Since quantum mechanics was established, its correctness has been well verified. But there exists serious controversy on its physical significance. Many people believe that quantum mechanics has not been well explained up to now days. However, the mathematical structure of quantum mechanics is commonly considered complete and perfect. It seems difficult to add additional things to it. Is it true? It is pointed out in this paper that the definition of momentum operator of quantum mechanics has several problems so that it should be improved.
In this paper, we first prove that using kinetic energy operator and momentum operator to calculate microparticle’s kinetic energies, the results are different. That is to say, kinetic energy operator and momentum Operator are not one to one correspondence. Secondly, in the curved coordinate system, momentum operator can not be defined well though we can define kinetic energy operator well. That is to say, except in the rectangular coordinates, the definition of momentum operator is still an unsolved problem in quantum mechanics.
With operators of quantum mechanics acting on the eigen functions, we obtain real eigen values. However, if operators act on the non-eigen functions, the results are complex numbers in general. We call theses complex numbers as non-eigen values. For non-eigen functions, the operators of quantum mechanics are not the Hermitian operators. In general, the average values of operators on non-eigen functions are complex number, unless they are zero.
Because the non-eigen values of complex numbers are meaningless in physics, the non-eigen functions have to be developed into the sum of the eigen functions of operators or the superposition of wave functions. The eigen function of momentum operator is the wave function of free particle. Hence, a very fundamental question is raised. We have to consider a non-free particle, for example an electron in the ground state of hydrogen, as the sum of infinite numbers of free electrons with different momentums. This result is difficult in constructing a physical image, thought it is legal in mathematics. Besides, it violates the Pauli’s exclusion principle. It is difficult for us to use it re-establishing energy levels and spectrum structure of hydrogen atoms.
Besides, some operators of quantum mechanics have no proper eigen functions, for example, angle momentum operator
, 
and 
in rectangular coordinate system. We can not develop arbitrary functions into the sum of their eigen functions. By acting them on arbitrary functions directly, we always obtain complex numbers. Can we say they are meaningless?
The descriptions of quantum mechanics are independent on representations. In momentum representation, the positions of momentum operator and coordinate can be exchanged with each other. It is proved that when the non-eigen wave functions in coordinate space are transformed in momentum space for description, the problems of complex non-eigen value and complex average value of coordinate operator occurs, though the problem of complex non-eigen value of momentum operator disappear.
In addition, there exists a famous problem of the EYP momentum paradox in quantum mechanics [1-3]. Because it can not be solved well, someone even thought that the logical foundation of quantum mechanics was inconsistent.
Because angle momentum operator is the vector product of coordinate operator and momentum operator, the problem also exists in the definition of angle momentum operator. For example, we can not define angle momentum operator in curved coordinate system well at present. The physical image and essence of micro-particle’s spin is still unclear at present.
Therefore, the momentum operator of quantum mechanics can not represent the real momentums of microparticles. It needs to be improved. The concept of universal momentum operator is proposed to solve theses problems in this paper.
Using universal momentum operator and kinetic operator to calculate the kinetic energy, we can explain the problem of inconsistency as mentioned before. In curved coordinate system, we can define momentum operator rationally. When universal momentum operator is acted on arbitrary non-eigen wave functions, the non-eigen values are real numbers. In coordinate space, the average value of universal momentum is real number. The EYP momentum paradox can also be resolved thoroughly.
After universal momentum operator is defined, we can define universal angle momentum operator. Because there is a difference between calculated value and real momentum value, the concepts of auxiliary momentum and auxiliary angle momentum are introduced. The relation between auxiliary angle momentum and spin is revealed. It is proved that spin is related to the supplemental angle momentum of micro-particle which orbit angel momentum operator can not describe. The fact that spin gyromagnetic ratio is two times of orbit gyro-magnetic ratio, as well as why the electron of ground state do not fall into atomic nuclear without orbit angle momentum can be explained well.
