MATHEMATICS & NATURE
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Article
.
Physics
Dongfang Angular Motion Law and Operator Equations
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Quantum mechanics based on Planck hypothesis and statistical interpretation of wave function
has achieved great success in describing the discrete law of micro motion. However, the idea of
quantum mechanics has not been successfully used to describe the discrete law of macro motion,
and the causality implied in the Planck hypothesis and the application scope of the basic principles
of quantum mechanics have not been clarified. Here, I first introduce the angular motion law and
its application, which seems to be of no special significance as a supplement to the very perfect
classical mechanics, but plays an irreplaceable role in testing whether the core mathematical program
of quantum mechanics of the operator evolution wave equation meets the unitary principle. Then, the
op erator evolution wave equations corresponding to the angular motion law are discussed, and the
necessity of generalized optimization of differential equations are illustrated by the form of ordinary
differential equations. Finally, the real wave equation which is superior to the Schr¨odinger equation
in physical meaning but not necessarily the ultimate answer is briefly introduced. The implicit
conclusion is that Hamiltonian can not be the only inevitable choice of constructing wave equation
in quantum mechanics, and there is no causal relationship between operator evolution wave equation
and quantized energy in bound state system, which indicates that whether the essence of quantum
mechanics can be completely revealed is the key step to unify macro and micro quantized theory.
Keywords: Centrifugal force equation; Angular motion law; Operator principle; Wave equations of
multiwave functions; Optimal differential equation; Existence and uniqueness of solution.
PACS number(s): 02.30.Jr—Partial differential equations; 03.65.Ge—Solutions of wave equations:
b ound states; 03.65.Pm—Relativistic wave equations
1 Introduction
From Planck’s hypothesis
[1, 2]
to Einstein’s pho-
ton theory
[3]
, to Bohr’s hydrogen atom theory
[4]
, to
Schr¨odinger’s equation
[5]
and Dirac’s equation
[6]
, quan-
tum mechanics
[7-11]
has achieved great success in ex-
plaining the laws of blackbody radiation
[12]
, photoelec-
tric effect
[13]
and hydrogen atom spectrum
[14]
, and has
also constantly made new breakthroughs in the practice
and applications
[15-19]
. Many problems of quantum me-
chanics have been clearly concluded.
However, the essence of quantum mechanics has not
been clarified. This fundamental factor has led to the
neglect of some fundamental problems which play a de-
cisive role in quantum mechanics. For example, can
or how to use basic principles to prove Planck’s quan-
tized energy hypothesis
[20]
? How to prove Bohr’s
[21]
or
Sommerfeld’s
[22, 23]
quantization conditions? How to de-
rive Schr¨odinger wave equation
[24, 25]
according to the
basic principle? What is the scope of application of
the basic principles of quantum mechanics such as the
representation of mechanical quantities by operators?
Can the fine structure constants
[26]
in the atomic sys-
tem change? Is Bowen’s statistical interpretation
[27]
the
ultimate physical meaning of wave function? Is the con-
cept of orbit compatible with the physical meaning of
probabilistic wave? Is the quantum mechanics operation
method of constructing wave equation by expressing me-
chanical quantities in the laws of mechanics as operators
suitable for describing the discrete law of macroscopic
motion? We need to find the answers to these question-
s and clarify the scop e of application of the principles
of quantum mechanics, so as to systematically describe
the macro discrete law and micro discrete law in the
same logical framework. Some basic principles and me-
chanical laws ignored by traditional theories will play an
important role.
Here I introduce the Dongfang angular motion law
and the concise method of solving Kepler’s orbital equa-
tion of motion by using the angular motion law, and
introduce the multi wave function wave equations of mi-
cro motion corresponding to the angular motion law, so
as to reveal the rich and colorful new mathematical and
physical problems hidden in quantum mechanics. The
solution of all these problems is the premise to solve the
basic difficulties of quantum mechanics, including reveal-
ing the essence of quantum mechanics, deciding the es-
tablishment of the basic principles of com quantum and
Citation: Dongfang, X. D. Dongfang Angular Motion Law and Operator Equations. Mathematics & Nature 1, 202105 (2021).
2 X. D. Dongfang Dongfang Angular Motion Law and Operator Equations
the development direction of com quantum theory, and
will also promote the further development of the analyt-
ical theory of differential equations.
2 Dongfang angular motion law
It is one of the important basic principles of quantum
mechanics that the mechanical quantities of classical me-
chanical laws are expressed by operators and used to
action on wave functions. New laws of mechanics or dif-
ferent expressions of existing laws of mechanics may have
different effects on the construction of the wave equation.
So, in addition to the equation describing the relation-
ship between momentum and energy, are there any other
classical mechanical equations that can be used to estab-
lish wave equations? What about the solutions of wave
equations established by new mechanical equations? In
order to find the answers to these questions, a universal
law of angular motion and its operator evolution wave
equations are proposed.
