k=1
(2k − l − 1) (2 k + l)
(2k + 1) 2k
(cos φ sin θ)
2j+1
l=1, 3, 5, ···
(2.6)
satisfies the steady-state 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
+
2αµc
~r
+
2µE
~
2
Υ = 0 (2.7)
Mathematics & Nature Vol. 3 (2023) 5
with energy eigenvalues
E = −
µα
2
c
2
2n
2
, (n = 1, 2, 3, · · ·)
The proof of the theorem is divided into the radial function section and the angular function
section. The radial function section can be found in the standard textbook theory. The proof
process for the two types of entangled spherical harmonic functions that appear in the angular
function section is only a simple but tedious calculation, written in the form of a mathematical
article. For any relatively small quantum number that satisfies the agreed conditions in the
theorem, the correctness of the theorem can be verified using a computer.
Exploring the essence of quantum mechanics and attempting to mathematically prove certain
hypothetical principles of quantum mechanics, it can be found that the fine structure constant
derived from Planck’s constant is not a true constant. The two should be renamed as Planck
parameters and fine structure parameters respectively. It is strongly recommended that exper-
imental physicists accurately measure the distribution of fine structural parameters or Planck
parameters.
The entangled wave function for the special case of magnetic quantum number m = 0 is very
convenient for exploring the applicability of the normalization conditions of the wave function
and whether the probability density definition based on the wave function is objective. By
setting n = 1, 2, 3, 4, and 5, and using (2.6) to calculate the first five complete sets of special
entangled wave functions for hydrogen atoms with magnetic quantum number m = 0, we can
obtain several specific forms with increasing undetermined coefficients. These results clearly
state the conclusions that have been overlooked throughout history, except for the ground state
case, the multiple undetermined coefficients of the solution to any other case of the hydrogen
Schr ¨o dinger equation cannot be determined, and therefore the specific form of the wave function
cannot be determined. The normalization coefficients of wave functions given in history are pseudo
coefficients disguised under descriptive techniques, rather than necessarily causal conclusions.
ψ
⟨0⟩
1
= β
1,0
e
−
cαµ
~
r
ψ
⟨0⟩
2
= e
−
cαµ
2~
r
β
2,0
1 −
cαµr
2~
+ b
2,1
r cos θ + r sin θ (d
2,1
sin φ + g
2,1
cos φ)
ψ
⟨0⟩
3
= e
−
cαµ
3~
r
β
3,0
1 +
2c
2
α
2
µ
2
r
2
27~
2
−
2cαµr
3~
+r
2
a
3,2
1 − 3cos
2
θ
+ c
3,2
1 − 3sin
2
θsin
2
φ
+f
3,2
1 − 3cos
2
φsin
2
θ
+
r −
cαµr
2
6~
[b
3,1
cos θ + sin θ (d
3,1
sin φ + g
3,1
cos φ)]
ψ
⟨0⟩
4
= e
−
cαµ
4~
r
β
4,0
1 −
c
3
α
3
µ
3
r
3
192~
3
+
c
2
α
2
µ
2
r
2
8~
2
−
3cαµr
4~
+
r
2
−
cαµr
3
12~
a
4,2
1 − 3cos
2
θ
+ c
4,2
1 − 3sin
2
θsin
2
φ
+f
4,2
1 − 3cos
2
φsin
2
θ
+
r
c
2
α
2
µ
2
r
2
− 20cαµ~r + 80~
2
80~
2
× [b
4,1
cos θ + sin θ (d
4,1
sin φ + g
4,1
cos φ)]
−
1
3
r
3
b
4,3
cos θ
5cos
2
θ − 3
+ sin θ
d
4,3
sin φ
5sin
2
θsin
2
φ − 3
+g
4,3
cos φ
5cos
2
φsin
2
θ − 3
(2.8)
Among them, β
n,0
is the reduced undetermined coefficient, also known as the undetermined
6 X. D. Dongfang Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen
integral constant.
