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Article
.
Mathematics
Dongfang Special Entangled Spherical Harmonic Functions
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Abstract: The spherical harmonic partial differential equation is usually solved by vari-
able separation. When the magnetic quantum number m of the obtained Legendre spher-
ical harmonic function is 0, the Legendre spherical harmonic function degenerates into the
Legendre function of angle θ. This is a deduction that may seem rigorous in mathematics
but is actually not true. Replace the cosine function of angle θ in the Legendre function
with the product of the sine function of angle θ and the sine function or cosine function
of angle φ, and obtain two sp ecial binary trigonometric series called special entanglement
functions that satisfy the spherical harmonic partial differential equation. According to
the basic principles of differential equation, the linear combination of three special spheri-
cal harmonic functions is a local special general solution of the spherical harmonic partial
differential equation with a magnetic quantum number m of 0. However, the normaliza-
tion condition cannot determine the three undetermined coefficients and therefore cannot
determine the specific spherical harmonic function. The established physical conclusions
based on spherical harmonic functions, especially the mathematical principles of quan-
tum mechanics, need to supplement the definite solution conditions of spherical harmonic
partial differential equations and rewrite them after determining specific exact solutions.
Keywords: Direct product function, Entanglement function, Laplace equation, spher-
ical harmonic partial differential equation, spherical harmonic function, wave function,
probability density.
MSC(2020) Subject Classification: 35J05, 43A90, 33E10
Contents
1 Legendre direct product spherical harmonic function · · · · · · · · · · · · · · · · · · · 1
2 Sp ecial entangled spherical harmonic function of the first kind · · · · · · · · · · · · 5
3 Sp ecial entangled spherical harmonic function of the second kind · · · · · · · · · · 10
4 Entangled wave function and probability density diagram · · · · · · · · · · · · · · · 15
5 Blending special entangled spherical harmonic functions · · · · · · · · · · · · · · · · 17
1 Legendre direct product spherical harmonic function
Definition 1 Direct Product Function: In a generalized space with n coordinate parameters
q
1
, q
2
, · · · , q
n
, a multivariate function ψ (q
1
, q
2
, · · · , q
n
) can be decomposed into the product of
several functions represented only by partial coordinate parameters, and the intersection of the
coordinate parameter sets of these functions is empty. This multivariate function is called a direct
product function, and the direct product function in the form of
n
j=1
ψ (q
j
) is a complete direct
product function.
Citation: Dongfang, X. D. Dongfang Special Entangled Spherical Harmonic Functions. Mathematics & Nature 202302 (2023).
2 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
Definition 2 Entangled Function: In a generalized space with n coordinate parameters
q
1
, q
2
, · · · , q
n
, a multivariate function ψ (q
1
, q
2
, · · · , q
n
) cannot be decomposed into the product
of several functions represented only by partial coordinate parameters, and the intersection of
the co ordinate parameter sets of these functions is empty. This multivariate function is called an
entangled function.
By using the method of variable separation to solve the Laplace equation
[1-5]
, various Schr¨odinger
equations
[6-8]
, etc. in a spherical coordinate system, second-order partial differential equations
with respect to two angles θ and φ can be obtained, which are called spherical harmonic partial
differential equations,
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
+
1
sin
2
θ
∂
2
Y
∂φ
2
+ l (l + 1) Y = 0 (1.1)
Among them, the variable separation universal constant λ has been written as λ = l (l + 1). The
spherical harmonic partial differential equation
[9]
is consistent with the quantum mechanics angular
momentum square operator equation. Further use the method of variable separation to solve the
spherical harmonic partial differential equation, obtaining the general analytical solution of the
spherical harmonic partial differential equation, which includes bounded functions with rotational
symmetry and unbounded functions without specific rotational symmetry.
Lemma 1 The direct product analytical function
Y
m
l
(θ, φ) =
α
⟨m⟩
l
sin mφ + a
⟨m⟩
l
cos mφ
sin
m
θ
×
∞
n=0
n
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2n
θ
+
β
⟨m⟩
l
sin mφ + b
⟨m⟩
l
cos mφ
sin
m
θ
×
∞
n=0
n
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2n+1
θ
(1.2)
with four undetermined coefficients α
⟨m⟩
l
, a
⟨m⟩
l
, β
⟨m⟩
l
, and b
⟨m⟩
l
satisfies the spherical harmonic
partial differential equation
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
+
1
sin
2
θ
∂
2
Y
∂φ
2
+ l (l + 1) Y = 0
for any constant m and l.
The analytical part of the direct product analytic function (1.2) is the usual expression obtained
by solving the associated Legendre equation, so Lemma 1 does not require a proof to be written.
The analytical function that satisfies the spherical harmonic partial differential equation may
not necessarily be the expected solution. Scientific theories such as mathematics and physics often
develop according to expectations, which may lead to certain biases. Usually, it is agreed to take
integers m = 0, ±1, ±2, · · · , and l = 0, 1, 2, · · · , so that the spherical harmonic function satisfies
the natural period and bounded function conditions, and m is called the magnetic quantum
number and l is the orbital angular momentum quantum number. One term in the analytical
function (1.2) of this condition is interrupted as a polynomial, which is called the associated
Mathematics & Nature Vol. 3 (2023) 3
Legendre polynomial
[10]
, represented by P
m
l
(cos θ) ,
P
m
l
(cos θ) =
χ
⟨m⟩
l
sin
m
θ
∞
n
=0
n
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2n
θ
l−m=0, 2, 4, ···
η
⟨m⟩
l
sin
m
θ
∞
n=0
n
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2n+1
θ
l−m=1, 3, 5, ···
(1.3)
The undetermined coefficients χ
⟨m⟩
l
and η
⟨m⟩
l
are usually represented by normalization coefficients,
which are only operational but not necessarily logical inferences. The solution of the spherical
harmonic partial differential equation is limited to a bounded function with rotational symmetry
expressed by the associated Legendre function,
Y
⟨m⟩
l
(θ, φ) =
A
⟨m⟩
l
sin mφ + B
⟨m⟩
l
cos mφ
P
m
l
(cos θ) (1.4)
This is the real number expression for the Legendre direct product spherical harmonic functions
[?, 11, 13]
. Real number expressions have two undetermined coefficients A
m
and B
m
, and cannot
be directly represented by normalization coefficients. Quantum mechanics writes the spherical
harmonic function in the form of a complex function, providing a unique normalization coefficient,
which is intriguing. More related issues will be gradually introduced later to illustrate that
normalization coefficients are not always effective.
