Deduction of Quantum Physics, Its Applications to the Eigenvalue Problems and Solving Two Puzzles of the Origins of Wave-Particle Duality and the First Quantization

1. Introduction

As far as quantum mechanics (QM) is concerned, there are still many problems that are very difficult to understand in terms of QM. However, classical mechanics is very well understood, and there is a very close correspondence between classical mechanics and QM. Therefore, studies on the correspondence between classical mechanics and QM enable people to have a deeper understanding of QM. Since QM is a statistical theory [1, 2], we naturally consider the correspondence relations between classical statistical mechanics and QM.

In fact, in the history of physics, people first studied classical mechanics [3], then further researched classical statistical mechanics with uncertainty [4], and finally developmentally investigated QM [5, 6], which is the inevitable result of steady development step by step. So, when we think about QM, we need to consider the classical limit of QM, which involves first transitioning to classical statistical mechanics and then to classical mechanics, rather than going straight to classical mechanics. Otherwise, this approach will lead to many problems that are hard to understand, resulting in some puzzles in the interpretations of the foundations of current QM [7, 8]. Many famous pioneers have conducted extensive exploratory work but still cannot resolve these puzzles – for example, the argument between Einstein and Bohr about the interpretation of QM [9–12].

In fact, in both classical statistical mechanics and QM, there are not only statistical average relations but also uncertainty relations, etc. [13]. Therefore, it is necessary to study how classical statistical mechanics can be directly extended to the situation of QM; this chapter intends to do just that.

It is now known that the uncertainty relations in both classical statistical mechanics and QM are important relations [13, 14]. In QM, the uncertainty relation is derived from the wave-particle duality of microscopic particles. It is of great practical significance to clarify the correspondence between classical statistical mechanics and QM for us to truly understand the more general physical essence of QM, especially the more general connection via QM between the stability and uncertainty of the world [13, 15].

In general, there is a correspondence between the general statistical sample space of classical statistical mechanics and the Hilbert space of QM. QM is a general theory, and classical mechanics can be regarded as the limit of QM when quantum effects are ignored. Therefore, there is a correspondence between QM and classical mechanics. In classical mechanics, the motion of a particle is determined by the solution of Newton’s equations and initial and boundary conditions. If the initial and boundary conditions are completely given, we get a completely deterministic description. If the initial and boundary conditions are given in the form of probability, the solution must also appear in the form of probability, so the form of motion obtained is also in the form of probability. Therefore, the description of classical statistical mechanics can be obtained [16].

The objects described by classical statistical mechanics can be an ensemble or some tiny particles [17]. For a system in classical statistical mechanics, a corresponding relationship with QM can be established.

As we all know, QM is based on the five basic axiom presumptions: wave function axiom presumption, operator axiom presumption, measurement axiom presumption, evolution axiom presumption, and identical axiom presumption. Using the five axiom presumptions, one can deduce QM [15, 18].

We now proceed from classical statistical mechanics without the basic presumptions to naturally deduce the present QM, providing solutions to the two puzzles of both the origin of wave-particle duality and the interpretations of first quantization. In addition, new important discoveries were made, and all the achieved conclusions satisfy the requirement that the classical limit of QM is classical statistical mechanics, and the macroscopic limit of classical statistical mechanics is classical mechanics, which are exactly as required by the self-consistency of physics and also serve as a flashback to the related cognitive development processes in physics.

After having tried to make a lot of efforts and experienced various explorations following the pioneers’ steps [19–22], and up to now, various efforts of almost all quantum theorists in quantum theory [23–31], we have finally overcome these large difficulties.

Especially, the development of quantum physics was reviewed [32]; probability, information, and statistical physics were discussed [33]; furthermore, fundamental principles of theoretical physics, concepts of quasi-averages, quantum protectorate, and emergence, as well as some different quantum physics theories, were studied [34–37].

Some important progress in the foundations of quantum theory in the last decades has been made, such as various good and interesting reconstructions of the quantum theory formalism [38–41], but this chapter is very different from them, offering a new approach that can return to classical reality. Also, the discussion of wave-particle duality has been present in the literature for decades, and wave-particle duality is just known as a fact of life, with quantum theory featuring complementarity [24, 40], but why is this so? Up to now, the reason has not really been provided. This chapter aims to address it.

