The Definition of Universal Momentum Operator of Quantum Mechanics and the Essence of Micro-Particle ’ s Spin — — To Reveal the Real Reason That the Bell Inequality Is Not Supported by Experiments

The definition of momentum operator in quantum mechanics has some foundational problems and needs to be improved. For example, the results are different in general by using momentum operator and kinetic operator to calculate microparticle’s kinetic energy. In the curved coordinate systems, momentum operators can not be defined properly. When momentum operator is acted on non-eigen wave functions in coordinate space, the resulting non-eigen values are complex numbers in general. In this case, momentum operator is not the Hermitian operator again. The average values of momentum operator are complex numbers unless they are zero. The same problems exist for angle momentum operator. Universal momentum operator is proposed in this paper. Based on it, all problems above can be solved well. The logical foundation of quantum mechanics becomes more complete and the EPY momentum paradox can be eliminated thoroughly. By considering the fact that there exist a difference between the theoretical value and the real value of momentum, the concepts of auxiliary momentum and auxiliary angle momentum are introduced. The relation between auxiliary angle momentum and spin is deduced and the essence of micro-particle’s spin is revealed. In this way, the fact that spin gyro-magnetic ratio is two times of orbit gyro-magnetic ratio, as well as why the electrons of ground state without obit angle momentum do not fall into atomic nuclear can be explained well. The real reason that the Bell inequality is not supported by experiments is revealed, which has nothing to do with whether or not hidden variables exist, as well as whether or not locality is violated in microcosmic processes.


Introduction
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 Opera-tor 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 ˆx L , ˆy L ˆ and z L   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][2][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.

Inconsistency in Calculating Kinetic Energy Using Momentum and Kinetic Operators
The Hermitian operators are used to represent physical quantities in quantum mechanics.The result of Hermitian operator acting on eigen function is a real constant.Momentum operator and its eigen function are


We have 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   ,t p x  is a constant and we call it as the non-eigen value of momentum operator.If it in the end of this section.
Because (2) is only a calculation formula of mathematics and the definitions of operator and wave function are alright, we should consider it is effective.We prove in this section that the results are different by using momentum operator and kinetic energy operator to calculate the kinetic energies of micro-particles.Taking ground state wave functions of hydrogen atom 100 It is obvious that 1 can not be the momentum of electron in ground state hydrogen atom.Despite

  p r
1 is an imaginary number, if it is electron's momentum, the kinetic energy should be Therefore, the kinetic energy of electron in ground state hydrogen atom is (7) is obviously different from ( 5). ( 7) is just the formula of energy conservation, in which 1 is the total energy of ground state electron and is potential energy.
According to (5), we have 1 1 T , i.e., electron's kinetic is equal to its total energy, so (5) is wrong.
If momentum operator is acted on the wave function  of linear harmonic oscillator, we obtain Momentum is also an imaginary number.Based on it, particle's kinetic energy is It means that particle's kinetic energy is a negative number which can not be true.Acting kinetic operator on it, we have Here 0 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.

The Difficulty to Define Momentum Operator in Curved Coordinate System
In the current quantum mechanics, the definition of momentum operator in curved coordinate system is an unsolved problem [4].Several definitions were proposed, but none of them is proper.If we claim that three partial quantities of momentum operator are commutative each other, the definition should be However, it is easy to prove that r 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 X. C. MEI, P. YU 454 According to the correspondence principle between classical mechanics and quantum mechanics, by considering the definition (11), the kinetic operator of quantum mechanics is However, the kinetic operator of quantum mechanics in spherical reference system is actually (13) and ( 14) are obviously different.Another definition of momentum operator is [ Substitute (15) in (12), we get We see that (16) has one item more than (14), so ( 15) is improper too.
The covariant differential operator i in differential geometry was also suggested to define momentum operator in the curved coordinate reference system [6].The action forms of operator on scalar i D  and vector By considering the metric s g q q  , we have According to this definition, the kinetic operator in curved coordinate reference system can be written as In spherical coordinate reference system, (19) is just (14).But this result was also criticized to have inconsistent for scalar and vector fields [7].Meanwhile, according to (17), the result of i acting on scalar field is (15).Therefore, the non-eigen values and average values of operators r and  may still be complex numbers.All problems existing in the Descartes coordinate system would appear in the spherical coordinate system.

