The Definition of Universal Momentum Operator of Quantum Mechanics

Journal of Modern Physics, 2012, 3, 451-470 http://dx.doi.org/10.4236/jmp.2012.36062 Published Online June 2012 (http://www.SciRP.org/journal/jmp)

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

Xiaochun Mei, Ping Yu Institute of Innovative Physics in Fuzhou, Department of Physics, Fuzhou University, Fuzhou, China Email: [email protected], [email protected]

Received March 21, 2012; revised April 12, 2012; accepted April 25, 2012

ABSTRACT 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 micro- particle’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.

Keywords: Quantum Mechanics; Universal Momentum Operator; Universal Angle Momentum Operator; Hermitian Operator; Self-Adjoint Operator; Spin; Bell Inequality; Hidden Variables

1. Introduction tor are not one to one correspondence. Secondly, in the curved coordinate system, momentum operator can not Since quantum mechanics was established, its correct- ness has been well verified. But there exists serious con- be defined well though we can define kinetic energy troversy on its physical significance. Many people be- operator well. That is to say, except in the rectangular lieve that quantum mechanics has not been well ex- coordinates, the definition of momentum operator is still plained up to now days. However, the mathematical struc- an unsolved problem in quantum mechanics. ture of quantum mechanics is commonly considered com- With operators of quantum mechanics acting on the plete and perfect. It seems difficult to add additional eigen functions, we obtain real eigen values. However, if things to it. Is it true? It is pointed out in this paper that operators act on the non-eigen functions, the results are the definition of momentum operator of quantum me- complex numbers in general. We call theses complex chanics has several problems so that it should be im- numbers as non-eigen values. For non-eigen functions, proved. the operators of quantum mechanics are not the Hermi- In this paper, we first prove that using kinetic energy tian operators. In general, the average values of operators operator and momentum operator to calculate micro- on non-eigen functions are complex number, unless they particle’s kinetic energies, the results are different. That are zero. is to say, kinetic energy operator and momentum Opera- Because the non-eigen values of complex numbers are

Copyright © 2012 SciRes. JMP 452 X. C. MEI, P. YU meaningless in physics, the non-eigen functions have to value of universal momentum is real number. The EYP be developed into the sum of the eigen functions of momentum paradox can also be resolved thoroughly. operators or the superposition of wave functions. The After universal momentum operator is defined, we can eigen function of momentum operator is the wave func- define universal angle momentum operator. Because there tion of free particle. Hence, a very fundamental question is a difference between calculated value and real mo- is raised. We have to consider a non-free particle, for ex- mentum value, the concepts of auxiliary momentum and ample an electron in the ground state of hydrogen, as the auxiliary angle momentum are introduced. The relation sum of infinite numbers of free electrons with different between auxiliary angle momentum and spin is revealed. momentums. This result is difficult in constructing a It is proved that spin is related to the supplemental angle physical image, thought it is legal in mathematics. Be- momentum of micro-particle which orbit angel momen- sides, it violates the Pauli’s exclusion principle. It is dif- tum operator can not describe. The fact that spin gyro- ficult for us to use it re-establishing energy levels and magnetic ratio is two times of orbit gyro-magnetic ratio, spectrum structure of hydrogen atoms. as well as why the electron of ground state do not fall Besides, some operators of quantum mechanics have into atomic nuclear without orbit angle momentum can no proper eigen functions, for example, angle momentum be explained well. ˆ ˆ ˆ operator Lx , Ly and Lz in rectangular coordinate By the clarification of spin’s essence, we can under- system. We can not develop arbitrary functions into the stand real reason why the Bell inequality is not supported sum of their eigen functions. By acting them on arbitrary by experiments. The misunderstanding of spin’s projec- functions directly, we always obtain complex numbers. tion leads to the Bell inequality. No any real angle mo- Can we say they are meaningless? mentum can have same projections at different directions

The descriptions of quantum mechanics are indepen- in real physical space. The formula ABab   1 dent on representations. In momentum representation, the does not hold in the deduction process of the Bell ine- positions of momentum operator and coordinate can be quality. The result that the Bell inequality is not sup- exchanged with each other. It is proved that when the ported by experiments has nothing to do with whether or non-eigen wave functions in coordinate space are trans- not hidden variables exist. formed in momentum space for description, the problems of complex non-eigen value and complex average value 2. The Necessity and Possibility to Introduce of coordinate operator occurs, though the problem of Universal Momentum Operator in complex non-eigen value of momentum operator dis- Quantum Mechanics appear. 2.1. Inconsistency in Calculating Kinetic In addition, there exists a famous problem of the EYP Energy Using Momentum and Kinetic momentum paradox in quantum mechanics [1-3]. Be- Operators cause it can not be solved well, someone even thought that the logical foundation of quantum mechanics was The Hermitian operators are used to represent physical inconsistent. quantities in quantum mechanics. The result of Hermitian Because angle momentum operator is the vector pro- operator acting on eigen function is a real constant. Mo- duct of coordinate operator and momentum operator, the mentum operator and its eigen function are problem also exists in the definition of angle momentum i pxEt operator. For example, we can not define angle momen- piˆ    x , tAe   (1) tum operator in curved coordinate system well at present. ˆ The physical image and essence of micro-particle’s spin We have pt xpx,, t. The momentum p is a constant. However, more common situation is that is still unclear at present. wave functions are not the eigin functions of operators. Therefore, the momentum operator of quantum me- In this case, we have chanics can not represent the real momentums of micro- particles. It needs to be improved. The concept of uni- ptiˆ xxpxx,,,,   t  t  t (2) versal momentum operator is proposed to solve theses problems in this paper. px ,t is a constant and we call it as the non-eigen Using universal momentum operator and kinetic opera- value of momentum operator. If   x describes a sin- tor to calculate the kinetic energy, we can explain the gle particle, according to definition, px should re- problem of inconsistency as mentioned before. In curved present the momentum of particle. Because momentum is coordinate system, we can define momentum operator the function of coordinate, is it consistent with the un- rationally. When universal momentum operator is acted certainty relation? Or is the function form of px  on arbitrary non-eigen wave functions, the non-eigen meaningful? This problem involves the understanding of values are real numbers. In coordinate space, the average real meaning of the uncertainty relation. We will discuss

it in the end of this section. According to (5), we have T11 E, i.e., electron’s ki- Because (2) is only a calculation formula of mathe- netic is equal to its total energy, so (5) is wrong. matics and the definitions of operator and wave function If momentum operator is acted on the wave function are alright, we should consider it is effective. We prove  0 of linear harmonic oscillator, we obtain in this section that the results are different by using mo- pxˆ  ixxpx 2  mentum operator and kinetic energy operator to calculate 000 (8) the kinetic energies of micro-particles. Taking ground pix 2 state wave functions of hydrogen atom 100 and linear Momentum p is also an imaginary number. Based harmonic oscillator  0 as examples, we have on it, particle’s kinetic energy is

1 ra0 100 re,,  2 2 42 22 πa3/2 pxmx  0 (3) T  (9) 1 22mm 2   22x  xN e2 00 It means that particle’s kinetic energy is a negative number which can not be true. Acting kinetic operator on Here a 22me and   m  in which a is 0  0 it, we have the Bohr first orbit radius of hydrogen atom and  is 22 the angle frequency of harmonic oscillator. When mo- ˆ  d Tx00 2  mentum operator is acted on 100 , we get 2m dx mx22 1 i r 00x Tx  (10) ra0 22 piˆ1 100   e  100 p 1 100  πa3/2 ar 0 0 (4) mx22 TEUx   i r 0 22 p1  ar0 Here E0 is the energy of ground state harmonic os-