By the clarification of spin’s essence, we can understand real reason why the Bell inequality is not supported by experiments. The misunderstanding of spin’s projection leads to the Bell inequality. No any real angle momentum can have same projections at different directions in real physical space. The formula 
does not hold in the deduction process of the Bell inequality. The result that the Bell inequality is not supported by experiments has nothing to do with whether or not hidden variables exist.
. The momentum 
is a constant. However, more common situation is that wave functions are not the eigin functions of operators. In this case, we have
is a constant and we call it as the non-eigen value of momentum operator. If 
describes a single particle, according to definition, 
should represent the momentum of particle. Because momentum is the function of coordinate, is it consistent with the uncertainty relation? Or is the function form of 
meaningful? This problem involves the understanding of real meaning of the uncertainty relation. We will discuss it in the end of this section.
and linear harmonic oscillator 
as examples, we have
and 
in which 
is the Bohr first orbit radius of hydrogen atom and 
is the angle frequency of harmonic oscillator. When momentum operator is acted on
, we get
can not be the momentum of electron in ground state hydrogen atom. Despite 
is an imaginary number, if it is electron’s momentum, the kinetic energy should be
, we obtain
is the total energy of ground state electron and 
is potential energy. According to (5), we have
, i.e., electron’s kinetic is equal to its total energy, so (5) is wrong.
of linear harmonic oscillator, we obtain
is also an imaginary number. Based on it, particle’s kinetic energy is
is the energy of ground state harmonic oscillator and 
is potential energy. It is obvious that the calculating results of two methods are different. According to (9), we have 
which is certainly wrong. In fact, this problem exists commonly in quantum mechanics. Kinetic operator and momentum operator do not have one-to-one correspondence, so that we can not determine the non-eigen values of momentum operator uniquely. Because kinetic operator is aright, we have to improve momentum operator to make it consistent with kinetic operator.
and 
are not the Hermitian operators. Their non-eigen values are imaginary numbers in general. Most fatal is that we can not construct correct kinetic operator based on (11). In classical mechanics, the Hamiltonian of free particles in spherical coordinate system is
in differential geometry was also suggested to define momentum operator in the curved coordinate reference system [
on scalar 
and vector 
are individually
, we have
acting on scalar field is (15). Therefore, the non-eigen values and average values of operators 
and 
may still be complex numbers. All problems existing in the Descartes coordinate system would appear in the spherical coordinate system.
and 
be arbitrary wave functions in coordinate spaces, according to the definition of quantum mechanics, if 
satisfies following relation
. It is easy to prove that the eigen value 
is a real constant We have
, i.e., 
is a real number. In this case, the average value 
of operator is also a real number. We have
and 
as the non-eigen value of operator. It is easy to prove that if wave function is not eigen function of operator, non-eigen value 
may be a complex number. In this case, operator will no longer be Hermitian. We have
is a real number. Obviously, if 
is a complex number, (25) and (26) are not equal to each other, so that (20) can not be satisfied. So 
is not the Hermitian operator. Because 
is a complex number, the average value
in coordinate space and 
in momentum space satisfies following Fourier transformation
is a complex number, 
is also a complex number too. Therefore, similar to the situation in coordinate space, the average values of momentum operator in momentum space is also a complex number which is meaningless in physics.
. When it is acted on the wave function in momentum space, we have
is a complex number in general. So the average value of coordinate operator in momentum space is a complex number. We have
is a real number. However, in coordinate space, the average value of coordinate operator is a real number. The problem of imaginary average value is transformed from coordinate space to momentum space. The problem exists still, unless (33) is equal to zero.