Assuming that the object moves in a curve, the mass
of the object ism, the size of the position vector isr, the
direction angle is θ, the radial velocity of the motion is
v
r
= v
r
e
r
, and the transverse velocity is v
θ
= v
θ
e
θ
,
then the radial force F
r
and the transverse force F
θ
of
the object satisfy the equations respectively,
F
r
=
m
r
v
θ
dv
r
dθ
− v
2
θ
F
θ
=
mv
θ
r
dv
θ
dθ
+ v
r
(1)
Among them, the central force F
r
is positive when it is
repulsive force and negative when it is attractive; the
transverse force F
θ
is positive when it is dynamic force
and negative when it is resistance force. Equation (1) is
called the Dongfang angular motion law.
The centripetal force F
n
of a body moving along a
curve is usually expressed by the rate v and the radius
ρ of curvature, F
n
= mv
2
ρ
−1
= p
2
(mρ)
−1
. The angular
motion law is the generalized centripetal force equation.
The common feature of the two methods is that they
contain momentum squares and have no relation to en-
ergy, so they can not be used directly to construct Hamil-
ton operators. This provides a basis for the comparison
demonstration of the applicable scope of the basic oper-
ation methods of quantum mechanics.
Now let’s prove the Dongfang angular motion law.
The physical quantities are represented by complex num-
bers in exponential form
[28]
. The polar coordinate sys-
tem or complex plane xoy is established, as shown in
Figure 1. To avoid confusion with the imaginary num-
ber unit i of quantum mechanics, j =
√
−1 is used here
to represent the imaginary number of units. The com-
plex exponential form of orbital equation r = r (t) is
r = r (t) e
jθ(t)
, where θ is the polar angle and t is the
time. The dynamic radial and transverse unit vectors
e
r
and e
θ
satisfy the right hand rule. The complex
exponential form of the velocity vector is obtained by
calculating the first derivative of r with respect to time
t,
v =
dr
dt
= (v
r
+ jv
θ
) e
jθ
(2)
Figure 1 The momentum, radial momentum and transverse
momentum of a particle in curvilinear motion on the plane of polar
coordinates
The real part and the imaginary part in parenthe-
ses correspond to the radial velocity v
r
= dr/dt and
the transverse velocity v
θ
= rdθ/dt, respectively. The
derivative of transverse velocity with respect to time t
is transformed into a derivative with respect to angle
θ. One has dt = rdθ/v
θ
, so dv
r
/dt = (v
θ
/r) dv
r
/dθ
anddv
θ
/dt = v
θ
dv
θ
/(rdθ). From this, the derivative of
velocity (2) with respect to time t is calculated, and the
following complex exponential form of the acceleration
vector is obtained,
a =
dv
dt
=
1
r
v
θ
dv
r
dθ
− v
2
θ
+ j
v
θ
dv
θ
dθ
+ v
r
v
θ
e
jθ
(3)
The real part and the imaginary part in curly brack-
ets are opposite to the radial acceleration a
r
and the
transverse acceleration a
θ
, respectively. The transverse
velocity formula rdθ/dt = v
θ
is used in the above ex-
pression.
The complex exponential representation of the central
force is F = (F
r
+ jF
θ
) e
jθ
. Where F
r
takes positive val-
ue to represent repulsive force and the direction is the
same as that of e
r
; F
r
takes negative value to represent
gravitation and the direction is opposite to that of e
r
;
F
θ
takes positive value to represent power and the direc-
tion is the same as that of e
θ
; F
θ
takes negative value
to represent resistance and the direction is opposite to
that of e
θ
. In the complex plane, the complex expo-
nential representation of Newton’s law of motion
[29]
is as
follows,
(F
r
+jF
θ
) e
jθ
=
m
r
[(
v
θ
dv
r
dθ
− v
2
θ
)
+j
(
v
θ
dv
θ
dθ
+v
r
v
θ
)]
e
jθ
(4)
Mathematics & Nature (2021) Vol. 1 3
Hence, from the definition of complex number equality,
the equation of the angular motion law (1) is obtained.
The angular motion law is a generalized form of cen-
tripetal force equation describing curve motion. The
characteristic of the angular motion law is that the
derivative of time can be transformed into the derivative
of angle. Linear motion has no angular motion, so the
angular motion law is not applicable to linear motion.
3 Angular motion law in central force field
From classical mechanics to quantum mechanics, the
problem of a central force is one of the standard subjects.