The arrangement of this chapter is as follows: Section 2 studies the general average representation and the related discussions of any observable physical quantity; Section 3 presents the general eigen equations and the new general Hellmann-Feynman theorem, along with the related discussions; Section 4 derives the general Schrödinger equation in QM and includes the related discussions; Section 5 provides applications to different commutation relations and the first quantization; Section 6 contains the summary and conclusions.

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2. The general average representation and the related discussions of any observable physical quantity

In current QM, the first basic axiom presumption is the wave function axiom presumption: the motion state of microscopic particles is described by the wave function (material wave), and the modulus square of the wave function is the probability density function of particles. Now, we don’t need the wave function axiom presumption and are able to derive both the wave function axiom presumption content and new, more physical contents.

In order to overcome and understand quantum mechanics’ many uncertainties and almost incredible strange novel phenomena, after teaching many times in more than 30 years, having taught quantum mechanics, classical statistical mechanics, advanced quantum mechanics, advanced quantum statistical theory, quantum field theory, gauge field theory, particle physics, superstring theory, and so on in college and graduate courses, and after conducting research on these puzzles repeatedly for a long time, we have discovered the following: because quantum mechanics is a statistical theory, its classical limit should be classical statistical mechanics, and the further macroscopic and deterministic limit of classical statistical mechanics is classical mechanics – not classical mechanics in one step. Namely, there is an intermediate process theory, that is, classical statistical mechanics. This is the key problem that people have not recognized so far (e.g., see [2, 6, 8, 24, 25]). This is the real process of transition from high abstract theory to reality theory, that is, from the top to the bottom, or simply from up to down.

Therefore, in order to start from the reality that we can understand and to develop upward to understand the very abstract general theory, we should start from classical statistical mechanics, that is, from the bottom to the top, to promote the general abstract and lofty quantum mechanics theory. Then, the classical limit of the probability density of quantum mechanics should be the probabilistic density of classical statistical mechanics. Therefore, we can start from the probabilistic density of classical statistical mechanics to extend upward to the statistical density of quantum mechanics, namely, probability density. After extensive research, we discover that we can actually and certainly deduce quantum mechanics in this way.

First, considering a general classical statistical physical system, in which a particle has a classical distribution density function ρ=ρ(r→,t) at any time t, according to the positive definition of classical distribution density function, there is 0≤ρr→,t≤1, and since it must be in the space V at any time, thus we have ∫Vρr→,tdr→=1, thus the average of any observable physical quantity satisfying the statistical classical mechanics of the particle is

So, we can give a description of the statistical property of a particle and in order to generalize to the general quantum theory from the classical statistical mechanics that we have, we can consider the density function of any positive definite real number ρ=ρ(r→,t), the density function can always be mathematically represented as the product of a complex number function and its complex Hermitian conjugate function

In order to do general research, we can generally consider when the density function takes discrete values ρr→,t=∑i=1nρir→,t, then we have ρr→,t=∑i=1nψi*r→,tψir→,t, this expression can be expressed as a matrix product ρr→,t=ψ+r→,tψr→,t, where ψ+r→,t=ψ1*r→,tψ2*r→,t … ψn*r→,t is the row matrix, which is the Hermitian conjugate of the column matrix ψr→,t. The general average value Eq. (1) of discrete values of observable physical quantities can be expressed again as [3, 4, 5, 14, 17, 21, 22, and 26].

where F^r→,t=diagF1r→,t,F2r→,t,…,Fnr→,t is a nxn diagonal square matrix (without differentiation) operator of classical general functions; for the third equality, according to linear algebra theory, a general unitary matrix Ur→,t satisfies the properties U+r→,tUr→,t=1 (there is U+r→,t=U−1r→,t), and we have defined ψ′r→,t=Ur→,tψr→,t, and then we have ψ′+(r→,t)=ψ+(r→,t)U+(r→,t) and

that is, the off-diagonal square matrix operator can be obtained by a general diagonal square matrix operator F^(r→,t)through the similarity transformation of a general unitary operator, in other words, F^′(r→,t) is a nxn general off-diagonal square matrix operator. In fact, Eq. (4) is an inverse operation on a general diagonalization of a matrix, and it is easy to prove that the general operator Eq. (4) must be Hermitian, that is, we quite naturally prove that the classical observable operator must be Hermitian.