. Let 
and
, in which 
and 
are the averages of coordinate and momentum, we have so-called uncertainty relation：
is meaningless.
in (2) is only the result of mathematical calculation. Because the definitions of operator and wave function have no problems, how can we consider the result meaningless? According to (2), as long as we know the concrete form of wave function, we know the momentum. It is unnecessary for us to measure momentum. How can we think that the coordinate and momentum of micro-particle can not be determined simultaneously? We need to discuss the real meaning of the uncertainty relation in brief.
describes the probability of a particle appears at the point 
at moment
. Therefore, 
is the accurate value of particle’s coordinate in theory. It is not the value of measurement, for measurements always have error. Therefore, 
is the difference between particle’s theoretical coordinate and average coordinate. It is not the measurements error of coordinate. In fact, it is actually the fluctuation of coordinate about average value as that defined in classical statistics physics. Similarly, 
is also the difference between particle’s accurate momentum and average momentum, or the fluctuation of momentum about the average value. It is not the measurements error of momentum. According to classical theory, fluctuation is also uncertainty. But this uncertainty is due to statistics, having nothing to do with measurement. In this meaning, (34) is not the uncertainty relation for a single particle recognized in the current quantum mechanics.
(35)
, 
, 
and 
are the average values, not the values for a single event. Therefore, (35) represents the product of mean square errors of coordinates and momentums. It is the result of statistical average over a large number’s of events. For a single event, we may have
, 
and
. Merely, their statistical average satisfies (35).
and
, its forms and meaning is completely different from (34). In the current quantum mechanics, (35) is simplified into (34), then the uncertainty relation is declared. This is improper. It is also the misunderstanding to consider the uncertainty relation as the foundational principle of quantum mechanics for (35) is only the deduced result of quantum mechanics.
. As long as we determine particle’s coordinates at different moments, we can determine particle’s velocity and momentum 
by calculation without doing measurements. The more accurate the measurement of particle’s coordinate, the more accurate the particle’s velocity and momentum. In this meaning, where is the uncertain relation?
and 
are commutative with
. According to (36), we can naturally obtain particle’s momentum 
after its kinetic energy is known. In fact, the kinetic operator and momentum operator are commutative with
commutates with
, 
and
, so particle’s energy, kinetic and potential energy can be determined simultaneously. For a particle in stationary state, we have certain energy
is potential energy. As long as particle’s coordinates are known, we know its kinetic energy, potential energy and momentum without direct measurements. However, on the other hand, because kinetic energy operator and potential operator do not commutate in general, we have
and get 
.
, p(x,t) represent microparticle’s momentum at time 
and position
.
describe a single particles, for example, an electron in the ground state of hydrogen atom, it represents the momentum distribution of an electron in hydrogen atom [
, 
and 
in the rectangular coordinate system. Because they have no proper eigen functions, we can not developed arbitrary functions into the sum of their eigen functions. Acted them on arbitrary functions directly, we always obtain complex numbers. Can we say they are meaningless? The universal momentum and angle momentum operators proposed in this paper can solve these problems well.
is the index of partial quantities, 
is real number with its form to be decided. 
is complex number in general to satisfy following relation
and the superposetion state of
. If we want to connect 
with particle’s momentum, it should be stationary state
. By solving the motion equation of quantum mechanics, we can obtain the concrete forms of wave functions, so we can determinate the forms of 
based on (44). By acting universal momentum operator on common wave functions and considering (45), we obtain the non eigne equation and the non-eigen values of real numbers
is the eigen function of momentum operator, we take
below. According to (44), the kinetic operator of microparticles is
is known complex number, we separate it into imaginary and real parts and write it as
needs to be determined, but we only have two equations (53) and (54). Therefore, one of them can be arbitrary. Let 