Coulomb’s law and the law of gravitation are both cen-
tral forces, which control the law of atomic spectrum and
the law of celestial motion respectively, and maintain
the dynamic stability of the micro-world and the macro-
world. The transverse force in the central force field is
zero. According to the second formula of equation (1),
we obtain the relation dv
θ
/dθ+v
r
= 0. So v
r
= −dv
θ
/dθ,
v
2
r
= v
r
v
r
= −v
r
(dv
θ
/dθ). From v
2
= v
2
r
+ v
2
θ
, we get
the formula v
2
θ
= v
2
+ v
r
dv
θ
/dθ, which is substituted for
the first formula of (1) to obtain several expressions of
the angular motion law of the central force field,
F
r
=
m
r
−v
θ
d
2
v
θ
dθ
2
− v
2
θ
F
r
=
m
r
v
θ
dv
r
dθ
− v
r
dv
θ
dθ
− v
2
r
− v
2
θ
F
r
=
m
r
dv
θ
dθ
2
− v
θ
d
2
v
θ
dθ
2
− v
2
r
− v
2
θ
(5)
The radial velocity of the circular motion is v
r
= 0, and
the centripetal force is directed to the center of the cir-
cle, so F
r
is negative. The centripetal force equation
F
r
= mv
2
θ
r is obtained by substituting this into the
first formula (1) or (5) formula. It can be seen that the
centripetal force equation is a special case of the angu-
lar motion law. Newton infers that the interaction force
between planets and the sun follows the law of inverse
square ratio by using the centripetal force equation of
circular motion. By using the angular motion law or
Newton’s law of motion, it can be proved that the inter-
action force governing the motion of a conic curve has
and only has the law of the inverse square ratio. The
proof process is omitted here.
Dongfang’s law of angular motion itself is not a break-
through discovery. Perhaps because the mathematical
process of proving the two-dimensional angular motion
law is too ordinary, we even think that it should not
be proposed as a new law or principle. However, if we
try to write and prove the three-dimensional angular
motion law, and apply the two-dimensional and three-
dimensional angular motion laws to quantum mechanics
to construct abundant wave equations, we will find that
the difficulties of accurate solutions of new wave equa-
tions and the unknown physical meaning of new wave
equations pose challenges to the operator principle of
quantum mechanics, and we can realize that the pro-
posal of Dongfang’s angular motion law is not only nec-
essary, but also of far-reaching significance.
According to Newton’s law of motion and vector op-
eration rules, we can derive the three-dimensional for-
m of the angular motion law which contains many
but not completely independent equations. The three-
dimensional form of the angular motion law in spherical
coordinate system of a central force field is a system of
equations containing multiple equations,
v
θ
dv
θ
dθ
+ v
θ
v
r
− v
2
φ
cot θ = 0
1
sin θ
dv
φ
dφ
+ v
r
+ v
θ
cot θ = 0,
dv
φ
dφ
+ v
r
sin θ + v
θ
cos θ = 0
v
φ
dv
θ
dφ
− v
θ
dv
φ
dφ
+ v
r
v
φ
dθ
dφ
−
v
2
φ
+ v
2
θ
cos θ − v
r
v
θ
sin θ = 0
F
r
=
m
r
v
θ
dv
r
dθ
− v
2
θ
− v
2
φ
, F
r
=
m
r
v
φ
sin θ
dv
r
dφ
− v
2
θ
− v
2
φ
F
r
=
m
r
v
θ
dv
r
dθ
− v
r
dv
θ
dθ
− v
2
r
− v
2
θ
+ m (v
r
cos θ − v
θ
sin θ) v
φ
dφ
dθ
F
r
=
m
r sin θ
v
φ
dv
r
dφ
− v
r
dv
φ
dφ
−
v
2
r
+ v
2
φ
sin θ − v
r
v
θ
cos θ − v
φ
v
θ
dθ
dφ
(6)
In the above equation, when F
r
is the repulsive force,
it takes a positive value, and when F
r
is the attractive
force, it takes a negative value. The momentum repre-
sentation of the three-dimensional angular motion law
is omitted here. Otherwise there are too many equa-
tions and the content appears redundant. The form of
Newton’s law of motion in the spherical coordinate sys-
tem can also be incorporated into the three-dimensional
angular motion law. On the other hand, if you discuss
the relativistic momentum form of the three-dimensional
angular motion law, you will find the difficulties in rela-
tivistic logic. However, this is not the focus of studying
the angular motion law and its operator evolution equa-
tion. The reader can discuss related issues separately.
4 X. D. Dongfang Dongfang Angular Motion Law and Operator Equations
It is very simple to solve Kepler orbital equation of
motion of an object under the action of the inverse
square ratio law F
r
= K
r
2
with the angular motion
law. Here F
r
denotes the central force, K is a constan-
t, K > 0 for repulsion, K < 0 for gravitation, and the
transverse force is F
θ
= 0. According to the first formula
of (5), there is
F
r
= −
m
r
v
θ
d
2
v
θ
dθ
2
+ v
2
θ
= −
mrv
θ
r
2
d
2
v
θ
dθ
2
+ v
θ
(7)
Where m denotes mass. mrv
θ
= L is an angular mo-
mentum constant, from which an expression of trans-
verse velocity v
θ
= L/mr is obtained. Then the second
order homogeneous linear differential equation with con-
stant coefficients is obtained by using the central force
F
r
= K
r
2
, and its simplified form is
d
2
dθ
2
L
mr
+
L
mr
= −
K
L
According to the general and special solutions of sec-
ond order homogeneous linear differential equation
with constant coefficients, the conic solution L/mr =
A cos (θ + α) −K/L is obtained. Among them, A and α
are undetermined coefficients, which are determined by
the orientation of the selected coordinate axis and the
initial value conditions. Take α = 0, and remember that
r = r
0
when θ = 0, one obtains A = L/mr
0
+ K/L.