It is obvious that pr1  can not be the momentum of cillator and Ux  is potential energy. It is obvious that electron in ground state hydrogen atom. Despite pr1   the calculating results of two methods are different. Ac- is an imaginary number, if it is electron’s momentum, the cording to (9), we have TUx  which is certainly kinetic energy should be wrong. In fact, this problem exists commonly in quantum mechanics. Kinetic operator and momentum operator do p2 24me 1 not have one-to-one correspondence, so that we can not T1  22 (5) 2m 22ma0  determine the non-eigen values of momentum operator uniquely. Because kinetic operator is aright, we have to It indicates that kinetic energy is a negative number. Of cause, this is impossible. If kinetic operator is acted improve momentum operator to make it consistent with kinetic operator. on the wave function 100 , we obtain

2 ˆ  2  1    2.2. The Difficulty to Define Momentum Tr100 2    sin  2mr rrsin    Operator in Curved Coordinate System 1 2  In the current quantum mechanics, the definition of mo-   (6) 22 100 mentum operator in curved coordinate system is an un- sin   solved problem [4]. Several definitions were proposed, 2  12 but none of them is proper. If we claim that three partial 2 100 T 1 100 2mara0 0 quantities of momentum operator are commutative each other, the definition should be Therefore, the kinetic energy of electron in ground state hydrogen atom is   piˆˆ pi  piˆ   (11) r r   2412 me e2 T    E Ur (7) 1221 However, it is easy to prove that pˆ and pˆ are not 2mara0 0 2 r r  the Hermitian operators. Their non-eigen values are ima- (7) is obviously different from (5). (7) is just the formula ginary numbers in general. Most fatal is that we can not of energy conservation, in which E1 is the total energy construct correct kinetic operator based on (11). In clas- of ground state electron and Ur is potential energy. sical mechanics, the Hamiltonian of free particles in

Copyright © 2012 SciRes. JMP 454 X. C. MEI, P. YU spherical coordinate system is kij1 gijgg ij ij ij g    (18) 2 qqq 112212 iii H Tppr 22 2p (12) 2m rrsin  According to this definition, the kinetic operator in According to the correspondence principle between curved coordinate reference system can be written as classical mechanics and quantum mechanics, by consi- 22    dering the definition (11), the kinetic operator of quan- Tgˆ ij  k (19) 2mqqqij  tum mechanics is ij k

22 2 2 In spherical coordinate reference system, (19) is just ˆ  11 T 222   22 2 (13) (14). But this result was also criticized to have inconsis- 2m rr rsin tent for scalar and vector fields [7]. Meanwhile, according to (17), the result of D acting on scalar field is (15). However, the kinetic operator of quantum mechanics i Therefore, the non-eigen values and average values of in spherical reference system is actually operators pˆ r and pˆ may still be complex numbers. 2 All problems existing in the Descartes coordinate system ˆ  2  1    Tr2    sin  would appear in the spherical coordinate system. 2mr rrsin    (14) 1 2  2.3. The Problems of Complex Number  22 sin   Non-Eigen Values of Momentum Operator The Hermitian operators are used to describe physical (13) and (14) are obviously different. Another defini- quantities in quantum mechanics. The eigen values of the tion of momentum operator is [5] Hermitian operators are real numbers. But the premise is  1 that the operators should be acted on eigen wave func- piˆ r  tions. However, we have seen many situations in quan- rr tum mechanics that wave functions are not the eigen  ctg functions of operators. For example, only the wave func- piˆ  (15)  2 tion of free particle is the eigen function of momentum  operator. All other non-free particle’s functions are not piˆ    the eigen function of momentum operator. In the coordinate space, when momentum operator is acted on non- Substitute (15) in (12), we get eigen functions, the obtained result, called as non-eigen values, are complex numbers in general. The average 2 ˆ  2  1    values of momentum operator in coordinate space are Tr2    sin  2mr rrsin    also complex numbers. These results are irrational and (16) can not be accepted in physics, unless the average values 11sin22  are zero. 22 2 sin 4sin  Let both   x,t and  x,t be arbitrary wave functions in coordinate spaces, according to the defini- We see that (16) has one item more than (14), so (15) tion of quantum mechanics, if Fˆ satisfies following is improper too. relation The covariant differential operator Di in differential geometry was also suggested to define momentum opera-  xxx,,dtFˆ t 3 tor in the curved coordinate reference system [6]. The  (20)  Ftˆxxx,, t d3 action forms of operator Di on scalar  and vector 

V j are individually we call it the Hermitian operator. The eigen equation of  ˆ the Hermitian operator is Fxx,,tF   t. It is Di qi easy to prove that the eigen value F is a real constant (17) We have  k DVijVVj ijk q  ˆ 3 i xxx,,dtF t (21) 2 ij  F  xxx,,tt d3 By considering the metric ddds  gqqij, we have   

  ˆ 3 1 ip'x/3 Ftxxx,,t d xp,,tt edp  (22)  3/2    3 2π   Fx ,,ttxx d (28) 1  xp,,tt  ep''x/3dp By considering (20), we get F   F , i.e., F is a 3/2     2π  real number. In this case, the average value Fˆ of opera-  The Fourier transformation and its inverse transforma- tor is also a real number. We have tion of non-eigen value of operator are ˆ  ˆ 3 Ft xx,,Ftdx 1  F xf,,tt peipx  d3 p (29)   32    Ftˆxx,,t d3 x (23) 2π   ˆˆ3 1  xx,,tF t dx F fp,,dtt Fxeipx 3 x (30)   32   2π  Suppose the action result of operator on the non-eigen function is Substitute formulas above in (27), we obtain the average value of operator on non-eigen function in momen- FˆxF,,ttt  x x,  (24) tum space. We call it as the non-eigen equation of operator Fˆ 1 Fˆ  pfpp,,ttte,i()pp p x p 3   and Fx,t as the non-eigen value of operator. It is 2π easy to prove that if wave function is not eigen function 333 3 of operator, non-eigen value F x,t may be a complex dddxpp d p number. In this case, operator will no longer be Hermi- 1  fp,,tt  p (31) 3   tian. We have 2π 3  xx,,tFˆ t dx  pp pdd33 pp  d 3 p  3 xFx x x (25) 1  33  ,,tt,t d fp,,,tt p p pt dd p p 3      Fx,,tt x d3 x 2π

 If F  x,t is a complex number, fp,t is also a 3  Ftˆxx,,t dx comp lex num ber too. Therefore, similar to th e situation in coordinate space, the average values of momentum  Fx*,,tt x x,t d3 x (26)   operator in momentum space is also a complex number   Fx*3,,tt  x dx which is meaningless in physics. In which the probability density 2.4. The Problems of Complex Values of xxx,,,ttt   is a real number. Obviously, Coordinate Operator in Momentum Space if Fx,t is a complex number, (25) and (26) are not equal to each other, so that (20) can not be satisfied. So On the other hand, the positions of coordinate and mo- Fˆ is not the Hermitian operator. Because Fx,t is a mentum exchanges each other when we describe physical complex number, the average value processes in momentum space. It is proved below that in momentum space, the problem of imaginary non-eigen Ftˆ   xxx,,Ftˆ  d3 value of momentum operator disappears, but the problem (27) of imaginary non-eigen value of coordinate operator 3   Fx,,tt x d x emerges. In momentum space, coordinate operator be-

comes xiˆ ~  p . When it is acted on the wave function is also a complex number too. Because the average value in momentum space, we have of operator in quantum mechanics is measurable quantity, complex average value is meaningless in physics, unless xˆ pp,,ti  p  t xpp,,t t (32) it is equal to zero. Let’s discuss the average values of momentum opera- Similar to non-eigen value of momentum operator in tor in momentum space. The wave functions   x,t in coordinate space, xp ,t is a complex number in ge- coordinate space and  p,t in momentum space satis- neral. So the average v alue of coordinate operator in mo- fies following Fourier transformation mentum space is a comple x number. We have