be the partial momentum of particle, we can decide its value by acting kinetic operator on stationary state wave function and have
and substitute it in (53) and (54), we can determine 
and
. In general, 
and 
are different from 
and
. We assume
and 
are known due to the fact that
, 
, 
and 
are known. In this way, we determine the concrete form of universal momentum operator (44) and explain why the results are not the same when we use kinetic operator and momentum operator to calculate the kinetic energy of microparticles. By consider (55), we have
and 
are determined just by two equations (53) and (54). Meanwhile, 
and 
are determined by kinetic operators. The relations between them are still shown in (56). If particles move in one dimensional space, only one 
needs to be determined, but we still have two equations (53) and (54) as follows
is not unique, unless two equations are compatible. From both equations, we obtain
, so we have 
and
. That is to say, 
is a purely imaginary number. In the case, universal momentum operator is
and momentum operator 
in momentum space as
is a real number and 
is a complex number in general to satisfy following equation
is the wave function in momentum space. The relation between 
and 
is the Fourier transform with
based on (67). Thus, we have
is
, 
and 
are real numbers, and
, 
and
are complex numbers which satisfy following relations
, 
and
. Substitute (73)-(75) into (12), we get
but only have two equations (81) and (82), so we 
can be chosen arbitrary. By solving the motion question of quantum mechanics, we know the form of wave functions. Let
, 
and 
are the partial momentum of a particle. Their forms can be determined by following formulas
and substitute it in (81) and (82), we can determine 
and
. In general, 
and 
are different from 
and
. We let
, 
, 
and 
are known, 
and 
are known too. So (63) can be written as
and 
do not represent real momentums of micro-particle too. In this way, we define universal momentum operator in spherical coordinate system. By the same method, we can define universal momentum operator in other curved coordinate systems.
. If we think that micro-particle’s momentums are unobservable so that its values are unimportant, the angle momentum of micro-particles are related to atomic magnetic moment and the magnetic moments are measurable directly. It is obvious that when angle momentum operators act on general wave function, their non-eigen values and average values may be complex numbers too. The angel momentum operators in the Descartes coordinate system is
is the eigen operator of stationary wave function 
of hydrogen atom with eigen values
, but 
and 
are not. Their non-eigen values and average values are complex numbers in general. The square of angle momentum operators is the eigen operator of angle momentum, we have
on
, we get real eigen values 
. In order to make the non-eigen values of 
and 
real numbers, we should refine they. Based on (44), universal angle operators are
is determined by (53) and (54) with
. By acting universal angle operators on common noneigen wave function, we get
and 
are also determined by (55) and (56). In this way, the non-eigen values and average values of universal angle momentum operators are real umbers. We do not discuss the forms of universal angle momentum operators in curved coordinate reference systems here.
to construct the motion equation of quantum mechanics but use 
to construct angle momentum, in quantum mechanics. So the kinetic energy 
and angle momentum 
are not one-to-one correspondent. In fact, according to quantum mechanics, the kinetic energy of electron in ground state hydrogen atom is not zero but its angle momentum is zero. This state can not exist stationary. The angle momentum should exist for electron moving around atomic nuclear, otherwise electron would fall into nuclear. By considering the error between calculated values of momentum using momentum operator and real values, auxiliary momentum and auxiliary angle momentum are introduced. By establishing relation between auxiliary angle momentum and spin, the essence of micro-particle’s spin can be revealed.
represents universal momentum operator, 
represents its value with
. It has been proved before that 
is still not real momentum of microparticle. Let 
represent real momentum operator and 
represent real momentum with
. 
represent auxiliary momentum operator and 
represents auxiliary momentum with
. Their relation is
is angle momentum operator in current quantum mechanics and 
is supplemental momentum operator. The real angle momentum operator 
is
is related to spin
. We discuss their relation below and prove that spin is related to the partial angle momentum of micro-particle which current momentum operator can not describe. By establishing the relation between them, the essence of spin can be explained well.