Therefore, we not only transition from the classical average representation to the QM average representation of the operator, which lies between the probability function and its complex conjugate function but also naturally observe that the average representation of different operators can vary in a similarity transformation, while their average value remains invariant. Furthermore, it has been proven in linear algebra that the Hermitian matrix is diagonalizable, and this diagonalizability is the necessary and sufficient condition for the Hermitian matrix to possess eigenvalues corresponding to its eigenvectors [15].

Therefore, the above research is consistent with the known QM theory, which is an important reason why we can generalize from classical statistical mechanics to QM. It can be seen, from the above discussions, that the three expressions in Eq. (3) are equivalent invariant expressions, and we show that the density is invariant under this transformation.

Therefore, we have obtained the general expression Eq. (3) of the average value of discrete observable values of any physical quantity in classical statistical mechanics, which is the same as that of QM. These are just the requirements of physics consistence. For any m and n, when there are F^mn′(r→,t)=1, the last equation of Eq. (3) can be written in normalized form

where we have used ψ′r→,t=Ur→,tψr→,t.

On the other hand, if we take the Hermitian conjugate of Eq. (3), we have:

Every physical quantity measured is a real value, thus we have F¯t=F¯+t, using Eqs. (3, 7), one has F^′r→,t=F^′+r→,t, that is, from an experimental observation point of view, all operators of observable quantities are Hermitian operators, and their eigenvalues are real.

In current QM, the second basic axiom presumption is the axiom presumption of operators [15, 18]: Classical mechanical quantities correspond to operators of QM, and the eigenvalue of operators is the measured value of their mechanical quantities. Quantization is the operatorization of mechanical quantities, and all the eigenfunctions of QM operators are complete.

In addition, we have proved that the operators of QM correspond to the observable physical quantities in classical statistical mechanics through the generalized expression of the operator mean value, Eq. (3), and the eigenvalues of the operators are the measured values of their mechanical quantities. Quantization is the operatorization of mechanical quantities because Eq. (3) is the generalized expression transitioning from the mean value of observable physical quantities in classical statistical mechanics to the mean value of operators, and the eigenfunctions of all operators of mechanical quantities in QM are complete (since the sample space of classical statistics is not only complete but also corresponds to the eigenfunctions of all quantum mechanical operators). Thus, we naturally derive the second axiom presumption, which is the axiom presumption of operators.

Mathematically, we have implemented the sum of observable values from the classical statistical mean value sum to the mean value sum of the QM matrix operators in the middle of a state vector and its complex conjugate. In fact, when we take the state of the microscopic particle system described by the general complex function of the square root of the density function, we begin to enter a more general description system that is different from the classical statistical mechanics system, namely the QM system.

Therefore, we not only obtain the statistical description of microscopic particles but also obtain the state vector description of microscopic particles, namely the probability vector description, and obtain the coherence of different components of the state vector Eq. (6). Later, we can more specifically prove that this description system is exactly the system of QM.

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3. Derivations of general eigen-equations and general Hellmann-Feynman theorems

Following the studies in literature [15, 18] and using the generally deduced expression Eq. (3), we derive the general eigen-equation and the general Hellmann–Feynman theorem.

In order to make general studies without losing generality, in Eq. (3), respectively denoting ψ′ as ϕ, then ψ′+=ϕ+, and taking the operator F^′ as A^, then Eq. (3) can be written as

Obviously, Eq. (8) is a direct extension of Eq. (3), dτ can be the measure of general representation, that is, dτ may be taken as the measures of different representations, such as the measures of coordinate, momentum and energy representations etc., and ∫ϕ+ϕdτ=1 can be obtained under the corresponding measures.

According to the probability theory, any particle must have the value under the corresponding measure, so it has its normalization. On the other hand, for the general function λ i(i=1,2,…,n), we generally have

where λ is a diagonal matrix. According to the variational theory, λii=1,2,…,n can be taken as the Lagrange multipliers, so the following function can be constructed

Using the variational theory, we have.