1 coordinate and momentum, we have so-called uncer- xpˆ   ,,,dttpxpt3 p (2π)3  tainty relation： (33) 1  xp,,tt pd3 p xp xx   pp   2 (34) 3   2π According to current understanding, (34) means that Here  p,t is a real number. However, in coordinate coordinates and momentums of micro-particle can not be space, the average value of coordinate operator is a real determined simultaneously. If it is true, the function rela- number. The problem of imaginary average value is tion px  is meaningless. transformed from coordinate space to momentum space. However, px  in (2) is only the result of mathe- The problem exists still, unless (33) is equal to zero. matical calculation. Because the definitions of operator All of these problems indicate that the definition of and wave f unction have no problems, how can we con- momentum operator in quantum mechanics should be sider the resu lt mea ningless? According to (2), as long as revised. The revised momentum operator should have we know the concrete form of wave function, we know real non-eigen values. The average values of momentum the momentum. It is unnecessary for us to measure mo- operator and coordinate operator should be real numbers mentum. How can we think that the coordinate and mo- no matter whether in coordinate representation or in momentum of micro-particle can not be determined simul- mentum representation. Using momentum operator and taneously? We need to discuss the real meaning of the kinetic operator to calculate the kinetic energies of mi- uncertainty relation in brief. cro-particles, the results should be consistent. If they are Firstly, the wave function   x,t describes the pro- not, we should have reasonable explanation. Only when bability of a particle appears at the point x at moment these are done, we can say that the definition of momen- t . Therefore, x is the accurate value of particle’s coor- tum operator is complete. dinate in theory. It is not the value o f measurement, for As mentioned before, when the non-eigen values of measurements always have error. Therefor e, x  x is operator are complex numbers, the operator is not the the difference between particle’s theoretical coordinate Hermitian operator any more. In fact, someone had seen and average coordinate. It is not the measurements error this problem and demanded that the operators in quantum of coordinate. In fact, it is actually the fluctuation of co- mechanics should be self-adjoint operators [8]. In fact, in ordinate about average value as that defined in classical his famous book “the principles of quantum mechanics”, statistics physics. Similarly, pp is also the difference Dirac only used real operator, instead of the Hermitian between particle’s accurate momentum and average operator. From the angle of mathematics, the relation be- momentum, or the fluctuation of momentum about the tween the self-adjoint operator and the Hermitian opera- average value. It is not the measu rements error of mo- tor is subtle. We do not discuss it in this paper. From the mentum. According to classical theory, fluctuation is also angle of physics, self-adjoint operator can be considered uncertainty. But this uncertainty is due to statistics, hav- as one which has the complete set of eigen functions [9], ing nothing to do with measurement. In this meaning, (34) so that its eigen values are certainly real numbers. But is not the uncertainty relation for a single particle recog- the Hermitian operator has no such restriction. The pro- nized in the current quantum mechanics. blem is that this restriction condition would greatly ef- Secondly, the strict uncertain relation in quantum me- fects the universality of operator and can not be satis- chanics is fied actually. For example, for all non-particle’s wave 2 222222 functions, momentum operator is not the one with self- xp xxpp       4 adjoint. However, we can prove that although we can not or (35) make momentum operators self-adjoint, we can make its 22 non-eigen functions to be real by redefinition. xx22 pp   It is proved below that though we can not make mo-     2 mentum operators is self-adjoint one, we can introduce 2 In the formula (35), x2 , x , pˆ 2 and p2 are the universal momentum operators to make their non-eigen average values, not the values for a single event. There- values be real numbers. fore, (35) represents the product of mean square errors of 2.5. The Problems Caused by Non-Commutation coordinates and momentum s. It is th e result of statistical of Operators in Quantum Mechanics average over a large number’s of events. For a single event, we may have xp  2 , xp 2 and As well know that momentum operator and coordinate xp 0 . Merely, their statistical average satisfies do not commutate with [,x piˆ ]  . Let x x x and (35). ppˆ p, in which x and p are the averages of In fact, because (35) only contains average values,

Copyright © 2012 SciRes. JMP X. C. MEI, P. YU 457 does n ot contain x and p , its forms and meaning is is the function of coordinates, if coordinates are known, completely different from (34). In the current quantum we can determine its kinetic energy. After that, we can mechanics, (35) is simplified into (34), then the uncer- determine its momentum. That is to say, we can deter- tainty relation is dec lared. This is improper. It is also the mine particle’s momentum by determining its coordinate. misunderstanding to consider the uncertainty relation as We consider particle’s energy operator again. Acting the foundational principle of quantum mechanics for (35) the operator on the non-eigen wave function of a single is only the deduced result of quantum mechanics. particle, we get: On the other hand, according to quantum mechanics,  time and micro-particle’s coordinates can be determined Etiˆxxxx,,,,  tEtt   (40) t simultaneously. The definition of velocity is V ddxt . As long as we determine particle’s coordinates at differ- It is easy to see that Eˆ commutates with xˆ , pˆ and ent moments, we can determine particle’s velocity and Tˆ , so particle’s energy, kinetic and potential energy can momentum pV m by calculation without doing mea - be determined simultaneously. For a particle in stationary surements. The more accurate the measurement of parti- state, we have certain energy cle’s coordinate, the more accurate the particle’s velocity p2  x and momentum. In t his meaning, where is the uncertain EUx (41) relation? 2m This kind of paradox exists commonly in quantum Here U  x is potential energy. As long as particle’s mechanics and the problem is more serious than what we coordinates are known, we know its kinetic energy, po- have imagined. Let’s discuss the commutation relation tential energy and momentum without direct measure- between coordinate and kinetic energy operator. The ments. However, on the other hand, because kinetic kinetic energy operator is energy operator and potential operator do not commutate

22 in genera l, we have ˆ pˆ  2 T  (36) 2 2m2m ˆ  2 Ur,, T rr U  r  2m Acting on the wave function of a single non-free parti-  (42) cle, we obtain 2  rr2U  0 2m 2 Ttˆxxxx,,,, 2  tTtt   (37) 2m For example, for hydrogen atom, we have Ur qr and get ˆ   It is easy to prove th at T and x are commutative 22Ur   q 10 r  q r  . with According to the current understanding, electron’s kinetic energy and potential energy could not be deter- xx,,Ttˆ    mined simultaneously (this is so-called quantum tunnel ˆˆ xxTt,,T xx  t (38) effect), (41) becomes meaningless. Howe ver, (41) is also deduced based the principle of quantum mechanics, how 2 xxxxTtTtˆ,,ˆ  x,t2 x 0can we say it is meaningless? If it is true, how can we 2m have the fine structure of hydrogen atom spectrum? In According to the understanding of quantum mechanics, fact, (41) is the formula of energy conservation. If it does the kinetic energy and coordinate of micro-particle can not hold, all theories and experiments of quantum me- be determined simultaneously, so it is meaningful to chanics become meaningless! write micro-particle’s kinetic energy as TT  x, t . In essence, quantum mechanics is a statistical theory According to (36), we can naturally obtain particle’s which involves a large number of micro-processes. Quan- tum measurement process always involves a large num- momentum ptxx,2, mTt  after its kinetic energy ber of micro-particles. Once statistical average is consi- is known. In fact, the kinetic operator and momentum dered, many problems do no exist any more. Our current operator are commutative with understanding on quantum mechanics may have same 2 foundational error. The standard explanation of quantum pTˆˆ,,0ˆ  p pˆ 2  (39) mechanics should have some essential changes. 2m  We will discuss the real meaning of the uncertainty So kinetic energy and momentum can be determined relation and the explanation of quantum mechanics fur- simultaneously. Because micro-particle’s kinetic energy ther in another paper. In this paper, we mainly discuss