The expression of Legendre’s direct product spherical harmonic function is concise, but a
simplified expression may lead us to overlo ok certain important principles. Let’s return to the
universal analytical expression and use (1.3) to represent the Legendre direct product spherical
harmonic function.
Lemma 2 For integers m and non negative integers l that satisfy the condition m 6 l, the
Legendre direct product spherical harmonic function
Y
⟨m⟩
l
(θ, φ) =
α
⟨m⟩
l
sin mφ + a
⟨m⟩
l
cos mφ
sin
m
θ
×
∞
n=0
n
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2n
θ
l−m=0, 2, 4, ···
β
⟨m⟩
l
sin mφ + b
⟨m⟩
l
cos mφ
sin
m
θ
×
∞
n=0
n
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2n+1
θ
l−m=1, 3, 5, ···
(1.5)
containing two undetermined coefficients α
⟨m⟩
l
and a
⟨m⟩
l
or β
⟨m⟩
l
and b
⟨m⟩
l
, or written in polynomial
4 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
form
Y
⟨m⟩
l
(θ, φ) =
α
⟨m⟩
l
sin mφ + a
⟨m⟩
l
cos mφ
sin
m
θ
×
(l−m+2)/2
n=0
n
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2n
θ
l−m=0,2,4,···
β
⟨m⟩
l
sin mφ + b
⟨m⟩
l
cos mφ
sin
m
θ
×
(l−m+1)/2
n=0
n
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2n+1
θ
l−m=1,3,5,···
(1.6)
satisfies the spherical harmonic partial differential equation
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
+
1
sin
2
θ
∂
2
Y
∂φ
2
+ l (l + 1) Y = 0
Now let’s focus on an issue that has been overlooked. From a purely deductive persp ective,
if m = 0, the spherical harmonic function degenerates into a Legendre polynomial, known as a
Legendre special spherical harmonic function. The Legendre special spherical harmonic function
is independent of angle φ and is a cosine series of angle θ
[14]
, which is a univariate function. But
in reality, there is no integer m in the spherical harmonic partial differential equation. This leads
to a new problem: Is there any missing spherical harmonic function with angle φ in the solution
set of the spherical harmonic partial differential equation given by the variable separation method
with m = 0?
Problem 1 When the magnetic quantum number m = 0, the Legendre spherical harmonic
functions degenerate into the one-dimensional Legendre special spherical harmonic function with
the angle θ
Y
0
l
(θ, φ) =
a
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2n
θ
l=0, 2, 4, ···
b
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2n+1
θ
l=1, 3, 5, ···
(1.7)
and without regard to angle φ, or written in polynomial form
Y
⟨0⟩
l
(θ, φ) =
a
l
(l+2)/2
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2n
θ
l=0,2,4,···
b
l
(l+1)/2
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2n+1
θ
l=1,3,5,···
(1.8)
So, is there a binary special spherical harmonic function with angles θ and φ that satisfies the
Mathematics & Nature Vol. 3 (2023) 5
spherical harmonic partial differential equation,
1
sin θ
∂
∂θ
sin θ
∂Y
∂θ
+
1
sin
2
θ
∂
2
Y
∂φ
2
+ l (l + 1) Y = 0
corresponding to a magnetic quantum number m = 0?
Table 1.1 lists the specific forms of Legendre special unary spherical harmonic functions corre-
sponding to different integers for l in (1.7) and (1.8) when magnetic quantum number m = 0.
Table 1.1 Some Legendre special univariate spherical harmonic functions
Y
0
0
(θ, φ) = a
0
Y
0
1
(θ, φ) = b
1
cos θ
Y
0
2
(θ, φ) = a
2
1 − 3cos
2
θ
Y
0
3
(θ, φ) = b
3
cos θ −
5
3
cos
3
θ
Y
0
4
(θ, φ) = a
4
1 − 10cos
2
θ +
35
3
cos
4
θ
Y
0
5
(θ, φ) = b
5
cos θ −
14
3
cos
3
θ +
21
5
cos
5
θ
Y
0
6
(θ, φ) = a
6
1 − 21cos
2
θ + 63cos
4
θ −
231
5
cos
6
θ
Y
0
7
(θ, φ) = b
7
cos θ − 9cos
3
θ +
99
5
cos
5
θ −
429
35
cos
7
θ
Y
0
8
(θ, φ) = a
8
1 − 36cos
2
θ + 198cos
4
θ −
1716
5
cos
6
θ +
1287
7
cos
8
θ
Y
0
9
(θ, φ) = b
9
cos θ −
44
3
cos
3
θ +
286
5
cos
5
θ −
572
7
cos
7
θ +
2431
63
cos
9
θ
Y
0
10
(θ, φ) = a
10
1 − 55cos
2
θ +
1430
3
cos
4
θ − 1430cos
6
θ
+
12155
7
cos
8
θ −
46189
63
cos
10
θ
2 Special entangled spherical harmonic function of the first kind
The answer to question 1 is yes. When the magnetic quantum number m = 0, replace cos θ in
(1.2) and (1.7) with sin φ sin θ and cos φ sin θ, respectively, to obtain special binary entanglement
analytic functions X
⟨0⟩
l
(θ, φ) and D
⟨0⟩
l
(θ, φ) that satisfy the spherical harmonic partial differential
equation. Extract the spherically symmetric parts, which are resp ectively called the first kind
of sp ecial entangled spherical harmonic function X
⟨0⟩
l
(θ, φ) with zero magnetic quantum number
and the second kind of special entangled spherical harmonic function D
⟨0⟩
l
(θ, φ), which correspond
one-to-one with the Legendre special spherical harmonic function.
Theorem 1 For any constant l, the first kind of special entanglement analytic function
X
⟨0⟩
l
(θ, φ) =
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
(2.1)
6 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
with two undetermined co efficients c
l
and d
l
satisfies the spherical harmonic partial differential
equation
1
sin θ
∂
∂θ
sin θ
∂X
∂θ
+
1
sin
2
θ
∂
2
X
∂φ
2
+ l (l + 1) X = 0 (2.2)
with magnetic quantum number m = 0.