The above Kepler orbital equation in polar coordinates
expressed by angular momentum is transformed into the
following form,
r =
L
2
mK
1 + L
2
mr
0
K
cos θ − 1
(8)
Dongfang’s angular motion law can also be used to
solve the common physical problems related to satel-
lite motion. Of course, in mechanics, because the de-
velopment of classical mechanics has been very perfect,
the angular motion law as a supplement to classical me-
chanics seems to have no special significance, although
its physical meaning is more clear than the radial force
formula of curvilinear motion because it describes radi-
al force and transverse force at the same time, and it
is more convenient to solve some complex problems of
curvilinear motion. However, the angular motion law
plays an irreplaceable role in testing the basic theoreti-
cal program of the operator evolution wave equation of
quantum mechanics and thus the logical basis of quan-
tum mechanics. This is because of the existence of a
new law of mechanics expressed by momentum. If the
mathematical art of wave equation evolution through
mechanical law operator is an inevitable physical logic,
then Hamiltonian is no longer the only choice for quan-
tum mechanics to construct wave equation.
4 Operator evolution wave equations of an-
gular motion law
The unitary principle is an important basic principle
generally applicable to the logical self consistency test
of natural science theory and social science theory
[30, 31]
.
I used the unitary principle to prove that the assump-
tion of constant speed of light can not be universally
established, and find out the morbid equation of quan-
tum numbers implied in quantum mechanics. As long
as a theory implies discordant logical contradictions, it
can be tested by the unitary principle. According to
the unitary principle, the exact solution of the operator
evolution wave equation of angular motion law must be
consistent with the exact solution of the operator evo-
lution wave equation of Hamiltonian. Otherwise, the
operator theory, which is the key mathematical art of
quantum mechanics, will be severely challenged because
it violates the unitary principle.
Now let’s discuss the operator evolution wave equa-
tion of angular motion law. The momentum p is decom-
posed into radial momentum p
r
= mv
r
and transverse
momentum p
θ
= mv
θ
. The momentum expressions of
equation (1) and (5) of the angular motion law are as
follows.
p
r
+
dp
θ
dθ
= 0
F
r
=
1
mr
p
θ
dp
r
dθ
− p
2
θ
F
r
= −
1
mr
p
θ
d
2
p
θ
dθ
2
+ p
2
θ
F
r
=
1
mr
p
θ
dp
r
dθ
− p
r
dp
θ
dθ
− p
2
(9)
Where the third formula is derived by substituting the
first formula into the second formula and eliminating
p
r
. The latter three formulas of this equation system
are equivalent. It is one of the basic principles of quan-
tum mechanics to replace mechanical quantities in me-
chanical laws by operators to construct the wave equa-
tion. Three-dimensional space usually constructs the
rectangular coordinate form of the wave equation first,
and then transforms it into spherical coordinate form.
The mechanical quantity of the angular motion law is
replaced by the operator to construct wave equation,
which will encounter the problem of expression of the ra-
dial momentum operator which has been controversial.
Historically, when solving Dirac equation of hydrogen
atoms in spherical coordinates, the radial momentum
operator was defined as ˆp
r
= −i~ (∂/∂r + 1/r). The re-
sult of the debate is to agree with this expression. How-
ever, the deduction of theoretical physics should follow
the rules of the mathematical operation. Different math-
ematical forms of operators in different coordinates can
be converted to each other, when they move from spher-
ical coordinates to rectangular coordinates. The expres-
Mathematics & Nature (2021) Vol. 1 5
sion of radial momentum op erators mentioned above will
be mathematically difficult. The correct form of trans-
lating momentum operators in rectangular coordinates
directly into radial and transverse momentum operators
in plane polar coordinates is as follows,
ˆp
r
= −i~
∂
∂r
, ˆp
θ
= −
i~
r
∂
∂θ
(10)
where h is Planck constant and ~ = (2π)
−1
h. The above
radial momentum operator and transverse momentum
operator (10) are necessary mathematical inferences. If
Dirac equation’s rectangular coordinate system is direct-
ly transformed into a polar coordinate system, for the
hydrogen atom, the expected solution of the standard
theory can still be obtained by solving the equation with
boundary conditions
[32, 33]
. To achieve this goal, it is not
necessary to redefine the independent conservative quan-
tity ~ˆκ in relativistic quantum mechanics. Constructing
the expected solution must not destroy any mathematics
rule. The redefinition of the radial momentum operator
~ = (2π)
−1
h violates the basic mathematical operation
rules and is unnecessary.