Copyright © 2012 SciRes. JMP 458 X. C. MEI, P. YU the definition of universal momentum operator. We do can not find proper eigen functions, for example, angle ˆ ˆ ˆ not consider the restriction of the uncertainty relation momentum operator Lx , Ly and Lz in the rectangu- again and think that it is meaningful to act operators on lar coordinate system. Because they have no proper eigen non-eigen wave functions directly. In operator equation functions, we can not developed arbitrary functions into pxˆ,,,ttt px  x, p(x,t) represent microparti- the sum of their eigen functions. Acted them on arbitrary cle’s momentum at time t and position x . functions directly, we always obtain complex numbers. Can we say they are meaningless? T he universal mo- 2.6. The Fourier’s Series of Non-Eigen Wave mentum and angle momentum operators proposed in this Functions of Momentum Operator paper can solve these problems well. According to quantum me chanics, if the wave functions are not the eigen functions of operators, they should be 3. The Definition of Universal Momentum developed into the eigen functions of operators. The ei- Operator gen function of momentum operator is that of free parti- 3.1. The Definition of Universal Momentum cle. For the stationary state wave functions of non-free Operator in Coordinate Space particles, we have In order to make all non-eigen values of momentum  ii pxk 1 px xpAepp ed3 operator be real numbers, we need to redefine the mo-   k 32   k 1 (2π) mentum operator of quantum mechanics. In the Cartesian (43) coordinate reference system, we write the partial quantities of universal momentum operator as This is actually the Fourier’s transformation of wave function which is legal in mathematics. It can be consi-  piˆ   Gxx  Q (44) dered as the principle of superposition principle of wave   x function in quantum mechanics. If  x describe a single particles, for example, an In which  1, 2, 3 is the index of partial quantities, electron in the ground state of hydrogen atom, it repre- Q  x is real number with its form to be decided. sents the momentum distribution of an electron in hy- G  x is complex number in general to satisfy follow- drogen atom [10]. But it does not mean that a non-free ing relation electron is equivalent to infinite number of free electrons  with di ffere nt momentums and energies. This is impossi- x G xx  (45) x  ble in physics. In fact, there are only two electrons with  opposite spins in the ground states of hydrogen. If (43) For the purpose of universality, the functions here can describes the hydrogen atom of ground state, it violates be both the stationary state  n x and the superpose- the Pauli’s exclusion principle. It is impossible for us to tion state of  n  x . If we want to connect Q  x use so many free electrons with different energies to with particle’s momentum, it should be stationary state construct hydrogen atom’s energy levels and the spec-  n  x . By solving the motion equation of quantum trum structures. mechanics, we can obtain the concrete forms of wave The reason we write the wave function of a single par- functions, so we can determinate t he forms of G  x ticle in the form of (43) is due to the definition of mo- based on (44). By ac ting universal momentum operato r mentum operator which is only effective to free particles. on common wave functions and considering (45), we It is ineffective when it is acted on the wave functions of obtain the non eigne equation and the non-eigen values non-free particles due to complex non-eigen values. If we of real numbers can find proper momentum operator to describer non-free pQˆ xxx (46) particle’s momentums, it is unnecessary for us always to   write the wave function of a single non-free particles in The average values of universal momentum operator the superposition form of free particle’s wave functions. are also real numbers In fact, the eigen function of kinetic energy operator is also the wave function of free particle. We do not need to ˆ  ˆ 3 pp  xxxd write arbitrary wave function as the form of (43) for ki-   xxxxQ d 3 (47) netic operator. The reason is just that when kinetic opera-     tor is acted on arbitrary wave functions, the results are 3   Q xxx d always real numbers. In fact, for some operators of quantum mechanics, we If   x is the eigen function of momentum operator,

we take Let Qp33 xx    and substitute it in (53) and (54),

we can determine Q1  x and Q2 x . In general, ip p x and p x are different from Q x and Q x . G xp Qx (48) 1   2   1  2    We assume (44) becomes the current definition of the Hermitian pQg111 xxx operator. W e now discuss the concrete forms of Q x (56)    pQgxxx below. According to (44), the kinetic opera tor of micro- 222   particles is g1  x and g2  x are known due to the fact that Q x , Q x , p x and p x are known. In 1 3 1   2   1   2  Tpˆ  ˆ 2 this way, we dete rmine the concrete form of universal 2m    1 momentum operator (44) and explain why the results are 3  2 1 22G not the same when we use kinetic operator and momen-    2GG  (49)  2 tum operator to calculate the kinetic energy of micro- 2mx 1  x x particles. By consider (55), we have 2  Q  Qi 22 Q  GQ    xx    piGQˆ11 nn    x    x1  However, practical kinetic operator should be actually Qpg111nn (57) 2223 2  ˆ pgˆ11nn p 1 T     2 (50) 22mm 1 x    By comparing (49) with (50) and considering (45), we piGQˆ 22 nn  x  2 x get  2 

Qpg222nn (58) 3  22GQ2   GQ  i0 (51) pgˆ 22nn p 2  1  xx    Because G x is known complex number, we sepa- piGQˆ33 nn  x  3 x rate it into imaginary and real parts and write it as  3 

Qp33nn (59) Ga xxx   ib  (52) ppˆ33nn Substituting it in (51), and dividing the equation into In this way, all non-eigen values of universal momen- imaginary and real pa rts again, we obtain two formulas tum operator are real numbers, but two of them are not 3 real momentums of micro-particles. Using universal mo- 2222a   ab Q 0 (53) mentum operator and kinetic operator to calculate parti-  1 x  cle’s kinetic energy, the results may be different. It is due to that three partial quantities of universal momentum 3  bQ operator do not commute with each other.  20ab (54)  1 xx If micro-particles moves in two dimensional spaces,

both Q1  x and Q2  x are determined just by two There are th ree Q x needs to be determined, but    Equations (53) and (54). Meanwhile, p1 x and p2  x we only have two Equations (53) and (54). Therefore, are determined by kinetic operators. The relations be- one of them can be arbitrary . Let p  x be the partial tween them are still shown in (56). If particles move in momentum of particle, we ca n decide its value by acting one dimensional space, only one Q x needs to be kinetic operator on stationary state wave function and determined, but we still have two Equations (53) and (54) have as follows 1 2 22 p TTx ˆ  n  2222abQ  n 2  ab Q 02 ab  0 (60) nn22mmx (55) xxx pmT 2  Therefore, Q x is not unique, unless two equations

Copyright © 2012 SciRes. JMP 460 X. C. MEI, P. YU are compatible. From both equations, we ob tain 1 ppˆ  pppp g 9/2       2π ba22 20ab b  a (61) i pp p  x xx  x e dd333xpdp d 3 p 1 The formula has infinite solutions. The simplest one is pgpp pp  32    a  0 , so we have Gib and Qb  . That is to say, 2π G is a purely imag inary number. In the case, universal  pp pdd33 pp d3p momentum operator is 1 pgppppppp dd33  32    piˆ  ibxbx   2π x (62) (65) pbˆ  xpxgx  The result is similar to (41). So in general situation s, we can no t define proper momentum operator in one dimensional space in quan- 3.3. The Definition of Universal Coordinate tum mechanics, though we can define proper kinetic Operator in Momentum Space operator. As mentioned above, the average value of coordinate According this kind of definition, universal momen- operator in momentum space is a complex number. So tum operator is not the Hermitian operator again. How- the coordinate operator in momentum space should also ever, as discussed before, the restriction of Hermitian be revised. Similar to (44), we define the universal coor- operator is neither necessary nor possible for non-eigen dinate operator xˆ and momentum operator pˆ in functions. Most important is that the non-eigen values   momentum space as and average values of operator should be real numbers. Only in this way, the descriptions of physical processes  xiˆ Rp x p can be consistent in different representations. The uni- p (66) versal momentum operator can do it. ppˆ  Although the deductions above are based on the wave  function of single particle, we can also do it for multi- Here x  p is a real number and R p is a com- particle’s functions. We do not discuss it here. According plex number in general to satisfy following equation to (44), the commutation relations between coordinate  pp R p (67) and momentum operators are unchanged with   p  Here   p is the wave function in momentum space. xpˆ  x, i  g xp x ,    The relation between   p and  x is the Fourier x (63) transform with i  px 1 3 xi , i pxx e  d (68) x  3/2    2π