Proof: The spherical harmonic partial differential equation (2.2) has the following equivalent
form
∂
2
X
∂θ
2
+
cos θ
sin θ
∂X
∂θ
+
1
sin
2
θ
∂
2
X
∂φ
2
+ l (l + 1) X = 0 (2.3)
The first and second partial derivatives of X
0
l
(θ, φ) with respect to angles θ and φ are as follows
∂X
∂θ
=
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2nsin
2n
φsin
2n−1
θ cos θ
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) sin
2n+1
φsin
2n
θ cos θ
∂X
∂φ
=
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2nsin
2n−1
φsin
2n
θ cos φ
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) sin
2n
φsin
2n+1
θ cos φ
∂
2
X
∂θ
2
=
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
θ − 1
(sin θ sin φ)
2n
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
θ − 1
(sin θ sin φ)
2n+1
∂
2
X
∂φ
2
=
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
φ − 1
(sin θ sin φ)
p
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
φ − 1
(sin θ sin φ)
2n+1
(2.4)
Substitute (2.1) and (2.4) into the left side of (2.3) to obtain
∂
2
X
∂θ
2
+
cos θ
sin θ
∂X
∂θ
+
1
sin
2
θ
∂
2
X
∂φ
2
+ l (l + 1) X
=
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
θ − 1
(sin θ sin φ)
2n
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
θ − 1
(sin θ sin φ)
2n+1
+
cos θ
sin θ
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2nsin
2n
φsin
2n−1
θ cos θ
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) sin
2n+1
φsin
2n
θ cos θ
Mathematics & Nature Vol. 3 (2023) 7
+
1
sin
2
θ
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
φ − 1
(sin θ sin φ)
p
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
φ − 1
(sin θ sin φ)
2n+1
+ l (l + 1)
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
= c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
θ − 1
(sin θ sin φ)
2n
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2nsin
2n
φsin
2n−2
θcos
2
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
φ − 1
sin
2n−2
θsin
2n
φ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
l (l + 1) (sin φ sin θ)
2n
+ d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
θ − 1
(sin θ sin φ)
2n+1
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) sin
2n+1
φsin
2n−1
θcos
2
θ
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
φ − 1
sin
2n−1
θsin
2n+1
φ
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
l (l + 1) (sin φ sin θ)
2n+1
(2.5)
Thefore,
∂
2
X
∂θ
2
+
cos θ
sin θ
∂X
∂θ
+
1
sin
2
θ
∂
2
X
∂φ
2
+ l (l + 1) X
= c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n)
2
− 2n
sin
2n−2
θsin
2n
φ
−(2n)
2
(sin θ sin φ)
2n
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2nsin
2n
φsin
2n−2
θ
−2n(sin φ sin θ)
2n
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n (2n − 1) (sin θ sin φ)
2n−2
−(2n)
2
sin
2n−2
θsin
2n
φ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
l (l + 1) (sin φ sin θ)
2n
8 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
+ d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2
− (2n + 1)
sin
2n−1
θsin
2n+1
φ
−(2n + 1)
2
(sin θ sin φ)
2n+1
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) sin
2n−1
θsin
2n+1
φ
− (2n + 1) (sin θ sin φ)
2n+1
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) 2n(sin φ)
2n−1
−(2n + 1)
2
sin
2n−1
θsin
2n+1
φ
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
l (l + 1) (sin φ sin θ)
2n+1
= c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n (2n − 1) (sin θ sin φ)
2n−2
−
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n − l) (2n + l + 1) (sin φ sin θ)
2n
+ d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) 2n(sin φ)
2n−1
−
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n − l + 1) (2n + l + 2) (sin φ sin θ)
2n+1
(2.6)
The above calculation uses the identity equation
(2n)
2
+ 2n − l (l + 1) = (2n − l) (2n + l + 1)
(2n + 1)
2
+ (2n + 1) − l (l + 1) = (2n − l + 1) (2n + l + 2)
(2.7)
Merge similar items in (2.6) and use the relationship equation
n+1
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
=
(2n − l) (2n + l + 1)
2 (n + 1) (2n + 1)
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
n+1
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
=
(2n − l + 1) (2n + l + 2)
(2n + 3) 2 (n + 1)
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2.8)
Mathematics & Nature Vol. 3 (2023) 9
One obtains,
∂
2
X
∂θ
2
+
cos θ
sin θ
∂X
∂θ
+
1
sin
2
θ
∂
2
X
∂φ
2
+ l (l + 1) X
= c
l
∞
n=0
n+1
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2 (n + 1) (2n + 1)
−
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n − l) (2n + l + 1)
(sin φ sin θ)
2n
+d
l
∞
n=0
n+1
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 3) 2 (n + 1)
−
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n − l + 1) (2n + l + 2)
(sin φ sin θ)
2n+1
= 0
(2.9)
So Theorem 1 holds. End of proof.
If the unbounded function term in the first kind of entangled analytical solution of the spherical
harmonic equation is removed, the result is a bounded polynomial function, that is, the first kind
of special entangled spherical harmonic function.
Inference 1 For any non negative integer l, a special class of entangled spherical harmonic
functions
X
⟨0⟩
l
(θ, φ) =
c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
l=0, 2, 4, ···
d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
l=1, 3, 5, ···
(2.10)
containing only one undetermined coefficient c
l
or d
l
, or written as polynomial form
X
⟨0⟩
l
(θ, φ) =
c
l
(l+2)/2
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
l=0, 2, 4, ···
d
l
(l+1)/2
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
l=1, 3, 5, ···
(2.11)
satisfies the spherical harmonic partial differential equation
1
sin θ
∂
∂θ
sin θ
∂X
∂θ
+
1
sin
2
θ
∂
2
X
∂φ
2
+ l (l + 1) X = 0 (2.12)
with magnetic quantum number m = 0.
Polynomials (2.10) and (2.11) are called the first kind of special spherical harmonic functions.
Table 2.1 lists the specific forms of the first kind of special entangled spherical harmonic functions
corresponding to different integers of l in (2.10) and (2.11) when the magnetic quantum number
m = 0.