. The projections of spin can only take two values 
at arbitrary direction in space. Spin seems an angle momentum but not real. Sometimes, we consider spin as that electron rotates around itself symmetry axis. But the calculation shows that if it is true, the tangential speed of electron’s surface would be 137 times more than light’s speed [
at arbitrary direction in space. In real physical space, such vector can not exist. The projection of a vector with mode 1 at any 
direction can only be 
with values
. Because spin is always coupled with magnetic field, the correct understanding should be that if we take 
axis as the direction of magnetic field, the projections of spin at 
axis direction take two values
. At other directions, spin’s projection should be related to
. We will see below that it is just due to the hypothesis that the projections of spin can only take two values, the Bell inequality can not be correct.
is
. However, according to (100), we have
. So spin gyro-magnetic ratio is two times of orbit gyro-magnetic ratio. It indicates that spin is not normal angle momentum. Let 
represent total magnetic moment of charged particle in magnetic field, as shown in 
precesses around total angle momentum
, so 
is considered as an immeasurable quantity in current theory. What can be measured directly is the partial quantity 
of 
at the direction of angle momentum
. We have
is the Lande factor. Let 
represent new total angle momentum after auxiliary angle momentum is considered, suppose that the relation between 
and 
is
, the direction of magnetic moment is the same as
. We do not need the assumption that 
precesses around angle total momentum 
again. In experiments, particle’s angle momentum can not be observed directly. What can be done is magnetic moment. We obtain angle momentum through measurement of magnetic moment actually. So introducing new total angle momentum does not cause inconsistent. Inversely, angle momentum theory of micro-particle becomes more rational.
, 
, 
, 
and 
are located on a plane. As shown in 
, 
and 
from quantum mechanics. The Lande factor 
can be obtained from experiment. So 
and 
can be determined based on (109) and (110). For example, when
, 
and
, we have 
and
. In this case, 
is just the angle momentum of electron in ground state hydrogen atom. That is to say, for ground state hydrogen atom with
, it is just angle momentum 
which ensures electron’s stable motion around atomic nuclear without falling into it. In this case, the magnetic moment caused by auxiliary angle momentum is
is real angle momentum, in stead of spin. We should consider 
as a kind of quantum number. Based on it, we can obtain real angle momentums of micro-particles. As shown in 
and
. It means that the partial quantity of angle momentums at 
axis direction is 
actually. The reason to write the projection of spin at any direction of space as 
is only for of mathematical convenience. Speaking correctly, we should consider 
as a kind of quantum number based on it we can obtain correct angle momentums of micro-particles.
, the angle momentum of micro-particles becomes complex. For example, for the states
, 
and 
of hydrogen atom, according to (109) and (110), we obtain
, 
and
, are only useful tools. By means of these, we can construct real angle momentum 
of micro-particles. Auxiliary angle momentum 
does not appear automatically in quantum mechanics. By means of it, the relation between real total angle momentum 
and magnetic moment 
becomes normal. We do not need to assume that magnetic moment precesses around
. We have only a real angle momentum 
for microparticle.
individually, so the total spin of system is zero. Spin operator is 
and we take 
as unite. Let 
represent the spin’s measurement value of particle 1 at 
direction, 
represent the spin’s measurement value of particle 2 at 
direction. The average value 
of association operator 
is
, we have
. Suppose that there exists hidden variable 
which makes deterministic motion possible for micro-particle. The ensemble distribution function of hidden variable is 
which is normalized with
and
, having nothing to do with
. Meanwhile, the measurement about particle 2 also depends on 
and
, having nothing to do with
. So for arbitrary 
and
, we have
and 
are at the same directions, according to (118), we have
. It means
be another direction vector, because of 
, we have
, from (123) we get
, 
and 
are vectors on the same surface. The angles between 
and 
is
, between 
and 
is also
, between 
and 
is
. According to quantum mechanics we have
,.
.
and substitute it into (124), however, we get absurd result
. It means that (124) can not coincide with the result of quantum mechanics (118). Since the Bell inequality is advanced, many experiments are completed. Most of them support quantum mechanics, not support the Bell inequality. So physicists do not think that hidden variables exist. The determined descriptions of micro-particles are considered impossible.
. However, this kind of vector can not exist in physical space. Suppose vector 
with mode
, if the projection of 
at direction 
is
, the projections at directions 
and 
can only be zero. No any physical vector can have same projections at different directions in real space.