As discussed before, the non-commutation of opera- We can determine R  p based on (67). Thus, we tors does not mean the uncertain of physical quantities have simultaneously.  xiˆpppp R   x p p (69) 3.2. The Average Values of Universal  x pp Momentum Operator   Therefore, the results that universal coordinate opera- The average value of universal momentum operator is a tors are acted on the non-eigen wave functions are real real number. According to (57)-(59), we have numbers. The average value s of universal coordinate ˆ  3 operators over non-eigen functions are also real numbers. p  pxxxx  d (64) We have  g xx  xd3 x   ˆ ˆ 33 xx pppppp  dd  x p (70) The first item on the right side of equal sign is the real 3  x ppp d average of particle’s momentum. By transforming into   momentum space, we have The Fourier transform of x ()p is

i px ˆ 1  3 T  xxpx ed (71) 32  22 2π   2 11      22 r sin  2 By considering (68) and (71), we transform (70) into 2smr rrsin     in   coordinate space for description and obtain   G2G  QQQ2222 22 G rr   G2 G i rrr   1 pxx x  rrr xxˆ   xx  e ddd333 pxxx d3  3  2 G 2π G cos  2  2 2 Gctg  G2 G G 1 4sin  r  x xxxxxxxx    dd33  d 3  2    QG2 Q 2π  rr  iQ22rrrGQQ2 1  rrr   x  xxxdd33 xx 2    Q 2π    Gctg 22 GQ Q 2 GQ  (72)   (78) The result is similar (41) and (65). By considering (13) and (76), we get 3.4. The Definition of Universal Momentum QQQ222 Operator in Spherical Coordinate System r   G G cos2  G  Based on the definition of universal momentum in the 22r 2 2   GGr 2 G (79) Descartes coordinate system, we can define universal mo-  r  4sin   mentum operator in spherical coordinate system. Similar Qr Q Q  to (44), we define i   0  r    1 piˆ rr  GQ  r (73) We take rr Gaib   ctg rr r ˆ  pi    GQ  (74) Gaib (80)  2   Gaib    piˆ   GQ  (75)  By substituting (80) into (79) and dividing it into imaginary part and real part, we have Here Qrr ,,  , Qr ,,  and Qr ,,  are

222222 a 22a real numbers, and Grr ,,  , Gr ,, and r  QQrQ  abr r  ab Gr,,  are complex numbers which satisfy follow-  r    (81) 2 ing relations cos  22a  2 ab  0  1 4sin    Gr rr br b b  ctg 22abrrab 2 ab    G (76) r    2  (82) QQQ  rrr  G 0   rrr We have three Q but only have two Equations (81) By acting (71)-(73) on general non-eigen wave func-  and (82), so we Q can be chosen arbitrary. By solving tions, we get non-eigen values of real numbers with  the motion question of quantum mechanics, we know the pQˆ    rr form of wave functions. Let pr , p and p are the

pQˆ   (77) partial momentum of a particle. Their forms can be determined by following formulas pQˆ   112  p2 Now let’s determine the concrete forms of Q , Q ˆ 2 r r  TTrrn2  r n (83) 22mrrmr  and Q . Substitute (73)-(75) into (12), we get nn

112 p2 By introducing spherical coordinate, (88) can also be TTˆ sin   rn 2 n written as nn22mmr sin  (84) ˆ  Lix sin ctg cos 22p2  11ˆ    TT rn22 2n  (85) nn2mr sin  2mˆ   Liy cos  ctg sin (89)  Let Qp   and substitute it in (81) and (82), we can determine Q and Q . In general, p and p are ˆ  r  r  Liz  different from Qr and Q . We let  ˆ pQgrrr In which Lz is the eigen operator of stationary wave function  of hydrogen atom with eigen values m , pQg  (86) nlm but Lˆ and Lˆ are not. Their non-eigen values and pQ x y  average values are complex numbers in general. The square of angle momentum operators is the eigen opera- Because pr , p , Qr and Q are known, gr and tor of angle momentum, we have g are known too. So (63) can be written as

22222 ppgˆ rn r r n ˆ LLLL xyz   ppgˆ   (87) 2 nn  11  (90) 2 sin  ppˆ   22 nn sin  sin    The non-eigen values of universal momentum operator ˆ2 Acting L on nlm , we get real eigen values pˆ r and pˆ do not represent real momentums of micro-particle too. In this way, we define universal mo- Lll22   1 . In order to make the non-eigen values of mentum operator in spherical coordinate system. By the ˆ ˆ Lx and Ly real numbers, we should refine they. Based same method, we can define universal momentum opera- on (44), universal angle operators are tor in other curved coordinate systems. ˆ  LizGyx   yzy   G  zQyQ  z (91) 4. The Definition of Universal Angle yz Momentum Operator and the Essence of  Micro-Particle’s Spin ˆ    LixGzy   zxz   G  xQzQ  x (92) 4.1. The D efinition of Universal Angle zx Momentum Operator ˆ  In quantum mechanics, angle momentum operator is re- Liyz  GxGyQxQxyx     y (93) lated to momentum operator with relation Lrpˆ ˆˆ. If xy we think that micro-particle’s momentums are unob- Here Q is determined by (53) and (54) with Qp . servable so that its values are unimportant, the angle  z z By acting universal angle operators on common non- momentum of micro-particles are related to atomic magnetic moment and the magnetic moments are measurable eigen wave function, we get directly. It is obvious that when angle momentum opera- ˆ LzQyQx nyzn  tors act on general wave function, their non-eigen values (94) and average values may be complex numbe r s too. The zp g yp  yy zn angel momentum operators in the Descartes coordinate system is ˆ LxQzQynzxn  (95)  ˆ xpz z pxx  g  n Lizyx     yz LyQxQˆ  ˆ  zn x y n Lixzy   (88) (96) zx  yp g xp g   x xyyn ˆ  Liyxz   x y g  x and p  x are also determined by (55) and

(56). In this way, the non-eigen values and average values of universal angle momentum operators are real um- bers. We do not discuss the forms of universal angle momentum operators in curved coordinate reference systems here.

4.2. Auxiliary Momentum and Auxiliary Angle Momentum We use the square of momentum operator pˆ 2 to construct the motion equation of qua ntum mechanics but use rpˆ ˆ to construct angle mome ntum, in quantum mechanics. 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 momentum should exist for electron moving around atomic nuclear, otherwise electron would fall into Figure 1. Auxiliary angle momentum, spin and magnetic nuclear. By considering the error between calculated moment. values of momentum using momentum operator and real values, auxiliary momentum and auxiliary an gle mo- 4.3. The Essence of Micro-Particle’s Spin mentum are introduced. By establishing relation between In order to explain the fine structure of light spectrum, auxiliary angle momentum and spin, the essence of mi- we assume that electron has a spin Sˆ . The projections cro-particle’s spin can be revealed. of spin can only take two values S  2 at arbitrary Let pˆ0 represents universal momentum operator, p0 direction in space. Spin seems an angle momentum but represents its value with pˆ00 p . It has been proved not real. Sometimes, we consider spin as that electron before that p0 is still not real momentum of micro- rotates around itself symmetry axis. But the calculation particle. Let pˆ represent real momentum operator and shows that if it is true, the tangential speed of electron’s

p represent real momentum with pˆ p . pˆ h re- surface would be 137 times more than light’s speed [11]. present auxiliary momentum operator and ph repre- It is difficult to understand the concept of micro-parti- sents auxiliary momentum with pˆ hh p . Their rela- cle’s spin from the point of view of classical mechanics. tion is Although quantum mechanics has been fully developed, the essence of spin is still an enigma at present day. ˆ ˆ ˆ pp0 ph (97) In quantum mechanics, spin is considered as angle mo- Let mentum actually. However, angle momentum is a kind of vector. So it is an unreasonable thing for spin vector to ˆ only take two projection values  2 at arbitrary direc- Lrp00ˆˆ (98) tion in space. In real physical space, such vector can not Lˆ rˆ pˆ hh exist. The projection of a vector with mode 1 at any direction can only be cos with values 1- 1 . Be- Here Lˆ is angle momentum operator in current 0 cause spin is always coupled with magnetic field, the quantum mechanics and Lˆ is supplemental momentum h correct understanding should be that if we take z axis operator. The real angle momentum operator Jˆ is p as the direction of magnetic field, the projections of spin ˆˆˆat z axis direction take two values  2 . At other J p rprˆˆ  ˆ pˆ00  pˆhhLL  (99) directions, spin’s projection should be related to cos . The relations among them are shown in Figure 1. We will see below that it is just due to the hypothesis that ˆ ˆ the projections of spin can only take two values, the Bell Auxiliary angle momentum Lh is related to spin S . We d iscuss their re lation below and prove that spin is inequality can not be correct. related to the partial angle momentum of micro-particle According to current quantum mechanics, magnetic which current momentum operator can not describe. By moment caused by spin in magnetic fie ld is establishing the relation between them, the essence of e μ  S (100) spin can be explained well. s mc