The statistical interpretation of wave functions enables them to be given normalization con-
ditions, thus only determining the integral constant of the ground state wave function. The
normalization coefficients of other wave functions are not the true results of normalization, but
are masked by the smoothness of specific descriptions. The general form of normalization condi-
tions is
∞
∫
0
π
∫
0
2π
∫
0
ψ
2
dφ
sin θdθ
r
2
dr = 1 (2.9)
Among the several complete set special entangled Schr¨odinger wave functions listed in (2.8), only
the total quantum number n = 1, so the undetermined coefficient β
1,0
of the ground state wave
function with orbital angular momentum quantum number l = 0 is unique and can be determined
by normalization conditions. The ground state wave function has specific forms,
ψ
⟨0⟩
1
=
c
3
A
3
µ
3
π~
3
e
−
cαµ
~
r
(2.10)
But the wave function with a total quantum number n > 1 has 3n − 2 independent undeter-
mined coefficients, and the normalization condition can only provide the relationship that these
undetermined coefficients satisfy, but cannot determine the multiple independent undetermined
coefficients. According to the so-called normalization condition, the relationship equations for
the undetermined coefficients of the entangled wave function of a complete set with n = 2, 3, 4,
respectively, are,
8π~
3
4~
2
b
2
2,1
+ d
2
2,1
+ g
2
2,1
+ α
2
µ
2
c
2
β
2
2,0
c
5
α
5
µ
5
= 1
27π~
3
8α
7
µ
7
c
7
27~
2
432~
2
a
2
3,2
+ c
2
3,2
− c
3,2
f
3,2
+f
2
3,2
− a
3,2
(c
3,2
+ f
3,2
)
+α
2
µ
2
c
2
b
2
3,1
+ d
2
3,1
+ g
2
3,1
+ 8α
4
µ
4
c
4
β
2
3,0
= 1
125π~
3
56c
11
α
11
µ
11
50000c
4
α
4
µ
4
~
4
a
2
5,2
+ 122500000000~
8
a
2
5,4
−50000c
4
α
4
µ
4
~
4
a
5,2
(c
5,2
+ f
5,2
)
+91875000000~
8
a
5,4
(c
5,4
+ f
5,4
)
+25~
2
(7c
6
α
6
µ
6
b
2
5,1
+ 109375c
2
α
2
µ
2
~
4
b
2
5,3
+2000c
4
α
4
µ
4
~
2
c
2
5,2
+ 4900000000~
6
c
2
5,4
+7c
6
α
6
µ
6
d
2
5,1
+ 109375c
2
α
2
µ
2
~
4
d
2
5,3
−2000c
4
α
4
µ
4
~
2
c
5,2
f
5,2
+ 2000c
4
α
4
µ
4
~
2
f
2
5,2
+3675000000~
6
c
5,4
f
5,4
+ 4900000000~
6
f
2
5,4
+7c
6
α
6
µ
6
g
2
5,1
+ 109375c
2
α
2
µ
2
~
4
g
2
5,3
) + 56c
8
α
8
µ
8
β
2
5,0
= 1
(2.11)
Within the framework of quantum mechanics, which is determined by the statistical interpretation
of wave functions, it is impossible to supplement the definite solution conditions that increase
with the total quantum number to determine these undetermined coefficients and thus determine
the specific form of the wave function. This fact provides a new conclusion for the Schr¨odinger
wave equation theory in quantum mechanics.
Inference 1: When the total quantum number is n, there are 3n − 2 undetermined integral
constants for the entangled eigensolutions of the complete set of the Schr¨odinger equation for
hydrogen atoms, and the specific form of the wave function cannot be determined. In short, there
is no specific functional solution to the Schr¨odinger equation for hydrogen atoms in the statistical
sense of wave functions.
Mathematics & Nature Vol. 3 (2023) 7
3 Problem of existence and uniqueness of probability density function
The abstract Born statistical interpretation of wave function
[14]
is actually very vague, and the
definition of probability density function has not been strictly proven. And the distortion from
the image of the modulus of the wave function to the image of the square of the modulus of the
redefined wave function representing probability density is a fatal flaw and one of the reasons why
the mathematical essence of quantum mechanics has always been difficult to reveal. Starting from
the physical meaning of probability density, derive the probability density function of interacting
microscopic particles appearing at any position in space in certain models, and then clarify the
relationship between this probability density function and the Schr¨odinger equation, which is the
best solution to solve the difficulties of quantum mechanics. The fact we are going to reveal is
that the wave function satisfying the Schr¨odinger equation is actually uncertain and can only
have a relatively abstract form. Specifically, except for the ground state wave function which can
have a definite form by determining the integral constant through normalization conditions, all
wave functions of other energy levels do not have corresponding definite solution conditions to
determine the integral constant and obtain a specific form.