There is a more complex relationship between radial
momentum and angular momentum in the momentum
representation of the angular motion law. When me-
chanical quantities are replaced by operators, the prob-
lem of composite operators arises in the equivalence
relations p
θ
(dp
r
/dθ) = (dp
r
/dθ) p
θ
, p
θ
d
2
p
θ
dθ
2
=
d
2
p
θ
dθ
2
p
θ
of mechanical quantity contained in equa-
tion (9). Because of the lack of corresponding math-
ematical basis, the standard answers to these ques-
tions are still unknown. According to the stochastic
expression of the system of equations (9), the form-
s of compound operators in the system of equation-
s are written by using the formula (10), respectively.
How to calculate one-step ˆp
θ
(dˆp
r
/dθ), ˆp
θ
d
2
ˆp
θ
dθ
2
and ˆp
r
(dˆp
θ
/dθ)? Is ˆp
θ
(dˆp
r
/dθ) and (dˆp
r
/dθ) ˆp
θ
commu-
tative, ˆp
θ
d
2
ˆp
θ
dθ
2
and
d
2
ˆp
θ
dθ
2
ˆp
θ
commutative,
ˆp
r
(dˆp
θ
/dθ) and (dˆp
θ
/dθ) ˆp
r
commutative? If only a ran-
dom expression of the system of equations (9) is taken,
the forms of the related composite operators in the sys-
tem of equations written by formula (10) are as follows,
ˆp
2
θ
=
(
−
i~
r
∂
∂θ
)(
−
i~
r
∂
∂θ
)
= −~
2
(
1
r
∂
∂θ
)(
1
r
∂
∂θ
)
ˆp
θ
dˆp
r
dθ
= −
i~
r
∂
∂θ
[
d
dθ
(
−i~
∂
∂r
)]
= −
~
2
r
∂
∂θ
[
d
dθ
(
∂
∂r
)]
ˆp
θ
d
2
ˆp
θ
dθ
2
= −
i~
r
∂
∂θ
d
2
dθ
2
(
−
i~
r
∂
∂θ
)
= −
~
2
r
∂
∂θ
d
2
dθ
2
(
1
r
∂
∂θ
)
ˆp
θ
dˆp
r
dθ
− ˆp
r
dˆp
θ
dθ
= ~
2
(
∂
∂r
)
d
dθ
(
1
r
∂
∂θ
)
−
~
2
r
∂
∂θ
d
dθ
(
∂
∂r
)
Since the basic method of quantum mechanics expressed
by operators is generally applicable, the radial and trans-
verse momentum in formula (9) should be replaced by
(10) radial and transverse op erators and used to action
on wave functions to describe the motion of the same
particle, which requires at least four wave functions ψ
1
,
ψ
2
, ψ
3
and ψ
4
. The corresponding four wave equations
constitute a wave equation system of real number,
∂ψ
1
∂r
+
d
dθ
1
r
∂ψ
1
∂θ
= 0
~
2
mr
2
∂
∂θ
1
r
∂ψ
2
∂θ
−
∂
∂θ
d
dθ
∂ψ
2
∂r
− F
r
ψ
2
= 0
~
2
mr
2
∂
∂θ
d
2
dθ
2
1
r
∂ψ
3
∂θ
+
∂
∂θ
1
r
∂ψ
3
∂θ
− F
r
ψ
3
= 0
~
2
mr
∇
2
ψ
4
+
∂
∂r
d
dθ
1
r
∂ψ
4
∂θ
−
1
r
∂
∂θ
d
dθ
∂ψ
4
∂r
− F
r
ψ
4
= 0
(11)
These equations are all linear equations, but unlike
Schr¨odinger equation and Dirac equation, the equation
does not contain time, and the central force replaces the
energy parameter.
At present, it is difficult to find the exact solution
of the wave equation system (11). This is not only be-
cause the exact solution method of each equation is not
clear, but also because there is no answer to the ques-
tion whether the wave functions in the equation system
are consistent or not and how to explain the physical
meaning of each wave function. Since the last two equa-
tions of the angular motion law equation (9) are derived
from the first two equations, the last three equations of
wave equation system (11) should be equivalent, but the
forms of the last three wave equations are quite differen-
t. The more difficult problem is whether the operators
are commutative or not because of the different writing
order of mechanics quantities. In terms of mathematical
form only, the third derivative of the last three equation-
s of wave equation system (11) seems to be out of line
with common sense. Whether the wave equation system
means that the scope of application of momentum oper-
ators is actually very limited, so the momentum in the
laws of mechanics can not always be replaced by opera-
tors to construct a wave equation. It is very interesting
and important to study the answers to the questions.
The wave equation system (11) is derived by applying
the operator op eration principle of quantum mechan-
ics to the angular motion law. The angular motion
law describes the motion of curves, so the wave equa-
6 X. D. Dongfang Dongfang Angular Motion Law and Operator Equations
tion system (11) is applicable to the motion of curves.