Magnetic moment caused by angle momentum L is 222 0 LJL0 ph2cos JL ph  (107)

e 222 μ00 L (101) LJS2cos JS  (108) 2mc 0 In atomic physics, the ratio that atomic ma gnetic mo- From (103), (105) and (106), we get ment divided by angle momentum is called as orbit LggJ11222 gLgS (109) gyro-magnetic ratio. According to (101), we have the h 0 ratio  Lemc 2 . However, according to (100), 00   From (107) and (108), we obtain we have s Semc200 L. So spin gyro-magnetic ratio is two times of orbit gyro-magnetic ratio. It indi- 21gJLS 22 2 cos   0 (110) cates that spin is not normal angle momentum. Let μ 1/2 21Jgg  J22+1 gL gS2 represent total magnetic moment of charged particle in  0  magnetic field, as shown in Figure 1, magnetic moment We know the values of J , S and L0 from quan- μ precesses around total angle momentum J , so μ tum mechanics. The Lande factor g can be obtained is considered as an immeasurable quantity in current from experiment. So Lh and  can be determined theory. What can be measured directly is the partial based on (109) and (110). For example, when L0  0 , quantity  of μ at the direction of angle momentum g g  2 and J  S , we have LSh  2 and   0 . In J . We have this case, Lh is just the angle momentum of electron in e ground state hydrogen atom. That is to say, for ground   gJ (102) state hydrogen atom with Ll0 , it is just angle g 2mc 0 momentum LSh  2 which ensures electron’s stable motion around atomic nuclear without falling into it. In Here g is the Lande factor. Let J p represent new total angle momentum after auxiliary angle momentum is this case, the magnetic moment caused by auxiliary angle momentum is considered, suppose that the relation between J p and J is ee μhhL S (111) J p  gJ (103) 2mc mc The formula above gives the Lande factor a new In this way, we explain that spin gyro-magnetic ratio is physical meaning. In this way, magnetic moment of par- two times of orbit gyro-magnetic ratio. It indicates that ticle becomes Lh is real angle momentum, in stead of spin. We should consider S as a kind of quantum number. Based o n it, ee μ JJg μ (104) we can obtain real angle momentums of micro-particles. p 22mc pgmc As shown in Figure 1, we have relation

By introducing new total angle mo mentum J p , the J p  g  LS00 LLh (112) direction of magnetic moment is the sam e as J p . We do not need the assumption that μ precesses around angle In fact, in quantum mechanics, we use the Pauli equations to describe the Zeeman effects of spectrum splits in total momentum J p again. In experiments, par ticle’s angle momentum can not be observed directly. What can magnetic fields. The equations are be done is m agnetic moment. We obtain angle momen- 2 eH 2Ur  Lˆ  E (113) tum through measurement of magnetic moment actually. 2211c z 11 So introducing new total angle momentum does not cause inconsistent. Inversely, angle momentum theory of 2 eH 2Ur  Lˆ    E (114) micro-particle becomes more rational. 2222c z 22 The relation between auxiliary angle momentum and ˆ spin is discussed below. Because the motions of objects In the for mulas, we have Lmz 11 and ˆ are restrained on a plane in center force fields, we sup- Lmz 221 . It means that the partial quantity of angle momentums at z axis direction is  pose that S p , J , J p , L0 and Lh are located on a plane. As shown in Figure 1, we have following relations actually. The reason to write the projection of spin at any direction of space as S   2 is only for of mathe- 222 SJL002cos JL (105) matical convenience. Speaking correctly, we should consider S   2 as a kind of quantum number based on 222 LJLhp002cos JL p  (106) it we can obtain correct angle momentums of micro-par-

Copyright © 2012 SciRes. JMP X. C. MEI, P. YU 465 ticles. ˆ ˆ ABab E  ab  In general situations L0  0 , the angle momentum of micro-particles bec omes complex. For example, for the Eabˆ ˆ ˆ (116) states 2P1/2 , 2P2/3 and 2D2/3 of hydrogen atom, ac- ˆ cording to (109) an d (110), we obtain  ˆ12abˆ ˆ  23 2:Pg ,,LLS2 ,  , The wave function of the system is 12 3h 0 2 1 3   A BAB    (117) JJ0.87,,  125 44 p  0.74 ; 2 2 33 By substituting (117) in (116), the calculating result of  2:Pg23 ,,,LLSh 20  2 , quant um mechanics is 42 (115) 15 Eabˆ  ˆ  abˆ ˆ (118) JJ,, 1.94   23 36,  2.60;    2 p ˆ ˆ ˆ ˆ 4215 3 When ab , we have Eab 1. Suppose that 2:Dg ,,LLS6,,  23 5h 50 2 there exists hidden variable  which makes determi- 15 nistic motion possible for micro-particle. The ensemble JJ,,, 1.94  135  2.82 . 2 p distribution function of hidden variable is   which is normalized with In this way, we reveal the essence of micro-particle’s d1  (119) spin. Spin is not real angle momentum though it related    to angle momentum. Due to the incompleteness of angle The average value of association operator on the en- momentum operator in quantum mechanics, we introduce semble function of hidden variable is the concept of spin. The quantities of S , L and Lh , ˆ ˆ are only useful tools. By means of these, we can con- Eab    ABab   d (120) struct real angle mo mentum J p of micro-particles. If theory is localized one without interaction at dis- Auxiliary angle mome ntum Lh does no t appear auto- matically in quantum mechanics. By means of it, the re- tance, the meas urement about particle 1 only depends on  and aˆ , having nothing to d o with bˆ . Meanwhile, lation between re al total angle momentum J p and the measurement about particle 2 also depends on  magnetic m oment μg becomes normal. We do not need and bˆ , having nothing to do with aˆ . So for arbitrary to assume that magnetic moment precesse s around J p . aˆ and bˆ , we have We have only a real angle momentum J p for micro- particle.  ABab  A a Bb (121)

5. The Real Reason That Bell Inequality Is Because two particles have opposites spins, when aˆ Not Supported by Experiments and bˆ are at the same directions, according to (118), we have Eaaˆˆ 1. It means 5.1. The Deduction of the Bell Inequality    AB  or AB 1 (122) Based on the clarification of spin’s essence, we can ex- aa    aa plain why the Bell inequality is not supported by experi- Let cˆ be another direction vector, because of ments. It is due to the misunderstanding of the spin’s AB    1, we have projections of micro-particles. ab Let’s first describe the deduction process of the Bell EabEacˆˆˆˆ  inequality briefly [12]. Suppose there is a system com-    posed of two particles with opposites spin  2 indi-  ABab  ABac  d vidually, so the total spin of system is zero. Spin operator ABab  ABAaba   Bc  d is ˆ and we take  2 as unite. Let Aa represent the   spin’s measurement value of particle 1 at a direction,  ABab 1dABac    B represent the spin’s measurement value of particle 2 b (123) at b direction. The average value Aab B of associ- ˆ ˆ ˆ ˆ ˆ ˆ ˆ Because of AB 1 , from (123) we get ation operator Eab12 a  b is ab   