The complete set special entangled wave function (2.3) is a representative of the missed solution
of the Schr¨odinger equation for hydrogen atoms, indicating the serious mathematical shortcomings
of quantum mechanics wave equation theory. Now, using the complete set special entangled wave
function further illustrates the logical problems hidden in the definition of probability density in
the context of Born’s statistical interpretation. It cannot be proven scientifically that quantum
mechanics uses the wave function modulus squared to represent the probability density of particles
appearing in the space. The initial definition of the probability density function may be based on
the selection of three features: 1) The square of the modulus function and the modulus function
are both positive definite functions; 2) The square of the modular function has the same zero
and extreme points as the modular function; 3) The square of the modulus function is more
convenient in calculation than the modulus function. However, these features only belong to
univariate functions, while multivariate functions do not have these features. This is one of the
major oversights in defining the probability density function. Another major oversight is that
any even power of a modular function is a positive definite function, and the zero and extreme
points of any even power of a univariate modular function remain unchanged. If the reason for
defining probability density in quantum mechanics is valid, then it cannot be ruled out to define
a density function for any positive integer k,
ρ (r, θ, ϕ) = |ψ (r, θ, ϕ)|
2k
(3.1)
However, there is no logical basis to prove that k = 1 is a unique value. The logical power of
counter evidence is beyond doubt.
In mathematics, functions can be defined with specific mathematical meanings. But in physics,
functions are used to describe natural laws and cannot be defined based on certain mathematical
characteristics. When it is impossible to prove the appropriateness of the definition of the proba-
bility density function through experimental observation, a sufficient number of similar definitions
lead to mutual negation of the uncertainty of the spatial distribution of physical quantities de-
scribed by the function, thus negating the initial narrow definition. The normalization condition
can be applied to any bounded function, but it may not necessarily be the standard for describing
natural laws. Taking ψ
⟨0⟩
3
in equation (2.8) as an example, taking all constants as one unit and
all undetermined coefficients as 1, the specific form of the agreed wave function is:
ψ
⟨0⟩
3
= e
−r/3
1 −
2r
3
+
2r
2
27
−
r
6
− 1
r [cos θ + sin θ (cos ϕ + sin ϕ)]
(3.2)
8 X. D. Dongfang Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen
Table 3D contour lines of the modulus and modulus square of the special entangled
Schr¨odinger wave function
Radial interval 0 6 r 6 10 0 6 r 6 25
Wave function
modulus
Wave function
modulus squared
Draw the three-dimensional contour lines of the mo dulus and its square of the wave function
(3.2), and the image is listed in the table. The contour lines intuitively and clearly indicate
that there is a significant difference in the spatial distribution of the two multivariate functions,
without a one-to-one correspondence. This leads to another new conclusion about the Schr¨odinger
wave equation theory in quantum mechanics.
Inference 2: Only any even power of a univariate wave function mo dule corresponds one-to-
one with zero and extreme points of the same function module. The distribution of even power
functions of multivariate wave function modules is related to even power, and there is no one-to-
one correspondence between zero and extreme points. It cannot be proven that the square of the
wave function module is the only choice to define the probability density function. The statistical
interpretation of quantum mechanics lacks sufficient logic and experimental basis, and is full of
irreconcilable contradictions.
4 Conclusions
This paper presents a special case of various missed exact solutions to the Schr¨odinger equation.