Schr¨odinger equation and Dirac equation do not seem
to be confined to the angular motion conditions, but
actually conceal more complex logic difficulties. For ex-
ample, Schrdinger equation of linear harmonic oscillator
has expected quantized energy solution, whereas Klein-
Gordon equation
[34, 35]
or Dirac equation of relativistic
linear harmonic oscillator has no quantized expected so-
lution; the orbital differential equation of curvilinear mo-
tion under the action of the law of inverse square ratio
has the exact solution of a conic curve, and the orbital
differential equation of linear motion under the action of
the law of inverse square ratio is a non-linear differen-
tial equation and can not give the exact solution of the
orbital equation. These difficult mathematical problems
of theoretical physics are more enlightening.
5 Ordinary differential form of wave equa-
tion
Wave equation system (11) contains composite dif-
ferential operation, which reveals an important problem
that has been neglected for a long time, that is, the
classical concept of orbital equation r = r (θ) is not a-
bandoned because of the principle of quantum mechan-
ics. To determine the spatial orientation of particles, r
and θ represent independent coordinate parameters. To
study the law of motion of objects governed by interac-
tion forces, the orbital equation r = r (θ) is a constraint
equation, and r and θ are not independent parameter-
s. However, in the Schrdinger equation, Dirac equation
and even other mathematical and physical equations, r
and θ have been treated as independent parameters and
achieved great success. Therefore, a wave equation ex-
pressed by a partial differential equation may actually be
an ordinary differential equation. Then, is the exact so-
lution of such an ordinary differential equation the same
as that of the original partial differential equation? This
is a new mathematical and physical problem worthy of
further study.
Considering two-dimensional hydrogen atom, the
Coulomb force between electrons and protons isF
r
=
−e
2
4πε
0
r
2
−1
= −α~ cr
−2
. Among them, r is the dis-
tance between electron and proton, e is the quantity of
elementary charge, ε
0
is the dielectric constant in vacu-
um, α is the fine structure constant, ~ = h (2π)
−1
and h
is the Planck constant, c is the speed of light in vacuum.
The polar coordinate form of the orbital equation of an
electron moving around a proton is,
r =
δ
1 − ς cos θ
(12)
where polar angle θ represents the direction of the posi-
tion vector. In order to avoid confusion caused by using
the same symbols for different physical quantities, δ is
used to represent the focus parameter and ς is used to
represent the eccentricity. The Coulomb force between
the electron and the proton is substituted for the wave
equation group (11), and the wave equation group of the
hydrogen atom is obtained,
∂ψ
1
∂r
+
d
dθ
1
r
∂ψ
1
∂θ
= 0
~
2
m
∂
∂θ
1
r
∂ψ
2
∂θ
−
∂
∂θ
d
dθ
∂ψ
2
∂r
−
e
2
ψ
2
4πε
0
= 0
~
2
m
∂
∂θ
d
2
dθ
2
1
r
∂ψ
3
∂θ
+
∂
∂θ
1
r
∂ψ
3
∂θ
+
e
2
ψ
3
4πε
0
= 0
~
2
mr
∇
2
ψ
4
+
∂
∂r
d
dθ
1
r
∂ψ
4
∂θ
−
1
r
∂
∂θ
d
dθ
∂ψ
4
∂r
+
e
2
ψ
4
4πε
0
r
2
= 0
(13)
This system of equations has no energy parameters,
so it is impossible to obtain quantized energy solutions
like Schr¨odinger equation or the Dirac equation directly.
However, since r and θ satisfy the constraint equation
(10) and are no longer independent coordinate parame-
ters, the energy parameters are actually expressed indi-
rectly by the focal parameters δ and eccentricity ς of the
elliptic equation. This is only a qualitative mathemat-
ical conclusion, but the quantitative problem is much
more complicated.
According to Dongfang unitary principle, if the prin-
ciples of quantum mechanics and the statistical interpre-
tation of wave functions are both correct, the solutions
of each equation of the wave equation group (13) should
be equivalent, otherwise, a reasonable choice must be
made among the non equivalent wave functions. This
requires negating other wave function equations without
scientific basis for trade-offs. Although the equation sys-
tem (13) is a system of linear differential equations, it
is still difficult to find the exact solutions of the equa-
tion system so as to obtain scientific conclusions. Are
their exact solutions consistent? Are there any indirect
quantized exact solutions representing energy? Whether
there is an indirect expression of the exact solution of
energy quantization? Does the exact solution of quanti-
zation conform to the prediction of quantum mechanics?
These problems puzzle us.