The Problems of Complex Number Non-Eigen Values of Momentum Operator
The Hermitian operators are used to describe physical quantities in quantum mechanics.The eigen values of the Hermitian operators are real numbers.But the premise is that the operators should be acted on eigen wave functions.However, we have seen many situations in quantum mechanics that wave functions are not the eigen functions of operators.For example, only the wave function of free particle is the eigen function of momentum operator.All other non-free particle's functions are not the eigen function of momentum operator.In the coordinate space, when momentum operator is acted on noneigen functions, the obtained result, called as non-eigen values, are complex numbers in general.The average values of momentum operator in coordinate space are also complex numbers.These results are irrational and can not be accepted in physics, unless the average values are zero.

 
Let both , t  x ˆ and be arbitrary wave functions in coordinate spaces, according to the definition of quantum mechanics, if F satisfies following relation we call it the Hermitian operator.The eigen equation of the Hermitian operator is It is easy to prove that the eigen value F  is a real constant We have Copyright © 2012 SciRes.JMP By considering (20), we get F F     , i.e., F  is a real number.In this case, the average value F of operator is also a real number.We have Suppose the action result of operator on the non-eigen function is We call it as the non-eigen equation of operator F 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 , , In which the probability density 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

  ,t F x
F is not the Hermitian operator.Because is a complex number, the average value is also a complex number too.Because the average value of operator in quantum mechanics is measurable quantity, complex average value is meaningless in physics, unless it is equal to zero.Let's discuss the average values of momentum operator in momentum space.The wave functions   The Fourier transformation and its inverse transformatio x p n of non-eigen value of operator are Substitute formulas above in (27), we obtain the average value of operator on non-eigen function in momentum space. is also a co lex num r to th x Values of ace On t -mp ber too.Therefore, simila e situation in coordinate space, the average values of momentum operator in momentum space is also a complex number which is meaningless in physics.

The Problems of Comple Coordinate Operator in Momentum Sp
he other hand, the positions of coordinate and mo mentum exchanges each other when we describe physical processes in momentum space.It is proved below that in momentum space, the problem of imaginary non-eigen value of momentum operator disappears, but the problem of imaginary non-eigen value of coordinate operator emerges.In momentum space, coordinate operator becomes ˆ~p x i   .When it is acted on the wave function in momentum space, we have Similar to non-eigen value of momentum co operator in ordinate space,   ,t x p is a complex number in general.So the average alue of coordinate operator in momentum space is a c x number.We have As mentioned before, when the non-eigen values of operator are complex numbers, the operator is not the Hermitian operator any more.In fact, someone had seen this problem and demanded that the operators in quantum mechanics should be self-adjoint operators [8].In fact, in his famous book "the principles of quantum mechanics", Dirac only used real operator, instead of the Hermitian operator.From the angle of mathematics, the relation between the self-adjoint operator and the Hermitian operator is subtle.We do not discuss it in this paper.From the angle of physics, self-adjoint operator can be considered as one which has the complete set of eigen functions [9], so that its eigen values are certainly real numbers.But the Hermitian operator has no such restriction.The problem is that this restriction condition would greatly effects the universality of operator and can not be satisfied actually.For example, for all non-particle's wave functions, momentum operator is not the one with selfadjoint.However, we can prove that although we can not make momentum operators self-adjoint, we can make its non-eigen functions to be real by redefinition.
It is proved below that though we can not make momentum operators is self-adjoint one, we can introduce universal momentum operators to make their non-eigen values be real numbers.