Based on the basic principles of differential equation theory, the abstract function form of the
special general solution of the complete set of zero magnetic quantum numbers for the Schr¨odinger
equation is clarified. The reason why the normalization condition cannot determine the specific
solution of the wave equation that is essentially a second-order linear partial differential equation is
Mathematics & Nature Vol. 3 (2023) 9
explained, and the non objectivity and hidden logical difficulties of the definition of the probability
density function in the statistical sense of the wave function are analyzed. The wave function
of hydrogen atoms is a microcosm of quantum mechanics theory. Even as a special case of
the large number of solutions missing from the Schr¨odinger equation for hydrogen atoms, the
special entangled wave function is sufficient to demonstrate the urgency of improving the theory
of quantum mechanical wave equations. The solutions of the Schr¨odinger equation for other
models are similar, but there are also irreversible conclusions such as the uncertainty of the
complete set wave function and the distortion of the shape from the wave function mode to the
wave function mode square, resulting in the loss of probability density significance of the wave
function mode square. The confidence level of quantum mechanics wave equation theory is very
low. All theories based on wave equations must be rewritten after improving the theory of partial
differential equations.
References
[1] Dongfang, X. D. Dongfang Special Entangled Spherical Harmonic Functions. Mathematics & Nature
3, 202302 (2023).
[2] Dongfang, X. D. Dongfang Special Solutions to Schr¨odinger Hydrogen Equation. Mathematics &
Nature 3, 202303 (2023).
[3] Griffiths, D. J. & Schroeter, D. F. Introduction to quantum mechanics. (Cambridge university press,
2018).
[4] Susskind, L. & Friedman, A. Quantum mechanics: the theoretical minimum. (Basic Books, 2014).
[5] Bes, D. R. Quantum mechanics: a modern and concise introductory course. (Springer, 2007).
[6] Zettili, N. Quantum mechanics: concepts and applications. (2009).
[7] Dirac, P. A. M. The principles of quantum mechanics. (Oxford university press, 1981).
[8] Schr¨odinger, E. Quantisierung als eigenwertproblem. Annalen der physik 385, 437-490 (1926).
[9] Manning, M. F. Exact solutions of the Schr¨odinger equation. Physical Review 48, 161 (1935).
[10] Shirley, J. H. Solution of the Schr¨odinger equation with a Hamiltonian periodic in time. Physical
Review 138, B979 (1965).
[11] Feit, M., Fleck Jr, J. & Steiger, A. Solution of the Schr¨odinger equation by a spectral method.
Journal of Computational Physics 47, 412-433 (1982).
[12] Dirac, P. A. M. Lectures on quantum mechanics. Vol. 2 (Courier Corporation, 2001).
[13] Schwartz, M. D. Quantum field theory and the standard model. (Cambridge university press, 2014).
[14] Born, M. Statistical interpretation of quantum mechanics. Science 122, 675-679 (1955).
Notes: The lemmas, theorems, examples, and conclusions in the paper have all been verified by the
computational software Wolfram Mathematica, so the proof process of the theorem is not written in the
pap er. For lengthy equations, formatting errors may occur during the process of writing Mathematica
co de. It is necessary to split the equations into several parts and process them separately, and then
combine them to perform operations.
• PS1: What is the significance of non renowned scientists submitting outstanding scientific
contributions to renowned journals?
This is a brief edition of the groundbreaking paper ‘‘Dongfang Special Entangled Solution of Schr¨odinger
Hydrogen Equation” that subverts the exact solution of the hydrogen atom Schr¨odinger equation. The
briefing version has added new scientific ideas on the basis of the detailed version.
The pap er was submitted to Nature and Physical Review Letters resp ectively, and the results, like other
groundbreaking papers previously submitted to Chinese scientific journals such as “SCIENCE CHINA
Mathematics”, “SCIENCE CHINA Physics, Mechanics & Astronomy, SCPMA”, “Science Bulletin”,
“Chinese Physics C”, “Acta Physica Sinica”, “Acta Mathematica Sinica,Chinese Series”, etc., once again
revealed to those who pursue scientific truth the reality of the scientific community: famous journals have
long been controlled by ignorant, immoral, and despicable people, widely used to promote lies under the
banner of science to maximize personal interests, without promoting any groundbreaking discoveries that
10 X. D. Dongfang Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen
are irrelevant to their personal or group interests. The outstanding scientific contributions of unknown
scientists will only be stifled and plagiarized there. In the era of capital, many scientists, editors of
academic journals, and peer reviewers do not have a true scientific belief, and they have even lost their
humanity. They only pursue fame and fortune and resort to any means necessary.