EabEacˆˆˆ  ˆ tain, the coupling between spin an d magnetic field is  certain. At other direct ions , we can not observe the physica l affec ti ons of spin. In fact, in current quantum  1dABac    (124) mechanics, matrix operators are used to describe spin ˆ ˆ 1 Ea b with

Let aˆ , bˆ and cˆ are vectors on the same surface. 01  0i   1 0  ˆ ˆ ˆ ˆ  ˆ xy    z    (125) The an gles between aˆ and b is 60 , between b 10 i 0   0 1  and cˆ is also 60 , between aˆ and cˆ is 120 . Ac- cording to quantum mechanics we have By acting them on spin wave functions 12  12

and 12 12 , we have ˆ ˆ ˆ ˆ  EabEbc  cos 60 1 2 ,. ˆˆxx12 12 12 12 0  (126) Eac ˆˆcos120 1 2 . ˆ xx12 12 12ˆ 12 0

Let EabEabˆ ˆ ˆ ˆ and substitute it into (124),   ˆ yy12 i  12 1/2ˆ  12 0 however, we get absurd result 112 . It means that (127) ˆˆ12i 12 12 12 0 (124) can not coincide with the result of quantum me- yy chanics (118). Since the Bell inequality is advanced, ˆˆ12 12 12 12 1 many experiments are completed. Most of them support zz (128) quantu m m e chanics, not support the Bell inequality. So ˆ zz12  12  12ˆ  12  1 physicists do not think t hat hid d en va riab les exist. The determined descript ions of micro-particles are c ons idered Therefore, only (128) is the eigen equation of spin impossible. operator ˆ z with eigen values 1 . ˆ y and ˆ z are not the eigen operators of 1/2 and 1/2 and do no t 5.2. The Real Reason That the Bell Inequality Is have certain eigen values. So we can not think particle’s Not Supported by Experiments spins have same projection values 1 at arbitrary directions. Because the square of spin operator is After the Bell inequality was established, a lo ts of ex- 22 perime nts were completed. Most of them support quan- 2 2223 2 Ss xyz   s 1 (129) tum mechanics does not support the Bell inequality. So 44 according to current point of view, hidden variables do so the value of spin is actually not exist and the deterministic descriptions of micro- particles are considered impossible. 31 Based on discussion above, we can say that real rea- Ssss 1 (130) 22 sons to make the Bell inequality impossible is the misunderstanding of spin’s projection. According to current It is more proper to consider S and s as a kind of understanding, the projections of spin at arbitrary direc- quantum nu mber, in stead of spin ang le momentum itself. tions are always 1 . However, this kind of vector can In light of mathematics strict ly, as a p rac tical physical not exist in physical space. Suppose vector quantity, the projection of spin operator at aˆ direction

AiAx AAyz jk  with mode A , if the projection of is ˆ aaˆ ˆˆx xyyzz ˆˆ a ˆˆ a. Th e projection at z

A at direction k is  A , the projections at direc- direction should be ˆ z aˆz . According to quantum me- tions i and j can only be zero. No any physical vec- chanics, we have tor can have same projections at different directions in real 0 a space. x ˆ xxaˆ  In fact, spin is always coupled with magnetic field ax 0 when we construct the interaction Hamiltonian. The cor- 0 iay ˆ aˆ  (131) rect understanding of spin is that if we chose z axis as the yy ia 0 direction of magnetic field, the projection of spin at z y axis direction of is  2 . That is to say, the projection a 0 ˆ aˆ z of spin at the direction of magnetic field is  2 , rather zz 0 az than at arbitrary direction! Speaking strictly, in quantum mechanics, spin is coupled with magnetic field in the Here ax , ay and az are the projections of unit vector value of  2 . If the direction of magnetic field is cer- aˆ at x , y and z axis directions. We have

ax  sin cos In fact, E. P. Wigner had posed a proof of the Bell ine- a  sin sin (132) quality which did not depend on hidden variables [13]. y But it still based on the hypothesis that the projections of az  cos spin at arbitrary direction of space w ere 1 . From this result we see again that the deductio n of Bell inequality So the formula (131) means that the projections of spin has nothing to do with hidden variables. The violation of operator at arbitrary direction take the values between the Bell inequality also has nothing to do with whether or 1- 1 , rather than 1 . When calculating the average not hidden variables exist. values of ˆ1 aˆ about the wave function of a single particle, we have 5.3. The Polarization Corr elati on of Photon and

ˆˆ1 aaz  cos   the Bell Inequality ˆ aaˆ cos  Most experiments for the verification of Bell inequality 1 z  (133) ˆˆaacos  are related to polarization correlation of photons [14]. In 1 z the deduction of Bell ine quality for these proce sses, ˆˆ 1 aa z cos  photon’s polarization valu es are cons idered to be 1 . When a photon passes through a polarize, its polarization  ˆˆ1 aaiaxysincosi sin  sin (134) value is considered to be 1 . When a photon does not  ˆ1 aaiaˆ xysin cos  i sin  sin pass, its polarization value is considered to be –1 [15]. The deduced Bell inequality can not be supported by Based on (133) and (134), we get (118). However, in experiments. The reason is the same as the projections of the deduction of the Bell inequality, we let spin. In fact, light’s polarization is macro-concept. It is AabB 1. It means that the projection of elec- meaningless to talk about polarization about a single tron’s spin at arbitrary direction can only take 1 . This photon. We can only discuss light’s polarization from the result is different from (131)-(134) an d can not be rea- macro-viewpoint of statistical average. lized in real physical space. So it is inevitable that the We know in classical optics that the polarization direc- Bell inequality can not be supposed by experiments. The tion of light is defined as the vibration direction of elec- Bell inequality is a misunderstanding of mathematics, tromagnetic field. When a beam of polarization light having nothing to do with hidden hypotheses. It neither passes through polarizer, the vibration direction of elec- coincides with quantum mechanics, nor coincides with tromagnetic field is changed. For example, when a beam classical mechanics and any logic of mathematics and of polarization light passes through calcite, it becomes physics. two lights named e light and o light. Their vibration di - In fact, according to this paper, spin is not real physic- rections are different from original one. If we must de- cal quantity which can be determined directly. What can fine the concept of polarization for a single photon, we be done in experiments is magnetic moment. Magnetic can only consider its polarization direction as the direc- moment is related with angle momentum directly. Ac- tion of electromagnetic field. When a photon pass through cording to (112), the projection of auxiliary angle mo- a polarizer with an angle  , the vibration direction of mentum at α direction is electromagnetic field turns an angle  . In this case, we  should think that photon’s polarization becomes cos . Lag 11 La gg Sa    La  g (135) h 002 That is to say, even in classical optics, for photons which pass through polarizer, their polarization is considered as ˆ2 2 The eigen values of L0 is ll1  . Suppose that cos , in stead of 1 in general. For photons which are the angle between L0 and a is  , (135) becomes reflected without passing through polarizer, their polarization values depend on the angle of reflection, in stead  Lah 21gll   1cos  g (136) of 1 in general. In fact, in quantum mechanics, when 2 calculating polarization correlation of photons, we use

Take  2 as unit, let   0 , for the electron at 2P1/2 some formulas similar to (126)-(128) which are related to state, we have Lah 1.64 or 0.27 . For the elec- the direction angle  of polarization. It is impossible tron at 2P2/3 state, we have Lah   2.27 or for photons always to have polarization values 1 or

Lah 0.39 . For the electron at 2D2/3 state, we have 1 under arbitrary situation.