The Problems Caused by Non-Commutation of Operators in Q
As well know that momentum operator and coordinate ot commutate with [ , ] x and e the averages of coordinate and momentum, we have so-called uncertainty relation： According to current understanding, (34) means that coordinates and momentums of micro-partic determined simultaneously.If it is true, the function relatio , in wh p  ich x and p ar le can not be n   p x is meaningless.
However,   p x in ( 2) is only the result of mathematical calculation.Because the definitions of operator and wa unction have no pr ve f oblems, how can we consi lt mea der the resu ningless?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.
Firstly, the wave function   , x t


describes the probability of a particle appears at the point x at moment t .Therefore, x is the accurate value of particle's coordi value o or nate in theory.It is not the f measurement, for measurements always have error.Theref e, x x  is e difference between particle's theoretical coordinate and average coordinate.It is not the measurements error of coordinate.In fact, it is actually the fluctuatio coordinate about average value as that defined in classical statistics physics.Similarly, th n of p p  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 rements 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.
Secondly, the strict uncertain relation in quantum mechanics is measu In the formula (35), p and x , 2 p are the average values, not the values for a single event.Therefore, (35) represents the product of mean square errors of co m e ordinates and momentu s.It is th result of statistical average over a large number's of events.For a single event, we may have

Merely, their statistical average satisfies (35).
In fact, because (35 lues, ) only contains average va doe ot c s n ontain x and p , its forms and meaning is completely different from (34).In the current quantum m c echanics, ( 35) is simplified into (34), then the uncertainty relation is de lared.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.
On the other hand, according to quantum mechanics, time and micro-particle's coordinates can be determined simultaneously.The definition of velocity is d dt  V x .A doing mea um.In t ined.Let's discuss the commutation relation be s long as we determine particle's coordinates at different moments, we can determine particle's velocity and momentum m  p V by calculation without surements.The more accurate the measurement of particle's coordinate, the more accurate the particle's velocity and moment his meaning, where is the uncertain relation?
This kind of paradox exists commonly in quantum mechanics and the problem is more serious than what we have imag tween coordinate and kinetic energy operator.The kinetic energy operator is Acting on the wave function of a single non-free particle, we obtain It is easy t prov at T and x are commutative with According to the understanding of quantum mechanics, the kinetic energy and coordinate of micro-particle can be determined simultaneously, so it is meaningful to w rite micro-particle's kinetic energy as According to (36), we can naturally obtain particle's after its kinetic energy is known.In fact, the kinetic operator and momentum tative with operator are commu So kinetic energy and momentum can be determined simultaneously.Because micro-particle's kinetic energy is the function of coordinates, if coordinat we can determine its kinetic energy.Af determine its momentum.That is to say, we can determ es are known, ter that, we can ine particle's momentum by determining its coordinate.We consider particle's energy operator again.Acting the operator on the non-eigen wave function of a single particle, we get: It is easy to see that Ê commutates with x , p and T , so particle's energy, kinetic and potential energy can be determined simultaneously.For a particle in statio state, we have certain energy nary   U x is potential energy.As long as particle's coordinates are known, we know its kinetic energy, potential energy and momentum without ments.However, on the other han energy operator and potential operator do not commutate in l, we direct measured, because kinetic genera have For example, for hydrogen atom, we have and get . According to the current understanding, electron's kinetic energy and potential energy could not be determ um tunnel effect), (41) b ess.ver, (41) is also un ined simultaneously (this is so-called quant ecomes meaningl Howe ntum deduced based the principle of qua mechanics, how can we say it is meaningless?If it is true, how can we have the fine structure of hydrogen atom spectrum?In fact, (41) is the formula of energy conservation.If it does not hold, all theories and experiments of quantum mechanics become meaningless!In essence, quantum mechanics is a statistical theory which involves a large number of micro-processes.Quantum measurement process always involves a large number of micro-particles.Once statistical average is considered, many problems do no exist any more.Our current derstanding on quantum mechanics may have same foundational error.The standard explanation of quantum mechanics should have some essential changes.
We will discuss the real meaning of the uncertainty relation and the explanation of quantum mechanics further in another paper.In this paper, we mainly discuss the definition of universal momentum operator.We do not consider the restriction of the uncertainty relation ag e be eiicle.
f non-free ain and think that it is meaningful to act operators on non-eigen wave functions directly.In operator equation x , p(x,t) represent microparticle's momentum at time t and position x .