The first equation in the wave equation system (11)
or (13) is the simplest. It has no controversial question
whether the operator is commutative or not. It can be
Mathematics & Nature (2021) Vol. 1 7
transformed into an ordinary differential equation by the
constraint equation, i.e. orbital equation (12). Because
r is a function of θ, the partial derivative of the equation
is rewritten to ordinary differential. From the orbital e-
quation (12), we get that,
dr
dθ
=
d
dθ
δ
1 − ς cos θ
= −
ςδ sin θ
(1 − ς cos θ)
2
∂ψ
1
∂r
=
dψ
1
dθ
dθ
dr
= −
(1 − ς cos θ)
2
ςδ sin θ
dψ
1
∂θ
d
dθ
1
r
∂ψ
1
∂θ
=
1 − ς cos θ
δ
d
2
ψ
1
dθ
2
+
ς sin θ
δ
dψ
1
∂θ
By substituting these relations into the first equation of
the wave equation system (11) or (13), the second order
ordinary differential equation with respect to the param-
eter θ is obtained,
d
2
ψ
1
dθ
2
−
(1 − ς cos θ)
2
− ς
2
sin
2
θ
ς sin θ (1 − ς cos θ)
dψ
1
dθ
= 0 (14)
Let χ = cos θ, then equation (14) is reduced to
dψ
1
dθ
= −sin θ
dψ
1
dχ
d
2
ψ
1
dθ
2
=
1 − χ
2
d
2
ψ
1
dχ
2
− χ
dψ
1
dχ
the equation (14) is transformed into
d
2
ψ
1
dχ
2
−
3ς
2
χ
2
− 3ςχ + 1 − ς
2
ς
2
χ
3
− ςχ
2
− ς
2
χ + ς
dψ
1
dχ
= 0 (15)
By using the orbital equation (12), the derivative of wave
function to θ can also be transformed into the derivative
to r, and then the first equation of wave equation sys-
tem (11) can be simplified to the ordinary differential
equation of the derivative to parameter r,
d
2
ψ
1
dr
2
−
1 − ς
2
r
2
− 2δr + 2δ
2
(1 − ς
2
) r
3
− 2δr
2
+ δ
2
r
dψ
1
dr
= 0 (16)
By using the above methods, the last three partial differ-
ential wave equations of the wave equation system (13)
can be transformed into ordinary differential equations.
The existence and uniqueness of the exact solutions of
these equations must be studied first. Only by finding
the exact solution of the equation and comparing the
solutions of the wave equation corresponding to the e-
quivalent mechanical equation of different forms, can we
know the true physical meaning of the wave function,
and then clarify the scope of application of the basic
principles of quantum mechanics in which mechanical
quantities are replaced by operators. In order to study
the exact solutions of equations (15) and (16), we need
to develop the analytical theory of differential equation-
s. The relevant mathematical basis is the optimization
theorem of differential equations
[36]
. We need to extend
the optimization theorem of differential equations to the
generalized optimization theorem to study the existence
and uniqueness of exact solutions of equations (15) and
(16). Of course, finding the exact solution of equation
(15) or ( 16) by any other method will also lead to impor-
tant new physical conclusions including but not limited
to the equation itself.
6 Comments and conclusions
In this paper, the law of angular motion is proposed,
and the operator evolution wave equations of the law
of angular motion of the central force field are written
by using the quantum mechanics calculation principle of
constructing the wave equation by replacing mechanical
quantities with operators. The simplest equations of the
wave equations of the hydrogen atom are simplified in-
to two kinds of ordinary differential equations. At the
same time, the necessity of generalized optimization of
the differential equations is pointed out.
However, does the exact solution of each wave equa-
tion exist? Or the exact solutions exist, but are they
consistent? These problems challenge the basic and
most important computational rule of operator evolu-
tion of the wave equation in quantum mechanics. For
any theory, it is only a necessary condition that the
principle
[37, 38]
, method and conclusion of the theory con-
form to the unitary principle. If it does not conform to
the unitary principle, there must be major defects or
even mistakes. The law of conservation of momentum,
the law of conservation of energy, the law of conservation
of angular momentum and the angular motion law are
all the inevitable inferences of Newton’s law of motion.
Applying the angular motion law to quantum mechan-
ics, the obtained wave equation set of the central force
field implies the existence of many different wave func-
tions. How to explain the physical meaning of these dif-
ferent wave functions is obviously an urgent topic. The
statistical interpretation of wave functions may not be
the only physical meaning of wave functions. Whether
it belongs to functions that contain different physical
meanings such as orbital density can not be affirmed or
negated at present. From the point of view of statistical
law, the concept of the electron cloud is very applicable,
but the probability of a probability wave does not mean
the negation of the concept of micro particle trajecto-
ry. The problem that quantized energy destroys the law
of conservation of energy implied in the morbid equa-
tion of quantum numbers
[31]
and the exact solutions of
the operator evolution equations of angular motion law
all show that quantum mechanics can not pass the logi-
cal test of the unitary principle, so it is actually a very
imprecise theory. But the idea of quantum mechanics
is meaningful, and quantum mechanics needs to be re-
vised systematically. I have tested some important ba-
8 X. D. Dongfang Dongfang Angular Motion Law and Operator Equations
sic physical theory problems with the unitary principle,
and found that the difficulty of fundamentally eliminat-
ing physical theory needs to correct or prove all basic
physical assumptions that lack logical basis.