According to Eq. (11) and using the independences of complex functions δϕ+ and δφ, we have

It follows from Eq. (13) that A^ϕ=∓λϕ, when the positive signs of Eqs. (12, 13) are taken, the eigenvalue equation in QM can be obtained, and it is also necessary to satisfy the condition that the first order linear Eq. (11) has a non-trivial solution

Equation (14) is the condition that any matrix in linear algebra can be diagonalized, or the condition that any matrix in linear algebra can be diagonalized is that the matrix has n linearly independent eigenvectors.

They also indicate that the eigenvalues of all operators are the values corresponding to the extreme values of the variational Eq. (11). From the last term in Eq. (11), we have:

Using δλ=δλδaiδai,δA^=δA^δaiδai and when δai are linearly independent parameters, Eq. (15) becomes

Where the positive sign has been taken in Eq. (15), that is, only the negative sign is taken in Eq. (10). Therefore, we get the general Hellmann–Feynman theorem Eq. (16). When the diagonal elements of the matrix are all the same, Eq. (16) is simplified to the usual Hellmann–Feynman theorem.

Equation (16) is the general Hellmann–Feynman theorem that we derived. This relation implies that ordinary Hermitian operators satisfy this important relation as well. In particular, the Hellmann–Feynman theorem is established under the condition that not only are the eigenvalues of all operators the values corresponding to the extreme values of their variational system but also the values of the rest of the system are the results of taking the extreme values for the variational system.

As a result, we find that not only are all the eigenvalues of the operators the values corresponding to their variational system’s taking extreme values, but we also find that the Hellmann-Feynman theorem is established on the condition that not only are all the eigenvalues of the operators the values corresponding to the extremum value of their variational system, but also the rest of this system is the result of taking extreme values for the variational system. That is, the whole variational system takes the extreme value among all possible values, i.e., the particle system will choose the optimal way to express the physical properties of the system. More detailed studies are shown in the next section.

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4. Derivation of general Schrödinger equation in QM

Using the general Eq. (12) derived from us, taking A^ and λ as the classical Hamiltonian operator H^ and the energy eigenvalue E (there is no differential operator but can include the matrix), respectively, then the Eq. (12) can be specifically written as the corresponding eigen equation

Namely, in E=∫ϕp→,E†H^p→,r→ϕ Equation (11) dτ can be taken as the measure of momentum representation, that is, H ^ ¯  ′ (T,V)= ∫ ϕ ( p → ,E) † H ^ (T,V)ϕ( p → ,E)d p → − ∫ ϕ ( p → ,E) † Eϕ( p → ,E)d p → = ∫ ϕ ( p → ,E) † H ^ ( p → , r → )ϕ( p → ,E)d p → − ∫ ϕ ( p → ,E) † Eϕ( p → ,E)d p → . . The four-dimensional momentum representation wave function ϕp→,E just reflects the system’s global property (having nothing to do with spacetime coordinates) that is just particle property for the single particle. Using e−itE/α/(2πα)1/2 times Eq. (18) and making the integral from minus infinity to positive infinity, namely, we make Fourier integral transformation of plane wave relevant to energy, we get

where α is a constant making their dimensions tE/α=1, and according to general mathematical principle, we have defined

Equation (20) is the Fourier transformation of a plane wave relevant to energy.

Equation (19)’s second line is still the incomplete Schrȍdinger equation for the classical Hamiltonian H^p→,r→ and the wave function φp→,t of the momentum representation, Because p→ is still a classical quantity not an operator. And from Eq. (19), we get H^p→,r→=iα∂/∂t. It can be found from the above studies, it is the presence of a weight factor e−itE/α that enables the classical Hamiltonian to be written in operator form H^p→,r→=iα∂/∂t.

It is well known from the mathematical properties of the Fourier transformation that any continuous physical quantity can be represented by the discrete Fourier basic vector expansion, which is equivalent to the projection of the physical quantity onto the discrete Fourier basic vector; i.e., the physical quantity is thus discretized (i.e., a discrete quantization).

Similarly, any continuous quantity can be represented by a continuous Fourier basic vector expansion. In particular, it should be pointed out that Eq. (18) uses e−itE/α/2πα1/2 times ϕp→,E and makes integral from minus infinity to positive infinity for E, that is, the wave function of the energy representation is deduced at time t via considering the E’s contribution to ϕp→,E from minus infinity to positive infinity, which is also the natural property of the Fourier integral transformation for plane wave energy part.