10 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
Table 2.1 Some special entangled spherical harmonic functions of the first kind
X
⟨0⟩
0
(θ, φ) = c
0
X
⟨0⟩
1
(θ, φ) = d
1
sin θ sin φ
X
⟨0⟩
2
(θ, φ) = c
2
1 − 3sin
2
θsin
2
φ
X
⟨0⟩
3
(θ, φ) = d
3
sin θ sin φ −
5
3
sin
3
θsin
3
φ
X
⟨0⟩
4
(θ, φ) = c
4
1 − 10sin
2
θsin
2
φ +
35
3
sin
4
θsin
4
φ
X
⟨0⟩
5
(θ, φ) = d
5
sin θ sin φ −
14
3
sin
3
θsin
3
φ +
21
5
sin
5
θsin
5
φ
X
⟨0⟩
6
(θ, φ) = c
6
1 − 21sin
2
θsin
2
φ + 63sin
4
θsin
4
φ −
231
5
sin
6
θsin
6
φ
X
⟨0⟩
7
(θ, φ) = d
7
sin θ sin φ − 9sin
3
θsin
3
φ +
99
5
sin
5
θsin
5
φ −
429
35
sin
7
θsin
7
φ
X
⟨0⟩
8
(θ, φ) = c
8
1 − 36sin
2
θsin
2
φ + 198sin
4
θsin
4
φ
−
1716
5
sin
6
θsin
6
φ +
1287
7
sin
8
θsin
8
φ
X
⟨0⟩
9
(θ, φ) = d
9
sin θ sin φ −
44
3
sin
3
θsin
3
φ +
286
5
sin
5
θsin
5
φ
−
572
7
sin
7
θsin
7
φ +
2431
63
sin
9
θsin
9
φ
X
⟨0⟩
10
(θ, φ) = c
10
1 − 55sin
2
θsin
2
φ +
1430
3
sin
4
θsin
4
φ − 1430sin
6
θsin
6
φ
+
12155
7
sin
8
θsin
8
φ −
46189
63
sin
10
θsin
10
φ
3 Special entangled spherical harmonic function of the second kind
Theorem 2 For any constant l, the first kind of special entanglement analytic function
D
⟨0⟩
l
(θ, φ) =
f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
(3.1)
with two undetermined coefficients f
l
and g
l
satisfies the spherical harmonic partial differential
equation
1
sin θ
∂
∂θ
sin θ
∂D
∂θ
+
1
sin
2
θ
∂
2
D
∂φ
2
+ l (l + 1) D = 0 (3.2)
with magnetic quantum number m = 0.
Proof: The spherical harmonic partial differential equation (3.2) has the following equivalent
form
∂
2
D
∂θ
2
+
cos θ
sin θ
∂D
∂θ
+
1
sin
2
θ
∂
2
D
∂φ
2
+ l (l + 1) D = 0 (3.3)
Mathematics & Nature Vol. 3 (2023) 11
The first and second partial derivatives of X
0
l
(θ, φ) with respect to angles θ and φ are as follows
∂D
∂θ
=
f
l
∞
n=0
n
k=1
(2k−l−2) (2k+l−1)
2k (2k−1)
2ncos
2n
φsin
2n−1
θ cos θ
+g
l
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n+1) cos
2n+1
φsin
2n
θ cos θ
∂D
∂φ
=
f
l
∞
n=0
n
k=1
(2k−l−2) (2k+l−1)
2k (2k−1)
2n
−cos
2n−1
φsin
2n
θ sin φ
+g
l
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n+1)
−cos
2n
φsin
2n+1
θ sin φ
∂
2
D
∂θ
2
=
f
l
∞
n=0
n
k=1
(2k−l−2) (2k+l−1)
2k (2k−1)
2n
(2n−1) cot
2
θ−1
cos
2n
φsin
2n
θ
+g
l
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n+1)
2ncot
2
θ−1
cos
2n+1
φsin
2n+1
θ
∂
2
D
∂φ
2
=
f
l
∞
n=0
n
k=1
(2k−l−2) (2k+l−1)
2k (2k−1)
2n
(2n−1) tan
2
φ−1
cos
2n
φsin
2n
θ
+g
l
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n+1)
2ntan
2
φ−1
cos
2n+1
φsin
2n+1
θ
(3.4)
Substitute (3.1) and (3.4) into the left side of (3.3) to obtain
∂
2
D
∂θ
2
+
cos θ
sin θ
∂D
∂θ
+
1
sin
2
θ
∂
2
D
∂φ
2
+ l (l + 1) D
=
f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
θ − 1
cos
2n
φsin
2n
θ
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
θ − 1
cos
2n+1
φsin
2n+1
θ
+
cos θ
sin θ
f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2ncos
2n
φsin
2n−1
θ cos θ
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) cos
2n+1
φsin
2n
θ cos θ
+
1
sin
2
θ
f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) tan
2
φ − 1
cos
2n
φsin
2n
θ
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ntan
2
φ − 1
cos
2n+1
φsin
2n+1
θ
+ l (l + 1)
f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
12 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
= f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) cot
2
θ − 1
cos
2n
φsin
2n
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2ncos
2n
φsin
2n−2
θcos
2
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n
(2n − 1) tan
2
φ − 1
cos
2n
φsin
2n−2
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
l (l + 1) (cos φ sin θ)
2n
+ g
l
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ncot
2
θ − 1
cos
2n+1
φsin
2n+1
θ
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) cos
2n+1
φsin
2n−1
θcos
2
θ
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1)
2ntan
2
φ − 1
cos
2n+1
φsin
2n−1
θ
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
l (l + 1) (cos φ sin θ)
2n+1
(3.5)
Thefore,
∂
2
D
∂θ
2
+
cos θ
sin θ
∂D
∂θ
+
1
sin
2
θ
∂
2
D
∂φ
2
+ l (l + 1) D
= f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n)
2
− 2n
cos
2n
φsin
2n−2
θ
−(2n)
2
cos
2n
φsin
2n
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2ncos
2n
φsin
2n−2
θ − 2ncos
2n
φsin
2n
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n)
2
− 2n
cos
2n−2
φsin
2n−2
θ
−(2n)
2
cos
2n
φsin
2n−2
θ
+
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
l (l + 1) (cos φ sin θ)
2n
+ g
l
+
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n+1)
2
−(2n+1)
cos
2n+1
φsin
2n−1
θ
−(2n+1)
2
cos
2n+1
φsin
2n+1
θ
+
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n + 1) cos
2n+1
φsin
2n−1
θ
−(2n+1) cos
2n+1
φsin
2n+1
θ
+
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
(2n+1)
2
−(2n+1)
cos
2n−1
φsin
2n−1
θ
−(2n+1)
2
cos
2n+1
φsin
2n−1
θ
+
∞
n=0
n
k=1
(2k−l−1) (2k+l)
(2k+1) 2k
l (l+1) (cos φ sin θ)
2n+1
Mathematics & Nature Vol. 3 (2023) 13
= f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
2n (2n − 1) cos
2n−2
φsin
2n−2
θ
−
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n − l) (2n + l + 1) (cos φ sin θ)
2n
+ g
l
+
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 1) 2ncos
2n−1
φsin
2n−1
θ
−
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n − l + 1) (2n + l + 2) (cos φ sin θ)
2n+1
(3.6)
The above calculation uses the identity equation
(2n)
2
+ 2n − l (l + 1) = (2n − l) (2n + l + 1)
(2n + 1)
2
+ (2n + 1) − l (l + 1) = (2n − l + 1) (2n + l + 2)
(3.7)
Merge similar items in (3.6) and use the relationship equation
n+1
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
=
(2n − l) (2n + l + 1)
2 (n + 1) (2n + 1)
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
n+1
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
=
(2n − l + 1) (2n + l + 2)
(2n + 3) 2 (n + 1)
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(3.8)
One obtains,
∂
2
D
∂θ
2
+
cos θ
sin θ
∂D
∂θ
+
1
sin
2
θ
∂
2
D
∂φ
2
+ l (l + 1) D
= f
l
∞
n=0
n+1
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n + 2) (2n + 1)
−
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(2n − l) (2n + l + 1)
(cos φ sin θ)
2n
+g
l
+
∞
n=0
n+1
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n + 3) (2n + 2)
−
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(2n − l + 1) (2n + l + 2)
(cos φ sin θ)
2n+1
= 0
(3.9)
So Theorem 1 holds. End of proof.