Think about how many great discoveries have been humiliated, ravaged, and stifled by renowned
scientific journals! However, the proceeds of theft or plagiarism can only be fragmented works and
cannot be systematic scientific ideas. For nearly a century, great discoverers who remained silent have
b een almost invariably stifled or plagiarized by ignorant and unethical editors and peer reviewers of
renowned scientific journals, yet scientific theories have never made substantial progress as a result. For
example, the simple problem of the mathematical essence of quantum mechanics has yet to be solved by
the powerful scientific community that has seized power. This is the fundamental reason why scientific
theories have been stuck in a dirty quagmire for 100 years. Famous scientific journals influence the
direction of scientific development, leading to a plethora of lies in the physics community and continuing
to move in the wrong direction.
The Chinese theoretical physics professors and academicians I have been in contact with do not have
any pioneering research abilities, but they try their b est to steal others’ breakthrough research results. Do
international scientists often mock the Chinese theoretical physics community? In history, Chinese people
have not made any breakthrough contributions to physics theory that can be included in textbooks, but
there are too many Chinese people who constantly promote themselves as excellent theoretical physicists
throughout the year. And the breakthrough research achievements that will truly impact the future sci-
entific world are stifled. Now I have to give up my plan to write 100 breakthrough papers in mathematics
and physics. This is not only pleasing to the Chinese official theoretical physics community constructed
by extremely foolish people, because they can continue to enjoy the vanity of so-called theoretical physics
academicians and theoretical physicists; This is also pleasing to the theoretical physics community in
Europ e and America, because a great scientist who could have been on par with the great Newton in the
future - the true master of theoretical physics in China - was silently and completely eliminated by the
Chinese themselves.
People who pursue truth should now understand a simple truth: only when the global scientific com-
munity establishes a sound system, so that famous scientific journals are no longer controlled by ignorant,
immoral, and despicable people, can science usher in a new era of brilliance.
• PS2: What would happen if an unknown scientist submits groundbreaking scientific
contributions to Nature?
Cover Letter
Beijing time: June 11, 2024 21:52 (Tuesday)
Dear Editors,
I am submitting this letter to your journal, which foreshadows significant changes in natural science
theory and experimental testing due to the discovery of serious flaws in partial differential equation theory.
It looks forward to promoting the rapid development of natural science theory in the right direction.
As an unknown discoverer from China, submitting groundbreaking papers to any journal is an insult
to oneself. Whether in the past, present, or future, such great discoveries have been or are destined to
b e stifled. Nevertheless, in response to readers’ doubts, I still often submit papers introducing major
discoveries to world-renowned journals, perhaps this is the last time. The expectation of unexpected luck
is not a rational behavior, and it is meaningful to make more readers aware of the scientific reality that
individuals cannot change.
May all good luck come to you. Everything go es smoothly.
Thank you!
Best wishes,
X. D. Dongfang
†Receipt of Nature manuscript 2024-06-11880
Beijing time: June 11, 2024 21:53 (Tuesday)
*Please ensure you delete the link to your author home page in this e-mail if you wish to forward it to
your coauthors.*
Dear Professor Dongfang,
Mathematics & Nature Vol. 3 (2023) 11
Thank you for submitting your manuscript, ”Dongfang Special Schr¨odinger Wave Function for Hydro-
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†NATURE: Your submission to Nature
Beijing time: June 12, 2024 18:28 (Wednesday)
Dear Professor Dongfang,
12 X. D. Dongfang Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen
Thank you for submitting your manuscript to Nature, which we are regretfully unable to offer to
publish.
It is Nature’s policy to return a substantial proportion of manuscripts without sending them to referees,
so that they may be sent elsewhere without delay. Decisions of this kind are made by the editorial staff
when it appears that papers are unlikely to succeed in the competition for limited space.
In the present case, while your findings may well prove stimulating to others’ thinking about such
questions, we are unable to conclude that the work provides the sort of firm advance in general under-
standing that would warrant publication in Nature. We therefore feel that the paper would find a more
suitable outlet in a specialist journal.