Substituting the classical operator eip→⋅r→/β/2πβ1/2 to multiply the second row of Eq. (19), and integrating from minus infinity to positive infinity, then we deduce

where we have already used that the kinetic energy Tp→ can be expanded into a series of p→ (e.g., Tp→=p→2/2m), β is a constant making their dimensionality p→⋅r→/β=1, and according to general mathematical rule, we can define

where we have substituted Eq. (20) into the first equation of Eq. (22), and any quantities can be expanded according to Fourier transformation.

The complex square root function ϕp→,E of the probability density of the classical particle in the complex field is transformed into ψr→,t through Eqs. (20, 22), which not only has the characteristics of the probability density’s relevant state vector of the classical particle, but also has the characteristics of the general wave eip→⋅r→/β−itE/α, that is, which make  ψr→,t have the characteristics of the wave-particle duality state vector at the same time. The second equation of Eq.  (22) shows superposition state’s representation of wave-particle duality, we discover the collapse theorem of superposition state of wave-particle duality: the collapse of superposition state of wave-particle duality into particle ϕp→,E state or general wave eip→⋅r→/β−itE/α state relevant to particle or wave property’s measure. These explain just why sometimes it is particle or why sometimes it is wave.

The Hamiltonian becomes an operator with differentiation form H^p→^,r→=T^p→^+Vr→ p→^=−iβ∇, it can be found from the above studies, it is the presence of a weight factor eip→⋅r→/β that enables momentum p→ to be written in operator form p→^=−iβ∇. By using Eq. (21) in a practical example or in comparison with the existing Schrȍdinger equation, we get α=β=ℏ, and substituting α=β=ℏ into Eq. (21), we finally get the Schrȍdinger equation in QM

Finally, from Eq. (23), we get that the Hamiltonian operator can be written as H^p→^,r→=iℏ∂/∂t.

It can be seen from the above derivation that Eq. (18) is the eigenvalue equation of the probability of classical particles from classical statistical mechanics, and the state function ϕp→,E is the complex square root of the probability density of classical particles in the complex number field, p→,E are the physical quantities of classical particles in classical statistical mechanics, and they reflect the properties of particles.

When Eq. (18) is related to Fourier transformation, Eqs. (19, 21) of the four-dimensional spacetime, Eq.(18) is projected onto a plane wave function eip→⋅r→/β−iEt/α of the four dimensional spacetime state vector and making integration, we get Schrȍdinger Eq. (23), Schrödinger Eq. (23) is reflecting both the particle property and the wave property, Eq. (18) is only reflecting the nature of the particle, and that Eq. (23) is turned into QM just reflects the material particle property and wave property in the wave-particle duality for Schrȍdinger equation.

Thus, the basic interpretation problem of QM has been debated for more than a century; for example, the puzzle of the dispute over the origin of the wave-particle duality of microscopic particles has been solved.

Consequently, Eqs. (18, 21), respectively, show the classical locality and quantum non-locality, ϕp→,E(from the complex square root function of the classical probability density of a particle in the complex domain in Eqs. (20, 22) directly shows the classical locality and quantum non-locality because ψr→,t has the characteristics of the wave-particle duality state vector at the same time.

Since, in the current QM, the third axiom presumption is the evolution axiom presumption: the evolution law of the wave function representing the state of the quantum isolated system before the measurement satisfies the wave equation (namely Schrödinger equation) relevant to the energy-momentum relation of classical mechanics. In the above studies, we do not need the third axiom presumption to derive not only the entire contents of the third axiom presumption but also more new physical contents.

It especially needs to be pointed out: we extensively take the complex square root function of the probabilistic density function in the most general mathematical form. Then, the probabilistic density square root state vector function is obtained, which similarly takes the square root of the Klein-Gordon equation to get the very different Dirac equation. We have obtained a general quantum mechanical function of complex numbers, which is completely different from the classical probabilistic density function of real numbers, and vice versa.

What is also special is that we have the classical statistical eigen-equation, which is only different from the Schrödinger equation of QM in terms of the Fourier transformation of the plane wave. This is what makes the classical statistical mechanics eigen-equation of particles become the Schrödinger equation with wave-particle duality. So, they are related and consistent, and thus we have determined the strict correspondence relationships from the classical statistical mechanics to QM, that is, from the down to the up, or from QM to the classical statistical mechanics, that is, from the up to the down.