In quantum mechanics, it is considered that the ener-
gy and angular momentum of the interactive system are
observable and can be expressed by a linear Hermitian
operator. From the point of view of the com quantum
phenomena of the macroscopic system, velocity is tru-
ly directly observable compared with angular momen-
tum and energy. From the so-called plane wave func-
tion ψ (r, t) = A exp [(i/~) (mv · r − Et)], we can write
the velocity operator ˆv = −(i~/m) ∇ or the velocity
square operatorˆv
2
= −
~
2
m
2
∇
2
. Therefore, a strict
proof is needed to make a choice between the veloci-
ty operator and the momentum operator. Essentially,
the steady wave equation means that the mathemati-
cal expression of the wave function independent of time
is ∂ψ/∂t = 0. But this accurate mathematical expres-
sion will lead to time-dependent Schr¨odinger equation
i~∂ψ/∂t = −
~
2
2m
∇
2
ψ +U (r) ψ can not be trans-
formed into the steady equation. The problem means d-
ifferent physical ideas, that is, the establishment of wave
equations may not be limited to Hamiltonian operators.
For example, considering the energy level transition of
a quantum system, the absorbed or radiated energy of
the system must follow the law of conservation of ener-
gy. Therefore, referring to the Maxwell equations, we
can find a more reasonable reason than the Hamiltonian
operator evolution method, but it may not be the true
portrayal of the natural law, and write the following real
number wave equation,
∇
2
ψ +
4π
2
σc
2
(E − U ) ψ −
1
c
2
∂
2
ψ
∂t
2
= 0 (17)
Among them, m denotes the mass of particle, E and U
denote energy and potential energy respectively, and σ
is a constant for com quantum theory. The real wave e-
quation is one of the corollaries of com quantum theory,
which shows that the virtual number is not an indispens-
able element of quantum mechanics. There is no doubt
about how to explain the concept of steady state in real
wave equation. Because ∂ψ/∂t = 0 corresponds to the
steady-state Schr¨odinger wave equation, and the solu-
tion of this steady-state equation is just that have been
given by quantum mechanics. Perhaps readers can write
other forms of real wave equations and find the intrinsic
relationship between different real wave equations.
It is generally believed that quantum mechanics is one
of the perfect and precise scientific theories in mathemat-
ics. However, problems such as the scope of application
of the basic principles of quantum mechanics and the na-
ture of quantum mechanics have not been solved. The
application of an operator operating method in quan-
tum mechanics to the new laws of mechanics such as the
angular motion law brings new mathematical and phys-
ical difficulties to be solved. Its scientific conclusion is
of great significance to the design idea of the assump-
tion that the quantum theory does not depend on the
assumption that it is impossible to prove or is finally
proved to be very limited. The assumption that accord-
s with the natural law must have exact causality and
can be proved and become a theorem. Otherwise, it
will not be regarded as the basic principle in the com
quantum theory. Even if some hypothetical form logic
deduction can solve some difficult problems locally. The
wave function of quantum mechanics is interpreted as
a probability function. Why can’t it be a new physical
quantity that has not yet been discovered? What are the
wave functions and wave equations of the com quantum
theory, which can be used to describe both macroscopic
and microscopic discrete laws? Let’s find the answers to
the questions together.
To sum up, Dongfang angular motion law and the
energy momentum equation constitute two metrics for
the unitary principle test of quantum mechanics. It is
one of the most important basic principles of quantum
mechanics to replace mechanical quantity with Hermite
operators acting on wave functions to construct wave e-
quations. Applying Dongfang angular motion law, there
is enough derived wave equations in the same quantum
system, and the existence of solutions of these equation-
s seriously deviates from the expectations of quantum
mechanics. Obviously, the operator principle of quan-
tum mechanics does not follow the unitary principle, so
it is not universally effective, which means that quan-
tum mechanics has serious mathematical defects. This
conclusion should have a far-reaching impact on physics
and promote theoretical physics to enter a new period
of vigorous development.
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Latest findings
Dongfang Com Quantum Equations of LIGO Signal
The signal waveform with monotonic frequency change detected by LIGO is implying the discrete law of macroscopic quantization. Here, I accurately analyzed the observation data of GW150914 signal and proved that the Livingston waveform of positive and negative strain reversal was 7.324218ms ahead of Hanford waveform. Then, the time of the positive and negative strain peaks of the main vibration part is corrected by using the superposition waveform, and the numerical results of the discrete frequency of GW150914 signal are calculated. Finally, a numerical analysis method using the minimum digit rational number solution of the characteristic Diophantine equation system was introduced to fit the quantized Lagrange frequency change rate and frequency jump rate of the GW150914 signal, providing a quantitative basis for inferring accurate information of the wave source.