. That is to say, the projection of spin at the direction of magnetic field is
, rather than at arbitrary direction! Speaking strictly, in quantum mechanics, spin is coupled with magnetic field in the value of
. If the direction of magnetic field is certain, the coupling between spin and magnetic field is certain. At other directions, we can not observe the physical affections of spin. In fact, in current quantum mechanics, matrix operators are used to describe spin with
and
, we have
with eigen values
. 
and 
are not the eigen operators of 
and 
and do not have certain eigen values. So we can not think particle’s spins have same projection values 
at arbitrary directions. Because the square of spin operator is
and 
as a kind of quantum number, in stead of spin angle momentum itself. In light of mathematics strictly, as a practical physical quantity, the projection of spin operator at 
direction is
. The projection at z direction should be
. According to quantum mechanics, we have
, 
and 
are the projections of unit vector 
at
, 
and 
axis directions. We have
, rather than
. When calculating the average values of 
about the wave function of a single particle, we have
. It means that the projection of electron’s spin at arbitrary direction can only take
. This result is different from (131)-(134) and can not be realized in real physical space. So it is inevitable that the Bell inequality can not be supposed by experiments. The Bell inequality is a misunderstanding of mathematics, having nothing to do with hidden hypotheses. It neither coincides with quantum mechanics, nor coincides with classical mechanics and any logic of mathematics and physics.
is
. Suppose that the angle between 
and 
is
, (135) becomes
as unit, let
, for the electron at 
state, we have 
or
. For the electron at
state, we have 
or 
. For the electron at 
state, we have 
or
. For the ground electron with
, we have
, so
. Let 
and
, we have 
in general.
. From this result we see again that the deduction of Bell inequality has nothing to do with hidden variables. The violation of the Bell inequality also has nothing to do with whether or not hidden variables exist.
. When a photon passes through a polarize, its polarization value is considered to be
. When a photon does not pass, its polarization value is considered to be –1 [
, the vibration direction of electromagnetic field turns an angle
. In this case, we should think that photon’s polarization becomes
. That is to say, even in classical optics, for photons which pass through polarizer, their polarization is considered as
, in stead of 
in general. For photons which are reflected without passing through polarizer, their polarization values depend on the angle of reflection, in stead of 
in general. In fact, in quantum mechanics, when calculating polarization correlation of photons, we use some formulas similar to (126)-(128) which are related to the direction angle 
of polarization. It is impossible for photons always to have polarization values 
or 
under arbitrary situation.
, the mistake is the same as made for particle’s spin when we deduct the Bell inequality of photon’s polarizations correlation. It is also inevitable that this kind of Bell inequality can not be supported by experiments. Therefore, the violation of Bell inequality of photon’s polarizations correlation also has nothing to do with hidden variables.
is particle’s mass. In the region
, wave function is
,
. The wave function of ground state can be written as
on (140), we obtain two egein values
. The result indicates that the wave function can be considered as the overlap of two wave functions with different momentums 
. So Einstein, Pauli and Yukawa thought that the particles in the ground state has only two independent momentum 
and 
with probability 
individually.
with probability 
individually. This is the so-called EPY momentum paradox is caused. Because this paradox can not be solved up to now days, some persons even thought that the logical foundation of quantum mechanics was inconsistent [
.
is the normalized wave function in a box, rather than the wave function of free particle in the region without boundary. The restriction of boundary condition would produce a great influence on the nature of wave functions. If 
are the wave functions with infinite boundary, after they are transformed in momentum space, the wave functions should be the 
function with
on wave function (140), we can only get
. It indicates that we only have two discrete momentums. The EPY momentum paradox has not eliminated really by considering boundary condition.
, so particle’s momentum is not 
in infinite trap. Because the Fourier transform of (68) is a pure mathematical one, its result is undisputed. Therefore, the momentum distribution (142) is correct. We see again that thought we can have rational definition of kinetic energy operator, we may not find proper momentum operator to match with kinetic energy operator sometimes.