We are sorry that we cannot respond more positively on this occasion but hope that you will rapidly
receive a more favourable response elsewhere.
Yours sincerely,
Manuscript Administration, Nature
This email has been sent through the Springer Nature Manuscript Tracking System NY-610A-SN&MTS
Confidentiality Statement: This e-mail is confidential and subject to copyright. Any unauthorised use
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authors/policies/confidentiality.html
Author’s inquiry letter
Beijing time: June 18, 2024 00:58 (Tuesday)
Dear Editor,
I don’t understand why there is no submission record for this groundbreaking paper manuscript in the
Nature system. Although the editor believes that they do not have professional knowledge and suggests
submitting the manuscript to a professional magazine, the record of submission should be kept in the
Nature system. Clicking on the link only displays information about previously rejected manuscripts.
Please explain. Thank you!
X. D. Dongang
†Nature’s automatic reply
Beijing time: June 18, 2024 00:58 (Tuesday)
Thank you for contacting Nature. This inbox is monitored on weekdays during UK and New York
office hours only. We will try to respond to your email with 48 hours. Thank you for your patience. Kind
regards, Nature Editorial Administration Team.
• PS3: What will be the outcome of submitting outstanding scientific contributions from
unknown scientists to Physical Review Letters?
Cover Letter
Dear Editor,
This paper titled “Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen” provides a
sp ecial case of a large number of missing solutions to the hydrogen atom Schr¨odinger equation that has
never appeared in history. It is hoped that it can quickly spread to the scientific community to promote
the revision and improvement of scientific theory.
The theory of partial differential equations has serious flaws, and the scientific theories described by
partial differential equations, especially the wave equation theory in quantum mechanics, are not entirely
correct. Natural science theory is facing a huge revolution.
There is an important experimental suggestion after equation (7). Expect the results of precise exper-
iments to confirm strict logical deductions rather than hypotheses that meet expectations.
If the journal does not accept this paper, please forward it to senior scientists for research on similar
issues and publication.
Thank you.
Sincerely,
X. D. Dongfang
Editorial Acknowledgment LT19318 Dongfang
Beijing time: June 26, 2024 at 01:26 (Wednesday)
Mathematics & Nature Vol. 3 (2023) 13
Re: LT19318
Dongfang special entangled Schrodinger wave function of hydrogen
by X. D. Dongfang
Dear Dr. Doongfang,
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PhySH Concepts:
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Title: Dongfang special entangled Schr\”{o}dinger wave function of
hydrogen
Collab oration:
1 Author(s):
X. D. Dongfang
Your manuscript LT19318 Dongfang
Beijing time: July 8th, 2024 23:17 (Monday)
Re: LT19318
Dongfang special entangled Schrodinger wave function of hydrogen
by X. D. Dongfang
Dear Dr. Dongfang,
Your manuscript has been considered. We regret to inform you that we have concluded that it is not
suitable for publication in Physical Review Letters.
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Latest findings
Dongfang Special Entangled Spherical Solution of Laplace Equation
The latest discovery of special entangled spherical harmonics indicates that traditional spherical harmonics are only a very narrow partial solution of spherical harmonic partial differential equations, and the theory of exact solutions for similar linear partial differential equations is therefore subject to disruptive effects. Here is a detailed introduction to the special entangled solution of the Laplace equation for the Dirichlet problem in a spherical region. Returning to the analytical form of the solution of spherical harmonic partial differential equations, it is first explained that in the special case where the magnetic quantum is zero, the spherical harmonic function partially degenerates into a Legendre function with respect to the angle θ, resulting in the existence of missing solutions in the Laplace equation. Then provide two special entangled solution sets for the Laplace equation with zero magnetic quantum number, including sine and cosine functions for angles θ and φ. The integral constant of the zero magnetic quantum number general solution in the special case formed by linear superposition of three types of special solution sets is not unique, indicating that the specific form of the exact solution of the Laplace equation cannot be determined. It can be seen that the conclusions of scientific theories described by partial differential equations are not absolutely reliable, and the theory of partial differential equations has serious flaws that urgently need to be improved.