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Journal of Modern Physics

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

Mei Xiaochun, Ping Yu

Published Jan 1, 2012• 4 citations
Publisher

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 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.

DOI
10.4236/jmp.2012.36062

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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.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.
The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the loss of reality, the behavior of micro particles shows randomness and hopping. The second is the loss of certainty, the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will change the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell's inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell's inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not parallel relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which show the wave-particle duality simultaneously and provide intuitionistic teaching materials for the new interpretation of quantum theory.
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
The EPR paradox is shown to be a result of the wrong definitions in quantum mechanics of such physical quantities as momentum and position. The corrected definitions show that momentum and position of particles can be precisely defined simultaneously and uncertainty relations do not prevent it. Therefore, with correct definition of physical quantities in quantum mechanics, the paradox disappears. The Bohm Aharonov version of the EPR paradox is illustrated by the example of measuring the polarization of photon pairs. The common view of the two photon state radiated by a source is presented. The common Bell’s inequality is discussed and the simplest version of it for a specific hidden variable is derived. A possibility of an experimental study of whether a photon and its polarization are preexistent before measurements is considered.
EPR paper contains an error. Its correction leads to a conclusion that position and momentum of a particle can be defined precisely simultaneously, EPR paradox does not exist and uncertainty relations have nothing to do with quantum mechanics. Logic of the EPR paper shows that entangled states of separated particles do not exist and therefore there are no nonlocality in quantum mechanics. Bell's inequalities are never violated, and results of experiments, proving their violation, are shown to be false. Experiments to prove absence of nonlocality are proposed where Bell's inequalities are replaced by precise prediction. Interpretation of quantum mechanics in terms of classical field theory is suggested. Censorship against this paper is demonstrated.
This paper proves that Bell's inequality is originated from classical probability theory,and thereby is related neither to locality nor to hidden variables.But it is also possible to derive spin correlation formula in quantum mechanics according to the classical probabilistic theory.This fact indicates that on cer- tain conditions,classical probabilistic theory would be inconsistent with quantum mechanics,but under other conditions,classical probabilistic theory and quantum mechanics would,reach the same goal by different means.Still further,this paper confirms that all of the probability operation rules in classical probability theo- ry,containing joint probability operation rules,are suitable for micro processes,but the event operation rules in classical probability theory are unsuitable.The reason obtaining Bell's inequality lies in the steps applying Boolean algebra rules on the non-Boolean micro event space.