If the unbounded function term in the second kind of entangled analytical solution of the
spherical harmonic equation is removed, the result is a bounded polynomial function, that is, the
second kind of special entangled spherical harmonic function.
Inference 2 For any non negative integer l, a special class of entangled spherical harmonic
14 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
functions
D
0
l
(θ, φ) =
f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
l=0, 2, 4, ···
g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
l=1, 3, 5, ···
(3.10)
containing only one undetermined coefficient f
l
or g
l
, or written as polynomial form
D
⟨0⟩
l
(θ, φ) =
f
l
(l+2)/2
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
l=0, 2, 4, ···
g
l
(l+1)/2
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
l=1, 3, 5, ···
(3.11)
satisfies the spherical harmonic partial differential equation
1
sin θ
∂
∂θ
sin θ
∂D
∂θ
+
1
sin
2
θ
∂
2
D
∂φ
2
+ l (l + 1) D = 0 (3.12)
with magnetic quantum number m = 0.
Polynomials (3.10) and (3.11) are called the second kind of special spherical harmonic functions.
Table 3.1 lists the specific forms of the second kind of special entangled spherical harmonic
functions corresponding to different integers of l in (3.11) when the magnetic quantum numb er
m = 0.
Table 3.1 Some special entangled spherical harmonic functions of the second kind
D
⟨0⟩
0
(θ, φ) = f
0
D
⟨0⟩
1
(θ, φ) = g
1
cos φ sin θ
D
⟨0⟩
2
(θ, φ) = f
2
1 − 3cos
2
φsin
2
θ
D
⟨0⟩
3
(θ, φ) = g
3
cos φ sin θ −
5
3
cos
3
φsin
3
θ
D
⟨0⟩
4
(θ, φ) = f
4
1 − 10cos
2
φsin
2
θ +
35
3
cos
4
φsin
4
θ
D
⟨0⟩
5
(θ, φ) = g
5
cos φ sin θ −
14
3
sin
3
θsin
3
φ +
21
5
cos
5
φsin
5
θ
D
⟨0⟩
6
(θ, φ) = f
6
1 − 21cos
2
φsin
2
θ + 63cos
4
φsin
4
θ −
231
5
cos
6
φsin
6
θ
D
⟨0⟩
7
(θ, φ) = g
7
cos φ sin θ − 9cos
3
φsin
3
θ +
99
5
cos
5
φsin
5
θ −
429
35
cos
7
φsin
7
θ
D
⟨0⟩
8
(θ, φ) = f
8
1 − 36cos
2
φsin
2
θ + 198cos
4
φsin
4
θ
−
1716
5
cos
6
φsin
6
θ +
1287
7
cos
8
φsin
8
θ
D
⟨0⟩
9
(θ, φ) = g
9
cos φ sin θ −
44
3
cos
3
φsin
3
θ +
286
5
cos
5
φsin
5
θ
−
572
7
cos
7
φsin
7
θ +
2431
63
cos
9
φsin
9
θ
Mathematics & Nature Vol. 3 (2023) 15
D
⟨0⟩
10
(θ, φ) = f
10
1 − 55cos
2
φsin
2
θ +
1430
3
cos
4
φsin
4
θ − 1430cos
6
φsin
6
θ
+
12155
7
cos
8
φsin
8
θ −
46189
63
cos
10
φsin
10
θ
4 Entangled wave function and probability density diagram
The spherical harmonic function is the solution of the spherical harmonic partial differential
equation, which is the arbitrary specified radius solution of the Laplace equation and the quan-
tum mechanics angular momentum square operator equation in the spherical coordinate system.
Usually, a bounded spherical harmonic function is taken, and the accuracy of the results is verified
through experimental observations. Some steady-state physical quantities represented by spheri-
cal harmonics can be accurately experimentally plotted for their spatial distribution, while others
represented by spherical harmonics cannot be experimentally plotted for their spatial distribu-
tion. The three-dimensional diagram of spherical harmonic functions can be used to theoretically
and intuitively analyze the credibility of describing physical phenomena using spherical harmonic
partial differential equations.
The Laplace equation is used to describe electrostatic fields, static magnetic fields, stable tem-
perature fields, and fluid fields. The spherical harmonic function is convenient for describing the
distribution of physical quantities such as electrostatic fields and static magnetic fields. This
type of application is theoretically perfect, so is there strict consistency between the actual dis-
tribution map of the electromagnetic field measured experimentally and the function map of the
spherical harmonic function? Here, we can draw a three-dimensional diagram of the Legendre
special spherical harmonic function, the first and second types of special entangled spherical
harmonic functions, in order to have a clear and intuitive understanding of the solution to the
Dirichlet problem of the Laplace equation. In the past, the solution to the spherical harmonic
partial differential equation with magnetic quantum numbers was limited to the Legendre special
spherical harmonic function. The first and second types of special entangled spherical harmonic
functions discovered now allow us to understand that different types of solutions to spherical
harmonic partial differential equations are not consistent. The established conclusion of using
spherical harmonic functions to describe the angular distribution of various stable fields is the-
oretically challenged. We hope that experimental physicists can provide clear conclusions in a
timely manner to indicate the direction of theory.