The Fourier's Series of Non-Eigen Wave Functions of Momentum Operator
According to quantum m chanics, if the wave functions are not the eigen functions of operators, they should developed into the eigen functions of operators.The gen function of momentum operator is that of free part For the stationary state wave functions o particles, we have This is actually the Fourier's transformation of wave function which is legal in mathematics.It can be con dered as the principle of superposition principle of function in quantum mechanics.
If    x describe a single particles, for example, el mber of free electrons wi ffere o complex non-eigen values.If we ca siwave an ectron in the ground state of hydrogen atom, it represents the momentum distribution of an electron in hydrogen atom [10].But it does not mean that a non-free electron is equivalent to infinite nu th di nt momentums and energies.This is impossible in physics.In fact, there are only two electrons with opposite spins in the ground states of hydrogen.If (43) describes the hydrogen atom of ground state, it violates the Pauli's exclusion principle.It is impossible for us to use so many free electrons with different energies to construct hydrogen atom's energy levels and the spectrum structures.
The reason we write the wave function of a single particle in the form of ( 43) is due to the definition of momentum operator which is only effective to free particles.It is ineffective when it is acted on the wave functions of non-free particles due t n find proper momentum operator to describer non-free particle's momentums, it is unnecessary for us always to write the wave function of a single non-free particles in the superposition form of free particle's wave functions.In fact, the eigen function of kinetic energy operator is also the wave function of free particle.We do not need to write arbitrary wave function as the form of (43) for kinetic operator.The reason is just that when kinetic operator is acted on arbitrary wave functions, the results are always real numbers.
In fact, for some operators of quantum mechanics, we can not find proper eigen functions, for example, angle momentum operator ˆx L , ˆy L and ˆz L in the rectangular coordinate system.Because they have no proper eigen functions, we can not developed arbitrary functions into th T op numbers, we need to redefine the mosian ntities e sum of their eigen functions.Acted them on arbitrary functions directly, we always obtain complex numbers.Can we say they are meaningless?he universal momentum and angle momentum operators proposed in this paper can solve these problems well.

The Definition of Universal Momentum Operator in Coordinate Space
In order to make all non-eigen values of momentum erator be real mentum operator of quantum mechanics.In the Carte coordinate reference system, we write the partial qua of universal momentum operator as For the purpose of universality, the functions here can be both the st The average values of universal momentum operator are also real numbers operator.e now discuss the concrete forms of

 
Q  x below.According to (44), the ki tor of microparticles is netic opera However, practical kinetic operator should be actually By comparing (49) with (50) and considering (45), we ge t into imaginary and real parts and write it as Substituting it in (51), and dividing the equation into im real pa aginary and rts again, we obtain two formulas ere are th x and substitute it in (53) and (54), we can     In this way, all non-eigen values of universal momentum operator are real numbers, but two of them are not real momentums of micro-particles.Using universal momentum operator and kinetic operator to calculate particle's kinetic energy, the results may be differe to that three partial quantities of universal momentum operator do not commute with each other.

 
Q x needs to be determined, but we still have two Equations ( 53) and (54) as follows Therefore,

 
Q x is not unique, unless two equations are compatible.From both equations, tain we ob The formula has infinite solutions.The simplest one is , so we G ib  ag is a purely inary number.In the case, universal momentum operator is we can define proper kinetic operator.
According this kind of definition, unive tum operator is not the Hermitian operator ever, as discussed before, the restriction of Hermitian op s, we t define proper omentum operator in one dimensional space in quantum mechanics, though (62) rsal momenagain.Howerator is neither necessary nor possible for non-eigen functions.Most important is that the non-eigen values and average values of operator should be real numbers.Only in this way, the descriptions of physical processes can be consistent in different representations.The universal momentum operator can do it.
Although the deductions above are based on the wave function of single particle, we can also do it for multiparticle's functions.We do not discuss it here.According to (44), the commutation relations between coordinate and momentum operators are unchanged with As discussed before, the non-commutation of operators does not mean the uncertain of physical quantities simultaneously.

The Average Values of Universal Momentum Operator
The average value of universal momentum operator is a real number.According to (57)-(59), we have e first item on the right side o  The result is similar to (41).