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5. Applications to different commutation relations and the first quantization

Using the deduced operators p→^=−iℏ∇ and H^p→^,r→=iℏ∂/∂t, we have

and

Therefore, for more operators, we naturally have.

where angular momentum components J^i=εijkx^jp^k,ε123=−ε132=1.

In fact, when people deduce wave functions, operator expressions of mechanical quantities, operators’ eigenvalues, and commutation relations of different operators, the first quantization of quantum theory has been achieved [15].

On the other hand, inserting the deduced momentum vector operator p→^=−iℏ∇ into the derived Hamiltonian operator H^p→^,r→=iℏ∂/∂t, we achieve H^(−iℏ∇,r→)=iℏ∂/∂t, using the deduced Hamiltonian operator H^−iℏ∇,r→=iℏ∂/∂t to act a general wave function ψr→,t, we derive a general Schrȍdinger equation

which displays that our studies are consistent with all the relevant investigations in this chapter.

According to all the investigations above in this chapter, we have done all the above these, consequently, we naturally show general quantum theory and its solutions to the two puzzles of both wave-particle duality origin and the first quantization ψr→,t have the characteristics of the wave-particle duality state vector at the same time (or interpretations).

Quantum effects are actually caused by the wave-particle duality. Because if there is no matter wave property, that is, there is no plane wave with the integral transformation, it is not possible, at the same time, for the presence of multiple points in space, and multiple points all having probability waves. This is because a probability wave can have a distribution. The matter probability wave function is used to describe wave-particle duality in current quantum theory, which is the basic hypothesis of QM, but in our studies, it is a naturally deduced consequence.

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6. Summary and conclusions

In classical statistical physics, the state of a microscopic particle or a microscopic particle system is described by a real density function, we extensively discover that the state of a microscopic particle or a microscopic particle system can be described by a general complex function (or call a state vector) ψr→,t from Fourier integral transformation of complex square root ϕp→,E of the real density function of classical statistical mechanics.

In fact, when we take the state ψr→,t, we begin to enter a general description system that is different from the classical statistical mechanics system, namely the QM system, which is, in some meaning, analogous to taking the square root of Klain-Gooden equation, we get a new system, Dirac equation Fermion system, which just means that we may say that QM is a generalization mechanic theory of the complex square root of real density function of classical statistical mechanics, namely, we, for the first time, factually discover that QM is a generalization mechanics theory of the complex square root of real density function of classical statistical mechanics, for short, QM is just a generalization theory of the complex square root of classical statistical mechanics, which can be seen from the overall deductions in this paper, and are both new physics and revolutionary discovery, these are affecting people’s deep philosophical thinking for modern physics development.

Especially this chapter and [42] naturally solve all the puzzles of QM, quantum information, and so on, and make QM have scientifically solid bases that are verified, with neither basic axiom presumption nor all the current strange and incomprehensible properties of quantum mechanics, because classical statistical mechanics and the complex square root of the density function of classical statistical mechanics have scientifically solid bases that are verified.

We naturally provide the inverse calculation of general matrix diagonalization for any observable physical quantity operator and prove that the general observable quantity operator must be a Hermitian operator. In linear algebra, it has been proven that any Hermitian matrix is diagonalizable, and the diagonalizable conditions are the sufficient and necessary conditions for the Hermitian matrices acting on their eigen vectors to have eigenvalues.

Therefore, the above research is consistent with the known QM theory, which is an important reason why we can generalize QM from classical statistical mechanics, from the down to the up.

Not only do we deduce that the eigenvalues of all the operators are the values corresponding to the extreme values of their variational system, but we also find that the Hellmann-Feynman theorem is true only if the eigenvalues of all operators are the values corresponding to the extreme values of their variational system, and the remaining values of the system are also the result of taking the extreme values for the variational system. All these belong to new physics.

From the research in the second and third sections, we have now reduced the third axiom presumption (evolution axiom presumption) to the evolution theorem; that is, the evolution theorem has been proved from classical statistical mechanics.