Now focus on the difference b etween the wave function and probability density definition in
quantum mechanics. The wave equation satisfied by the angular wave function is exactly the
spherical harmonic partial differential equation. The angular wave function with magnetic quan-
tum number m = 0 is either a real function or an imaginary function. Quantum mechanics
defines the square of the wave function modulus as the probability density of a particle appearing
in space, so the square of the angular wave function modulus means the probability density of a
particle app earing in the angular direction. Because experiments cannot observe the probability
density of particles appearing in space or determining their orientation, the description of quan-
tum mechanics largely belongs to the realm of consciousness, as evidenced by the inconsistency
between the distribution space of any wave function and the square distribution space of the
corresponding wave function modulus representing probability density.
Referring to Table 4.1, when the orbital angular momentum quantum number l > 2, comparing
the three-dimensional diagram of the modulus of the special spherical harmonic function with
m = 0and the three-dimensional diagram of the square of the modulus, the two are clearly
inconsistent. The early definition of particle probability density using the square of the wave
function modulus was only proposed to eliminate imaginary numbers, but it did not take into
account that the fundamental change in the properties of the function had occurred from the
16 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
wave function modulus to the square of the wave function modulus.
Mathematics can freely define functions and assign them specific meanings. But physics uses
functions to describe natural laws, and functions cannot be defined arbitrarily. There is no logical
basis for defining the probability density function ρ (θ, φ) as the square of the wave function ψ (θ, φ).
This can be proven by the method of proof to the contrary. If the square of the wave function
modulus is defined as the probability density because it has the same extreme point as the wave
function modulus, then because any even power of the wave function can ensure that the zero and
extreme points of the wave function are consistent, it cannot be ruled out to define a probability
function for any positive integer n
ρ (θ, φ) = [ψ (θ, φ)]
2n
But in reality, the specific value of n cannot be determined, and the probability density func-
tion is uncertain. Therefore, the definition of probability density function is not an inevitable
logical inference. The excessive abstraction of quantum mechanics theory in describing implicit
fuzzy logic problems should also be perfectly solved from a mathematical perspective. Admit-
tedly, discussing special spherical harmonic functions does not require addressing this issue at the
moment.
The theory describ ed by Legendre’s special spherical harmonic function in the past is about
to be completely changed due to the discovery of entangled spherical harmonic functions. Even
in the special case of magnetic quantum number m = 0, for any orbital angular momentum
quantum number LL, there are three special angular entanglement functions that can be used to
describe the solutions of the Laplace equation or wave equation. The three-dimensional diagram
shows that the distribution of physical quantities corresponding to the probability density of
quantum mechanics for the three special angular wave functions is completely different, while
the solutions of Schr¨odinger equation and Klein Gordon equation depend on the solutions of
spherical harmonic partial differential equations. How can quantum mechanics embrace these
new mathematical challenges and more undiscovered ones?
Table 4.1 A three-dimensional diagram of some special spherical harmonic functions and their
squared modulus
l
Y
⟨0⟩
l
Y
⟨0⟩
l
2
X
⟨0⟩
l
X
⟨0⟩
l
2
D
⟨0⟩
l
D
⟨0⟩
l
2
0
1
2
3
Mathematics & Nature Vol. 3 (2023) 17
l Y
⟨0⟩
l
Y
⟨0⟩
l
2
X
⟨0⟩
l
X
⟨0⟩
l
2
D
⟨0⟩
l
D
⟨0⟩
l
2
4
5
6
7
8
9
10
5 Blending special entangled spherical harmonic functions
The Legendre special bounded spherical harmonic function from a single angle, the first kind of
special bounded entangled spherical harmonic function, and the second kind of special bounded
entangled spherical harmonic function all have only one undetermined coefficient. Normaliza-
tion conditions can be used to obtain the so-called normalization coefficient for the coefficients.
But the previous list of special b ounded spherical harmonic functions and the drawing of special
bounded spherical harmonic functions did not provide the so-called normalization coefficients.
There are two considerations for this. Firstly, for a spherical harmonic function with only one
undetermined coefficient, the function graph drawn by arbitrarily assigning undetermined coeffi-
cients is a completely similar graph, without the need for normalization coefficients; Secondly, for
any principal quantum l, there are three different special spherical harmonic functions with m = 0
that satisfy the spherical harmonic partial differential equation. Using normalization conditions
to obtain the coefficients of any special spherical harmonic function has neither mathematical nor
physical significance.
The basic principle of the solution of ordinary differential equations is that the linear combina-
tion of all solutions that satisfy the same ordinary differential equation is the general solution of
this ordinary differential equation. This principle also holds true when applied to partial differ-
ential equations. However, the solutions of partial differential equations are relatively complex,
and different solution methods give different solutions. Usually, the variable separation method is
used to construct solutions for some linear partial differential equations, which belong to special
18 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
general solutions that conform to the construction characteristics. The Legendre spherical har-
monic function is a bounded function obtained by constructing solutions to spherical harmonic
partial differential equations using the method of separating variables. Since two special types
of entangled spherical harmonic functions can be constructed to satisfy the spherical harmonic
partial differential equation for the case of magnetic quantum number m = 0 , the question arises
whether there exists a general entangled spherical harmonic function that satisfies the spherical
harmonic partial differential equation. How many undiscovered solutions satisfy the same def-
inite solution conditions for the Laplace equation of the same physical model. The answers to
these questions will have more impact on the established conclusions of physics and also have a
positive impact on mathematical theory. It can now be clarified that in the construction method
of known partial differential equation solutions, the linear combination of all solution sets of the
same partial differential equation is the local general solution of this linear partial differential
equation.
The linear combination of the Legendre special unary analytic function with magnetic quan-
tum number m = 0, the first kind of special entanglement analytic function, and the second kind
of special entanglement analytic function is the local general solution of the spherical harmonic
partial differential equation. The Legendre special bounded spherical harmonic function with
magnetic quantum number m = 0, the linear combination of the first kind of special bounded en-
tangled spherical harmonic function and the second kind of special bounded entangled spherical
harmonic function form a special local universal spherical harmonic function. From any per-
spective, physical and mathematical conclusions based on spherical harmonic partial differential
equations need to be rewritten, and today’s scientific theories cannot absolutely dominate the
future scientific world, and only the correct parts can sustain development.