The Definition of Universal Coordinate Operator in Momentum Space
As mentioned above, the average value of coordinate operator in momentum space is a complex number.So the coordinate operator in momentum space shoul be revised.Similar to (44), we define the universal di

 
x  p is a real number and   R  p is a complex number in general to satisfy following equation Therefore, the results that universal coord tors are acted on the non-eigen wave functions are real nu ue ors over non-eigen functions are also real numbers.We have inate operambers.The average val s of universal coordinate operat The Fourier transform of ( ) By considering (68) and (71), we transform (70) into coordinate space for description and obtain The result is similar (41) and (65).

The Definition of Universal Momentum Operator in Spherical Coordinate System
Based on the definition of universal momentum in the Descartes coordinate system, we can define universal momentum operator in spherical coordinate system.Similar to (44), we define Here By considering ( 13) and (76), we get We take By substituting (80) into (79) and dividing it into imaginary part and real part, we have We have three Q  but only have two Equation and (82), so we Q s (81)  can be chosen arbitrary.By solving the motion question of quantum mechanics, we know the form of wave functions.Let r p , p and p   are the partial momentum of a particle.Their forms can be determined by following formulas Copyright © 2012 SciRes.JMP X. C. MEI, P. YU 462

ngle . The efinition of Universal Angle Momentum Operator
In quantum mechanics, angle momentum operator is related to momentum operator with relation ˆˆL r p   .If we think that micro-particle's momen servable so that its values are unimportant, th momentum of micro-particles are related to atomic magtic moment and the magnetic moments are measurable ectly.It is obvious that when tors act on general wave function, their non-ei and average values may be complex numb r angel momentum operators in the Descartes coordinate system is By introducing spherical coordinate, (88) can also be written as ˆsin cos ˆcos si

In which ˆz
L is the eigen operator of stationary wave function nlm  of hydrogen atom with eigen values m , but ˆx L and ˆy L 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 . In order to make the non-eigen values of ˆx L and ˆy L real numbers, we should refine they.Based on (44), universal angle operators are ˆx Here Q  is determined by ( 53) and (54) with z z Q p  .By acting universal angle operators on common no eigen wave function, we get n-     g  x and p  x are also determined by (55) and Copyright © 2012 SciRes.JMP (56).In this way, the non-eigen values and average values of universal angle momentum opera bers.We do not discuss the forms of universal angle momentum operators in curved coordinate reference systems here.

Auxiliary Momentum and Auxiliary Angle Momentum
We use the square of mome operator struct the motion equation of ntum mechanics but use ˆr p  to construct angle mo ntum, in quantum me-tors are real umntum qua me 2 p to conchanics.So the kinetic energy T and angle momentum L 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 m 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 [11].
The relations among them ar Auxiliary angl omentum ˆh L is related to spin S .W iscuss th lation below and p that spin is related to the partial angle momentum of micro-particle which current momentum opera establishing the relation between them, the essence of spin can be explained well.Although the essence of spin is still an enigma at present day In quantum mechanics, spin is considered as ang mentum actually.However, angle momentum is a kind of vector.So it is an unreasonable thing for spin vector to only take two projection values 2 at ar  bitrary 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 cos with values 1-1  .Because spin is always coupled with magnetic field, the correct understanding should be that if we take z axis as the direction of magnetic field, the projections of spin at z axis direction take two values 2  .At other directions, spin's projection should be related to cos .We will see below that it is just due to the hypothesis that t tic fie he projections of spin can only take two values, the Bell inequality can not be correct.
According to current quantum mechanics, magnetic moment caused by spin in magne ld is .So spin gyro-magnetic ratio is two times of orbit gyro-magnetic ratio.It indicates that spin is not normal angle momentum.Let μ represent total magnetic momen magnetic field, as shown in Figu t of charged particle in re 1, magnetic moment μ precesses around total angle momentum J , so μ considered as an immeasurable quantity in current is theory.What can be measured directly is the partial quantity g  of μ at the direction of angle momentum J .We have or.Let is the Lande fact p J represent new total angle momentum after auxiliary angle momentum is considered, suppose that the relation between p J and J is The formula above gives the Lande factor a new physical meaning.In this way, magnetic moment of particle becomes By introducing new total angle mentum mo p J , the on m directi of magnetic moment is the sa e as p J .We do not need the assumption that μ precesses around angle total momentum p J again.In experiments, pa ticle's angle momentum can not be observed directly.What can done is m r be agnetic moment.We obtain angle momentum through measurement of m So int between auxiliary angle momentum and spin is discussed below.Because the mo are restrained on a plane in center force agnetic moment actually.roducing new total angle momentum does not cause inconsistent.Inversely, angle momentum theory of micro-particle becomes more rational.
The relation tions of objects fields, we suppose that p S , J , p J , 0 L and h L are located on a plane.As shown in Figure 1, we have following relations From ( 103), ( 105) and (106), we get From ( 107) and (108), we obtain We know the values of J , S an tum mechanics.The Lande factor d 0 L from quang can be obtained from experiment.So h L and  can based on (109) and (110).For example, when 0 0 L be determined In this way, we explain that spin gyro-magnetic ratio is two times of orbit gyro-magnetic ratio.It indicates that h L is real angle momentum, in stead of spin.We should consider S as a kind of quantum number.Based n it, we can obta real angle momentums of micro-particles.As shown in Figure 1, we have relation In fact, in quantum mechanics, we use the Pauli equations to describe the Zeeman effects of spectrum splits in magnetic fields.T In the f mulas, we have   It means that the partial quantity of angle momentums at z axis direction is  actually.The reason to write the projec direction of space as tion of spin at any 2 S   is only for m of matheatical convenience.Speaking correctly, we should consider 2 S   as a kind of quantum number based on it we can obtain correct angle momentums of micro-par-ticles.
In general situations 0 0 L  , the angle mome micro-pa omes complex.For example, for the 2P and