Lah 1.78 or 0.10 . For the ground electron with Because photon’s polarization values are always taken l  0 , we have Lah 22S , so Lah 1 . Let 1 , the mistake is the same as made for particle’s spin ˆ Laha ~ A and LhbbB~ , we have ABab   1 when we deduct the Bell inequality of photon’s polariza- in general. tions correlation. It is also inevitable that this kind of Bell

Copyright © 2012 SciRes. JMP 468 X. C. MEI, P. YU inequality can not be supported by experiments. There- The possibility distribution of momentum is fore, the violation of Bell inequality of photon’s polari- 3 2 zations correlation also has nothing to do with hidden 2 π cos ap  Pp11  p (142) variables. 2a3222 pp 1

6. The Elimination of EPY Momentum It is not the distribution of two momentums  p1 with Pa radox in Quantum Mechanics probability 12 individ ually. Thi s is the so-called EPY momentum paradox is caused. Because this paradox can The momentum paradox of Einstein-Pauli-Yukawa is a not be solved up to now days, some persons even thought confusing problem in quantum mechanics. Based on the that the logical foundation of quantum mechanics was definition of universal momentum operator in this paper, inconsistent [16]. we can eliminate it. Let’s first repeat this problem. We The problem is that according to discussion above, we discuss a micro-particle’s motion in the infinite potential can not define rational momentum operator for quantum trap of one dimension with form mechanics in the situation of one dimension. If we act 0 axa  current momentum operator on wave functions (139) or Ux  (137) (140), the obtained non-eigen value is an imaginary  x  a number. We have By solving the motion equation of quantum mechanics, p1 ip11 x// ip x we obtain particle’s energy pxˆ11  e e 2 a (143) n222π  p2 ππ x E n n 2 itgxpx3/2 111  ma 28 m (138) 2a 2a nπ p  ππ x n 2a px i tg (144) 1 3/2 22 aa Here m is particle’s mass. In the region axa  , According to discussion before, we can not find a wave function is proper momentum operator for micro-particle which moves  1 nxπ in one dimensional space. That is to say, in one dimen- cos n odd  a 2a sional infinite trap, particle’s momentums can not be  n x   (139)  p . 1 nxπ 1  sin ne ven Some one proposed the explanation of boundary con-  a 2a dition trying to eliminate EPY paradox [6]. According to this explanation, when (139) is written in the form of In the region x  a ,   0 . The wave function of n (140), eip1 x  is the normalized wave function in a box, ground state can be written as rather than the wave function of free particle in the re-

1 πx 1 ip11 x// ip x gion without boundary. The restriction of boundary con- 1 xecos e (140) aa2a 2 dition would produce a great influence on the nature of wave functions. If eip1 x  are the wave functions with By acting momentum operator pˆ idx d on (140), infinite boundary, after they are transformed in momen- we obtain two egein values  p1 . The result indicates tum space, the wave functions should be the  function that the wave function can be considered as the overlap with of two wave functions with different momentums  p pa1 π 2 . So Einstein, Pauli and Yukawa thought 1   that the particles in the ground state has only two inde-  1 ip() px ip () px pendent m omentum p and  p with probability 12 ee11d x 1 1    individually. 22π a  a On the other hand, by substituting (140) in (68), the 1 i()p px ip() px lim ee11dx(145) a   wave function in momentum space is 22πa a a sin ppasin ppa  1 ip()1pk ip()1 px π 11 1 p eed x lim    a 22πa a 2apppp ππ11 (141)   π32 cos ap   π 32 22 pp11 pp 2a pp 1 2a

It represents two mome ntums which is the same with the premises of ensuring kinetic operator to be invariable, the result in coordinate space. But if particles are located non-eigen values and average values of universal mo- in infinite trap with the restriction of boundary condition, mentum operator are real numbers. In this way, the de- (145) can not hold. scription of physical processes can be equivalent really in This kind of explanation has its reason but has not coordinate representation and momentum representation. touch the essence. Because (138) is the wave function in For eigen wave function, universal momentum operator coordinate space, the definition of momentum operator restores to the current Hermitian operator. For general of quantum mechanics is unrelated to boundary condition, situations, universal operator is not the Hermitian opera- no matter whether boundary conditions are finite or infi- tor because it is unnecessary. The most important thing nite, the actions of momentum operator on wave function in physics is that the average values of operator should are effect and certain. In fact, the boundary conditions be real numbers. Using universal momentum operator have been considered when we solve the motion equation and kinetic operator to calculate micro-particle’s kinetic of quantum mechanics. That is to say, the influence of energy, the results are still different, but we can get con- boundary condition has been contained in the wave func- sistent result through proper method. Only in this way, tions. So it is unnecessary for us to consider boundary we can reach logical consistency for quantum mechanics. condition. When we act momentum operator on wave The problems of momentum operator’s definition in the function, the result we get is what it should be. By acting curved coordinate reference systems can be solved well.

pˆ i on wave function (140), we can only get  p1 . Therefore, we need to introduce the concept of auxi- It indicates that we only have two discrete momentums. liary momentum and auxiliary angle momentum. The The EPY momentum paradox has not eliminated really relation between auxiliary angle momentum and spin is by considering boundary condition. deduced and the essence of micro-particle’s spin is re- According this paper, though (140) represents the vealed. Spin is related to auxiliary angle momentum of wave function in coordinate space, the momentum opera- micro-particle which angle momentum operator can not tor is not id d x, so particle’s momentum is not  p describe. We understand real reason why the Bell ine- in infinite trap. Because the Fourier transform of (68) is a quality is not supported by experiments. It is due to the pure mathematical one, its result is undisputed. Therefore, misunderstanding of spin’s projections and photon’s po- the momentum distribution (142) is correct. We see again larizations. No any real angle momentum can have same that thought we can have rational definition of kinetic projections at different directions in real physical space. energy operator, we may not find proper momentum The violation of the Bell inequality has nothing to do operator to match with kinetic energy operator some- with whether or not the hidden variables exist actually. In times. this way, the EPY momentum paradox can also be eliminated thoroughly. The logical foundation of quan- 7. Conclusions tum mechanics becomes more perfect. According to current quantum mechanics, when the operator is acted on non-eigen function, non-eiegn values and REFERENCES average values of momentum operator are complex num- [1] A. Einstein, “Science Paper Presented to Max Born, on bers in general. In theses cases, momentum operator is no His Retirement from the Tait Chair of Natural Philosopby longer the Hermitian operator. Though we can make the in the University of Edinburgh,” 1953. average values real numbers in momentum representa- [2] W. Pauli, “Pauli Lecture on Physics,” MIT Press, Cam- tion, it leads to in consistency of coordinate space and bridge, 1973. momentum space. Using momentum operator and kinetic [3] H. Yukawa, “Quantum Mechanics,” 2nd Edition, YanBo, operator to calculate momentum of micro-particles, the Bookshop, Tokyo, 1978. results may be different. It means that kinetic operator [4] X. L. Ge, “Quantization of Canonical Coordinates,” 2001. and momentum operator of quantum mechanics are not [5] Y. B. Zhang, “Momentum Operator and Kinetic Operator one-to-one correspondence. Besides momentum operator, in Curved Coordinates,” 1988. other operators in quantum mechanics, just as angle mo- [6] Z. Xu, “Discuss on Canonical Operators,” 1991. mentum operator, also have the same problems. These [7] C. B. Liang, “Quantization of Classical Systems,” Jour- problems have not caused the attention of physicists at nal of Beijing Normal University, Vol. l, 1994, p. 67. present day. Because these problems involve the ration- [8] H. Guang, “Foundation of Quantum Mechanics,” 1996. ality of logical foundation of quantum mechanics, we [9] J. M. Domingos and M. H. Caldeira, “Self-Adjiontness of should treat them seriously. Momentum Operators in Generalized Coordinates,” Foun- By introducing the concept of universal momentum dation of Physics, Vol. 14, No. 2, 1984, pp. 147-154. operator, all of these problems can be solved well. Under doi:10.1007/BF00729971

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