When making four-dimensional spacetime Fourier integral transformations Eqs. (19, 21) of Eq. (18), that is, when Eq. (18) is projected onto the state vector of the plane wave function eip→⋅r→/β−iEt/α of the four-dimensional spacetime and is integrated, we get Schrȍdinger Eq. (23) reflects both the particle nature of matter and the wave nature of matter. According to the exact deducing logic, in other words, Eq. (18) only reflects the properties of particles, while Eq. (23) is transformed into Schrȍdinger equation reflecting the wave-particle duality in QM. The complex square root function ϕp→,E of the probability density of the classical particle in the complex field is converted to ψr→,t by Eqs. (20, 22), it not only has the property of the probability density’s relevant state vector of classical particles but also has the property of the wave, which make ψr→,t have characteristics of wave-particle duality state vector at the same time.

However, the puzzle of the fundamental interpretation of QM, which has been debated for more than a century, still isn’t solved, but, for example, the puzzle of the debates on the origin of the wave-particle duality of microscopic particles has been rigorously solved in this chapter.

Therefore, Eqs. (18, 21), respectively, show the classical locality and quantum non-locality, ϕp→,E in Eq. (20) and ψr→,t in Eq. (22) directly shows the classical locality and quantum non-locality respectively. Furthermore, we discover the collapse theorem of superposition state of wave-particle duality.

Using Eq. (19), we deduce H^p→^,r→=iℏ∂/∂t, therefore, we discover that it is the existence of a projection weight factor e−itE/ℏ that makes the Hamiltonian be written in operator form H^p→^,r→=iℏ∂/∂t, which just shows, in Schrȍdinger equation, taking derivative of the wave function is a partial derivative about time t rather than a full derivative, that is, it is the reason why people don’t take the time derivative of space coordinates of the wave function. People haven’t found out the reason for more than 80 years, because Schrȍdinger equation is still an axiom presumption up to now, but this paper deduces general Schrȍdinger equation via two kinds of general methods, anyway, up to now, it has not been rigorously derived in the most general way.

Because we have deduced wave functions, operator expressions of mechanical quantities, operators’ eigenvalues, and commutation relations of different operators, the first quantization of quantum theory has not only been achieved, but we very naturally also give the origin of the first quantization of quantum theory. Consequently, the puzzle of the origin of the first quantization of quantum theory is naturally solved.

Especially, the main reason why QM has so many strange and incomprehensible properties is that QM is currently based on a few basic presumptions. In this chapter, we find that QM must return to the classical limit, and the most direct classical limit for QM is classical statistical mechanics. Therefore, we should be able to generalize classical statistical mechanics to derive QM. In this chapter, we discover that the square root of the density function of classical statistical mechanics can derive the entirety of QM. In particular, classical statistical mechanics and the square root of the density function of classical statistical mechanics are based on exact scientific deductive logic and reasoning, consequently eliminating the basic axiomatic presumptions of current QM. Thus, we can provide exact scientific explanations for all the strange phenomena in quantum physics. As a result, all the strange and incomprehensible phenomena of QM are eliminated in QM in this chapter and in the following reference [42].

This chapter demonstrates the deduction of quantum physics, particularly presenting its new and significant applications to eigenvalue problems. Therefore, we deduce the three most important axiomatic presumptions of the five in QM, and then these three axiomatic presumptions are transformed into three fundamental theorems of QM. Consequently, we present a general quantum theory and its solutions to two puzzles: the origins of both wave-particle duality and the first quantization. Due to the length limitations of this paper, studies on the other two axiomatic presumptions of QM, as well as the origins and classifications of entanglements, wave collapse, and related topics, will be provided in a subsequent paper [42]. In other words, this chapter and references [42, 43] deduce quantum physics without relying on basic axiomatic presumptions and without involving all the strange, incomprehensible phenomena of current QM.

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Acknowledgments

The work is supported by the U.S. Department of Energy, contract no. DE-AC02-05CH11231; NSF through grants PHY-08059; DOE through grant DEFG02-91ER40681; and the National Natural Science Foundation of China (No. 11875081).

The first version was published as a preprint under the title “General Quantum Theory: I Deducing Quantum Physics and Solving Two Origin Crisises of Wave-Particle Duality and the First Quantization” on 5 January 2021, and it is available on the following https://www.preprints.org/manuscript/202002.0091/v3.