Theorem 3 The special mixed entanglement analytic function
Ω
⟨0⟩
l
(θ, φ) =
a
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2n
θ
+c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
+f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
+b
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2n+1
θ
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
(5.1)
with 6 undetermined coefficients a
l
b
l
c
l
d
l
f
l
, and g
l
for any constant l and magnetic quantum
number m = 0 satisfies the spherical harmonic partial differential equation
1
sin θ
∂
∂θ
sin θ
∂Ω
∂θ
+
1
sin
2
θ
∂
2
Ω
∂φ
2
+ l (l + 1) Ω = 0 (5.2)
Theorem 4 For any non negative integer l and magnetic quantum number m = 0, a special
Mathematics & Nature Vol. 3 (2023) 19
mixed bounded angled spherical harmonic function
Υ
⟨0⟩
l
(θ, φ) =
a
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2n
θ
+c
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
+f
l
∞
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
l=0, 2, 4, ···
b
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2n+1
θ
+d
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
+g
l
∞
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
l=1, 3, 5, ···
(5.3)
with three undetermined coefficients a
l
c
l
, and f
l
(or b
l
d
l
, and g
l
), or written in polynomial form
Υ
⟨0⟩
l
(θ, φ) =
a
l
(l+2)/2
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2n
θ
+c
l
(l+2)/2
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2n
+f
l
(l+2)/2
n=0
n
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(cos φ sin θ)
2n
l=0, 2, 4, ···
b
l
(l+1)/2
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
cos
2n+1
θ
+d
l
(l+1)/2
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2n+1
+g
l
(l+1)/2
n=0
n
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(cos φ sin θ)
2n+1
l=1, 3, 5, ···
(5.4)
satisfies the spherical harmonic partial differential equation
1
sin θ
∂
∂θ
sin θ
∂Υ
∂θ
+
1
sin
2
θ
∂
2
Υ
∂φ
2
+ l (l + 1) Υ = 0 (5.5)
Theorem 3 is a summary of Lemma 1, Theorem 1, and Theorem 2, while Theorem 4 is a
synthesis of Problem 1, Corollary 1, and Corollary 2, and does not require re proof.
The special mixed bounded entangled spherical harmonic function has three undetermined coef-
ficients, and the normalization condition alone cannot determine these undetermined coefficients,
20 X. D. Dongfang Dongfang Special Entangled Spherical Harmonic Functions
that is, the normalization condition cannot determine the solution of the spherical harmonic par-
tial differential equation. The normalization coefficient in the past was a theoretical bias. Two
more definite solution conditions are needed to determine the special mixed bounded entangled
spherical harmonic function. Therefore, the key to future research on the solutions of partial
differential equations such as quantum mechanics wave equations and Laplace equations is to
provide at least two new definite solution conditions for spherical partial differential equations.
Table 5.1 Some special mixed bounded entangled spherical harmonic functions
Υ
⟨0⟩
0
(θ, φ) = a
0
+ c
0
+ f
0
Υ
⟨0⟩
1
(θ, φ) = b
1
cos θ + d
1
sin θ sin φ + g
1
cos φ sin θ
Υ
⟨0⟩
2
(θ, φ) = a
2
1 − 3cos
2
θ
+ c
2
1 − 3sin
2
θsin
2
φ
+ f
2
1 − 3cos
2
φsin
2
θ
Υ
⟨0⟩
3
(θ, φ) =
b
3
cos θ −
5
3
cos
3
θ
+ d
3
sin θ sin φ −
5
3
sin
3
θsin
3
φ
+g
3
cos φ sin θ −
5
3
cos
3
φsin
3
θ
Υ
⟨0⟩
4
(θ, φ) =
a
4
1 − 10cos
2
θ +
35
3
cos
4
θ
+ c
4
1 − 10sin
2
θsin
2
φ +
35
3
sin
4
θsin
4
φ
+f
4
1 − 10cos
2
φsin
2
θ +
35
3
cos
4
φsin
4
θ
Υ
⟨0⟩
5
(θ, φ) =
b
5
cos θ −
14
3
cos
3
θ +
21
5
cos
5
θ
+d
5
sin θ sin φ −
14
3
sin
3
θsin
3
φ +
21
5
sin
5
θsin
5
φ
+g
5
cos φ sin θ −
14
3
sin
3
θsin
3
φ +
21
5
cos
5
φsin
5
θ
Υ
⟨0⟩
6
(θ, φ) =
a
6
1 − 21cos
2
θ + 63cos
4
θ −
231
5
cos
6
θ
+c
6
1 − 21sin
2
θsin
2
φ + 63sin
4
θsin
4
φ −
231
5
sin
6
θsin
6
φ
+f
6
1 − 21cos
2
φsin
2
θ + 63cos
4
φsin
4
θ −
231
5
cos
6
φsin
6
θ
Υ
⟨0⟩
7
(θ, φ) =
b
7
cos θ − 9cos
3
θ +
99
5
cos
5
θ −
429
35
cos
7
θ
+d
7
sin θ sin φ − 9sin
3
θsin
3
φ +
99
5
sin
5
θsin
5
φ −
429
35
sin
7
θsin
7
φ
+g
7
cos φ sin θ − 9cos
3
φsin
3
θ +
99
5
cos
5
φsin
5
θ −
429
35
cos
7
φsin
7
θ
Υ
⟨0⟩
8
(θ, φ) =
a
8
1 − 36cos
2
θ + 198cos
4
θ −
1716
5
cos
6
θ +
1287
7
cos
8
θ
+c
8
1 − 36sin
2
θsin
2
φ + 198sin
4
θsin
4
φ
−
1716
5
sin
6
θsin
6
φ +
1287
7
sin
8
θsin
8
φ
+f
8
1 − 36cos
2
φsin
2
θ + 198cos
4
φsin
4
θ
−
1716
5
cos
6
φsin
6
θ +
1287
7
cos
8
φsin
8
θ
Mathematics & Nature Vol. 3 (2023) 21
Υ
⟨0⟩
9
(θ, φ) =
b
9
cos θ −
44
3
cos
3
θ +
286
5
cos
5
θ −
572
7
cos
7
θ +
2431
63
cos
9
θ
+d
9
sin θ sin φ −
44
3
sin
3
θsin
3
φ +
286
5
sin
5
θsin
5
φ
−
572
7
sin
7
θsin
7
φ +
2431
63
sin
9
θsin
9
φ
+g
9
cos φ sin θ −
44
3
cos
3
φsin
3
θ +
286
5
cos
5
φsin
5
θ
−
572
7
cos
7
φsin
7
θ +
2431
63
cos
9
φsin
9
θ
Υ
⟨0⟩
10
(θ, φ) =
a
10
1 − 55cos
2
θ +
1430
3
cos
4
θ − 1430cos
6
θ
+
12155
7
cos
8
θ −
46189
63
cos
10
θ
+c
10
1 − 55sin
2
θsin
2
φ +
1430
3
sin
4
θsin
4
φ − 1430sin
6
θsin
6
φ
+
12155
7
sin
8
θsin
8
φ −
46189
63
sin
10
θsin
10
φ
+f
10
1 − 55cos
2
φsin
2
θ +
1430
3
cos
4
φsin
4
θ − 1430cos
6
φsin
6
θ
+
12155
7
cos
8
φsin
8
θ −
46189
63
cos
10
φsin
10
θ
How abstract conclusions describing natural phenomena based on imperfect mathematical de-
ductions have passed experimental tests is a meaningful topic. For the special case of magnetic
quantum number m = 0, the solution of the spherical harmonic partial differential equation, also
known as the angular momentum square operator equation, is a special mixed spherical harmonic
function. The special mixed spherical harmonic function has three undetermined coefficients,
which can not be determined only by the normalization condition. Describing the distribution of
various fields using the Laplace equation and the probability of microscopic particles appearing
in space using the Schr¨odinger equation are both based on treating a single type of solution set
constructed using the variable separation method as a general solution and drawing one-sided
conclusions that cannot b e truly confirmed by experimental observations. The theory that hides
irreconcilable contradictions often claims to be widely experimentally confirmed, which is one
of the important forms of public opinion orientation. The analysis of spherical harmonic func-
tions
[15-17]
in physics and applied sciences needs to be reconstructed based on further improved
spherical harmonic function theory
[18-22]
.