The Real Reason That Bell Inequality Is
Not Supported by Experiments

The Deduction of the Bell Inequality
Based on the clarification of spin's essence, we can explain why the Bell inequality not supported by experiments.It is due to the misunderstanding of the spin's projections of micro-particles.Let's first descri the deduction process of the Bell inequality briefly [12].Suppose there is a system composed of two particles with opposites spi 2  individually, so the total spin of system is zero.Spin operator is  and we take The wave function of the system is By substituting (117) in ( 116), the calculating result of qua um mechanics is nt The average value of association operator on the ensemble function of hidden variable is Because two particles have opposites spins, when â and b are at the same directions, according to (118), we have Let ĉ be another direction vector, because of and substitute it into (124), get absurd result 120

, a lo rime
Most of them support quantum mechanics does not support the Bell inequality.So according to current point of view, hidden variables do not exist and the deterministic descriptions of microparticles are considered impossible.
Based on discussion above, we can say that real reasons to make the Bell inequality impossible is the misunderstanding of spin's projection.According to current understanding, the projections of spin at arbitrary directions are alw not exist in physical space.Suppose vector with mode A , if the projection of A at direction k is  A , the projections at direcchose z axis as the di tions i and j can only be zero.No any physical vector can have same projections at different directions in real space.
In fact, spin is always coupled with magnetic field when we construct the interaction Hamiltonian.The correct understanding of spin is that if we rection of magnetic field, the projection of spin at z axis direction of is 2  .That is to say, the projection of spin at the direction of magnetic field is 2  , rather than at arbitrary direction!Speaking strictly, in quantum mechanics, sp is coupled with magnetic field in the value of in 2  .If the direction of magnetic between spin d magnetic field is rtain.At other direct , we can not observe the physic l affec ons of spin.In fact, in current quantum mechanics, matrix operators are used to describe spin with field is cer-tain, the coupling an ce ions a ti By acting them on spin wave functions 1 2 1 2 so the value of spin is actually It is more proper to consider S and s as a kind of quantum n mber, in stead of spin a le momentum itself.In light of mathematics stric ly, as a p tical physical quantity, the projection of spin operator at â direction is ˆˆˆˆˆˆˆû ng t rac    .This result is different from ( 131)-( 1d 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 m having nothing to do with hidden hypotheses coincides with quantum mechanics, nor coincides with classical mechanics and any logic of mathematics and physics. In fact, according to this paper, spin is not real p cal quantity which can be determined directly.What can be 34) an athematics, .It neither hysicdone in experiments is magnetic moment.Magnetic moment is related with angle momentum directly.According to (112), the projection of auxiliary angle momentum at α direction is The eigen values of 2 0 L is  2 1 l l   .Suppose that the angle between 0 L and a is  , (135) becomes  [14].In eduction of quality for sses, photon's polariza es are cons  .When a photon passes through a polarize, its polarization value is considered to be 1  .When a photon does not pass, its polarization value is considered to be -1 [15].The deduced Bell inequality can not be supported by experiments.The reason is the same as the projections of spin.In fact, light's polarization is macro-concept.It is meaningless to talk about polarization about a single photon.We can only discuss light's polarization from the macro-viewpoint of statistical average.

