Quantum mechanics based on Planck hypothesis and statistical interpretation of wave function has achieved great success in describing the discrete law of micro motion. However, the idea of quantum mechanics has not been successfully used to describe the discrete law of macro motion, and the causality implied in the Planck hypothesis and the application scope of the basic principles of quantum mechanics have not been clarified. In this paper, we first introduce the angular motion law and its application, which seems to be of no special significance as a supplement to the perfect classical mechanics, but plays an irreplaceable role in testing whether the core mathematical procedure of quantum mechanics of operator evolution wave equation satisfies the unitary principle. Then, the operator evolution wave equations corresponding to the angular motion law are discussed, and the necessity of generalized optimization of differential equations are illustrated by the form of ordinary differential equations. Finally, the real wave equation which is superior to the Schr\{o}dinger equation in physical meaning but not necessarily the ultimate answer is briefly introduced. The implicit conclusion is that Hamiltonian can not be the only inevitable choice of constructing wave equation in quantum mechanics, and there is no causal relationship between operator evolution wave equation and quantized energy in bound state system, which indicates that whether the essence of quantum mechanics can be completely revealed is the key to unify macro and micro quantized theory.
Quantum mechanics based on Planck hypothesis and statistical interpretation of wave function has achieved great success in describing the discrete law of micro motion. However, the idea of quantum mechanics has not been successfully used to describe the discrete law of macro motion, and the causality implied in the Planck hypothesis and the application scope of the basic principles of quantum mechanics have not been clarified. In this paper, we first introduce the angular motion law and its application, which seems to be of no special significance as a supplement to the perfect classical mechanics, but plays an irreplaceable role in testing whether the core mathematical procedure of quantum mechanics of operator evolution wave equation satisfies the unitary principle. Then, the operator evolution wave equations corresponding to the angular motion law are discussed, and the necessity of generalized optimization of differential equations are illustrated by the form of ordinary differential equations. Finally, the real wave equation which is superior to the Schr{o}dinger equation in physical meaning but not necessarily the ultimate answer is briefly introduced. The implicit conclusion is that Hamiltonian can not be the only inevitable choice of constructing wave equation in quantum mechanics, and there is no causal relationship between operator evolution wave equation and quantized energy in bound state system, which indicates that whether the essence of quantum mechanics can be completely revealed is the key to unify macro and micro quantized theory.
The relation of matter wave, which is well-known as a hypothesis proposed by de Broglie in 1923, gave basis for establishing the quantum mechanics. After that, experimental results revealed that a micro particle has a wave nature. However, the theoretical validity of the relation itself has never been revealed since his proposal. Theoretical basis that a micro particle has a wave nature has been thus disregarded in the unsolved state. The diffusion equation having been accepted as Fick’s second law was derived from the theory of Markov process in mathematics. It was then revealed that the diffusivity D depends on an angular momentum of a micro particle in a local space. The fact being unable to discriminate between micro particles in a local space resulted in having to accept the existence of minimum time t0 (>0) in the quantum mechanics. Based on t0 and D obtained here, the theoretical validity of relation of matter wave was confirmed. Denying the density theorem in mathematics for time in physics indicates that the probabilistic interpretation is essentially indispensable for understanding the quantum mechanics. The logical necessity of quantum theory itself is thus understandable through introducing t0 into the Newton mechanics. It is remarkable that the value of t0 between 1.14×10-17 s ≤ t0 ≤1.76×10-14 s obtained here is extremely larger than that of the well-known Planck time tp=5.396×10-44 s.
In reality, micro objects and macro objects coexist harmoniously, and each macro object depends on micro particles to exist (macro is a collection of micro objects, indicating that they have a very extensive relationship and are symbiotic). However, classical mechanics and quantum mechanics have long been regarded as binary oppositions. This is a serious disconnect between theory and phenomena, requiring the establishment of new theories and methods that are compatible between quantum and classical. Prior to this, many reasons and methods for combining quantum theory with classical theory have been found (providing numerous computational examples). The wave mechanics equations of classical mechanics laws have been written. The technical method used is still the conversion rule of the Hermitian operator in the fundamental postulates of quantum mechanics. There are no new assumptions. The key is to use the relationship between force and energy in the planetary model to obtain wave mechanics equations that include force, Newton's second law, and the law of conservation of momentum. Thus, quantum mechanics, which is compatible with classical mechanics, truly becomes a form of 'mechanics'. This kind of mechanics is generalized wave mechanics. Quantum and classical are coordinated and compatible, resulting in no clear boundary between quantum and classical. It will seriously affect the credibility of quantum decoherence and wave function collapse.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.
INTRODUCTION A theory of classical quantum mechanics (CQM), derived from frst principles, successfully applies physical laws on all scales. The classical wave equation is solved with the constraint that a bound electron cannot radiate energy. The mathematical formulation for zero radiation based on Maxwell's equations follows from a derivation by Haus. 2 The function that describes the motion of the electron must not possess spacetime Fourier components that are synchronous with waves traveling at the speed of light. CQM gives closed form solutions for the atom, including the stability of the n=l state and the instability of the excited states, the equation of the photon and electron in excited states, the equation of the free electron, and photon which predict the wave particle duality behavior of particles and light. The current and charge density functions of the electron may be directly physically interpreted. For example, spin angular momentum results from the motion of negative
In this chapter, we finally introduce time evolution. The development is carried out for finite, enumerable systems (i.e., systems whose states live in a finite dimensional, inner product space) which, of course, allows a straight forward extension to the continuum and infinite dimensional state spaces. The necessary mathematical background of unitary operators, and their properties relevant to quantum mechanics are first introduced. In particular, the fact that preservation of norm under the action of an operator implies unitarity, and that every unitary operator can be expressed as the exponential of i timesa Hermitian operator are introduced. Time evolution is then introduced in its integral form through the time evolution operator, and the Hamiltonian operator is introduced in this context. This approach, however, necessitates the restriction to conservative systems. Next, the differential form of the law of time evolution, i.e., Schrodinger equation, is derived from the time evolution operator, and this is postulated to be the general law of time evolution. We then demonstrate how, for conservative systems, the solution of the Schrodinger equation essentially reduces to solving the eigenvalue problem of the Hamiltonian. Constants of motion and Heisenberg picture have been relegated to the problem section.