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Notes: All new theorems, proof processes for theorems, and listed examples in the article have passed
the calculation verification requirements of the scientific computing software Wolfram Mathematica. If
you are unable to verify using scientific software, the possible reason may be that the input code does not
comply with the coding rules of the software. Please consult a professional or seek help on the internet
to obtain the correct calculation code.
PS: This groundbreaking mathematical paper has been reviewed by nearly ten journals for about a year
and has not been published, with the longest review time being 50 days. It is unknown whether it has
b een plagiarized and rewritten like pioneering physics papers in the past. In today’s world, mathematics
and physics are monopolized by ignorant and immoral despicable people, and great discoveries have
nowhere to be published. Free publication on the internet often suffers from their attacks and slander.
1. Dongfang, X. D. On the relativity of the speed of light. Mathematics & Nature 1, 202101 (2021).
2. Dongfang, X. D. The Morbid Equation of Quantum Numbers. Mathematics & Nature 1, 202102
(2021).
3. Dongfang, X. D. Relativistic Equation Failure for LIGO Signals. Mathematics & Nature 1, 202103
(2021).
4. Dongfang, X. D. Dongfang Modified Equations of Molecular Dynamics . Mathematics & Nature 1,
202104 (2021).
5. Dongfang, X. D. Dongfang Angular Motion Law and Operator Equations. Mathematics & Nature 1,
202105 (2021).
6. Dongfang, X. D. Dongfang Com Quantum Equations of LIGO Signal. Mathematics & Nature 1,
202106 (2021).
7. Dongfang, X. D. Com Quantum Proof of LIGO Binary Mergers Failure. Mathematics & Nature 1,
202107 (2021).
8. Dongfang, X. D. Dongfang Modified Equations of Electromagnetic Wave. Mathematics & Nature 1,
202108 (2021).
9. Dongfang, X. D. Nuclear Force Constants Mapped by Yukawa Potential. Mathematics & Nature 1,
202109 (2021).
Mathematics & Nature Vol. 3 (2023) 23
10. Dongfang, X. D. The End of Yukawa Meson Theory of Nuclear Forces. Mathematics & Na-
ture 1, 202110 (2021).
11. Dongfang, X. D. The End of Klein-Gordon Equation for Coulomb Field. Mathematics & Nature 2,
202201 (2022).
12. Dongfang, X. D. The End of Teratogenic Simplified Dirac Hydrogen Equations. Mathematics &
Nature 2, 202202 (2022).
13. Dongfang, X. D. Dongfang Solution of Induced Second Order Dirac Equations. Mathematics &
Nature 2, 202203 (2022).
14. Dongfang, X. D. The End of Isomeric Second Order Dirac Hydrogen Equations. Mathematics &
Nature 2, 202204 (2022).
15. Dongfang, X. D. The End of True Second Order Dirac Hydrogen Equation. Mathematics & Nature
2, 202205 (2022).
16. Dongfang, X. D. Dongfang Challenge Solution of Dirac Hydrogen Equation. Mathematics & Nature
2, 202206 (2022).
17. Dongfang, X. D. Neutron State Solution of Dongfang Modified Dirac Equation. Mathematics &
Nature 2, 202207 (2022).
18. Dongfang, X. D. Ground State Solution of Dongfang Modified Dirac Equation. Mathematics &
Nature 2, 202208 (2022).
19. Dongfang, X. D. The End of Dirac Hydrogen Equation in One Dimension. Mathematics & Nature
2, 202209 (2022).
20. Dongfang, X. D. Multiple Morbid Mathematics of Dirac Electron Theory. Mathematics & Nature2,
202210 (2022).
21. Dongfang, X. D. Dongfang Brief General Solutions to Congruence Equations. Mathematics & Nature
3, 202301 (2023).
Latest findings
Dongfang Special Entangled Solution of Schrödinger Hydrogen Equation
The textbook solutions of Schrödinger equations are usually represented as the combination of multiple special function symbols, resulting in a large number of missing exact solutions not being discovered. Taking the Schrödinger wave function of a hydrogen atom as an example, regressing to the analytical expression that can be extended and programmed for testing, two types of special entangled wave functions that have not been described by established theories have been discovered. Specifically, when the magnetic quantum number is 0, the traditional Schrödinger wave function degenerates 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 Schrödinger equation for hydrogen atoms has two types of special 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 functions that satisfy the Schrödinger equation and even affect the energy eigenvalue formula. The three-dimensional function image of the special entangled solution of the hydrogen atom Schrödinger 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 modulus as the probability density function lacks causality, and it is an urgent problem to derive the probability density function according to the basic principle.