trom beam of polarization light di
We know in classical optics that the polarization direction of light is defined as the vibration direction of elecagnetic field.When a passes through polarizer, the vibration direction of electromagnetic field is changed.For example, when a beam of polarization light passes through calcite, it becomes two lights named e light and o light.Their vibrationrections are different from original one.If we must define the concept of polarization for a single photon, we can only consider its polarization direction as the direction of electromagnetic field.When a photon pass through a polarizer with an angle  , the vibration direction of electromagnetic field turns an angle  .In this case, we should think that photon's polarization becomes cos .That is to say, even in classical optics, for photons which pass through polarizer, their polarization is considered as cos , in stead of 1  in general.For photons which are reflected without passing through polarizer, their polarization values depend on the angle of reflection, in stead of 1   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 1   or 1  under arbitrary situation.
Because photon's polarization values are always taken 1  , the mistake is the same as made for particle's spin when we deduct the Bell ine uality of photon's polarizations correlation.It is also inevitable that this kind of Bell q Copyright © 2012 SciRes.JMP X. C. MEI, P. YU 468 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 les.

The Elimination of EPY Momentum P radox in Quantum Mechanics
The momentum paradox of Einstein By solving the motion equation of quantum mechanics, we obtain particle's energy  [16].
The problem is that according to discussion above, we can not define rational momentum operator for q mechanics in the situation of one dimension.If current momentum operator on wave functions (139) or (140), the obtained non-eigen value is an imaginary nu   t be solved up to now days, some persons even thought that the logical foundation of quantum uantum we act mber.We have According to discussion before, we can not find a proper momentum operator for micro-particle which moves in one dimensional space.That is to say, in one dimensional infinite trap, particle's momentums can not be 1 p  .
Some one proposed the explanation of boundary condition trying to eliminate EPY paradox [6].According to this explanation, when (139) is written in the form of (140), 1 e is the normalized wave function in a box, rather than the wave function of free particle in the region without boundary.The restriction of boun dition would produce a great influence on the nature of wave functions.If This kind of explanation has its reason but has not touch the essence.Because (138) is the wave function in coordinate space, the definition of momentum operator of quantum mechanics is unrelated to boundary condition, no matter whether boundary conditions are finite or infinite, the actions of momentum operator on wave fu are effect and certain.In fact, the boundary conditions have been considered when we solve the motion equation of quantum mechanics.That is to say, the influence of boundary condition has been contained in the wave functions.So it is unnecessary for us to consider boundary condition.When  , so particle's momentum is not  p 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.

Conclusions
According to current quantum mechanics, when the operator is acted on non-eigen function, non-eiegn values and average values of momentum operator are complex numbers in general.In theses cases, momentum operator is no longer the Hermitian operator.Though we can make the average values real numbers in momentum representation, it lea consistency of coordinate space momentum space.Using momentum operator and kinetic operator to calculate momentum of micro-particles, the results may be different.It means that kinetic operator and momentum operator of quantum mechanics are not one-to-one correspondence.Besides momentum operator, other operators in quantum mechanics, just as angle momentum problem ds to in and operator, also have the same problems.These s have not caused the attention of physicists at e these problems involve the ration-ntum mechanics.Th echanics," 1996.
By introducing the concept of universal momentum operator, all of these problems can be solved well.Under the premises of ensuring kinetic operator to be invariable, non-eigen values and average values of universal momentum operator are real numbers.In this way, the description of physical processes can be equivalent really in coordinate representation and momentum representation.For eigen wave function, universal momentum operator restores to the current Hermitian operator.For general situations, universal operator is not the Hermitian operator because it is unnecessary.The most important thing in physics is that the average values of operator should be real numbers.Using universal momentum operator and kinetic operator to calculate micro-particle's kinetic energy, the results are still different, but we can get consistent result through proper method.Only in this way, we can reach logical consistency for qua e problems of momentum operator's definition in the curved coordinate reference systems can be solved well.
Therefore, we need to introduce the concept of auxiliary momentum and auxiliary angle momentum.The relation between auxiliary angle momentum and spin is deduced and the essence of micro-particle's spin is revealed.Spin is related to auxiliary angle momentum of micro-particle which angle momentum operator can not describe.We understand real reason why the Bell inequality is not supported by experiments.It is due to the misunderstanding of spin's projections and photon's polarizations.No any real angle momentum can have same projections at different directions in real physical space.The violation of the Bell inequality has nothing to do with whether or not the hidden variables exist actually.In this way, the EPY momentum paradox can also be eliminated thoroughly.The logical foundation of quantum mechanics becomes more perfect.


the Bohr first orbit radius of hydrogen atom and  is the angle frequency of harmonic oscillator.When momentum operator is acted on 100 and explain why the results are not the same when we use kinetic operat tum operator to calculate the kinetic en particles.By consider (55), we have or and momenergy of micro-

1 Q x and   2 Q
nt.It is dueIf micro-particles moves in two dimensional spaces, both   x are determined just by two Equations (53) and (54).Meanwhile, determined by kinetic operators.The relations between them are still shown in (56).If particles move in one dimensional space, only one

3 p
average of particle's momentum.By transforming into momentum space, we have operator x and momentum operator p in momentum space as

(
71)-(73) on general non-eigen wave functions, we get non-eigen values of real numbers with ˆr define univers moentum operator in spherical coordinate system.By the e method, we can define universal momentum

Figure 1 . 4 . 3 .
Figure 1.Auxiliary angle momentum, spin and magnetic moment.4.3.The Essence of Micro-Particle's SpinIn order to explain the fine structure of light spectrum, we assume that electron has a spin Ŝ .The projections of spin can only take two values

e shown in Figure 1
can not describe.By It is difficult to understand the concept of micro-particle's spin from the point of view of classical mechanics.
, the ratio that atomic gnetic moment divided by angle momentum is called as gyro-magnetic ratio.According to (101), we have the ratio

2 a
 as unite.Let A represent the , b at spin's measurement value of particle 1 at a direction B represent the spin's measurement value of particle 2 b direction.The average value a b A B  of association operator   1 2 ˆÊ a b a b . Suppose that there exists hidden variable  which m nistic motion possible for micro-particle.The ensemble distribution function of hidden variable is akes determi-


and a , having nothing to o with b .Meanwhile, the measurement about partic 2 also depends on ˆ d le  and b , having nothing to do with â .So for arbitrary â and b , we have

Copyright
a and z a are the projections of unit vector â at x , y and z axis directions.We hav e So the formula (131) means that the projections of spin operator at arbitrary direction take the values between 1


Based on (133) and (134), we get (118).However, in the deduction of the Bell inequality, we let .It means that the projection of electron's spin at arbitrary direction can only take1 are the wave functions with infinite boundary, after they are transformed in momentum space, the wave functions should be the  function with ntums which is the same with the result in coordinate space.But if particles are located in infinite trap with the restriction of boundary condition, (145) can mome not hold.
In this way, we reveal the essence of micro-particle's spin.Spin is not real angle momentum though it related to angle momentum.Due to the incompleteness of angle momentum operator in quantum mechanics, we introduce the concept of spin.The quantities of S , L and h

The Real Reason That the Bell Inequality Is Not Supported by Experiments
we act momentum operator on wave function, the result we get is what it should be.By acting p It indicates that we only have two discrete momentums.The EPY momentum paradox has not eliminated really by considering boundary condition.According this paper, though (140) represents the wave function in coordinate space, the momentum opera-