In quantum mechanics, the probabilistic interpretation, the complementarity principle (wave-particle duality), and the uncertainty principle form the foundation of the orthodox Copenhagen interpretation. However, this interpretation occasionally produces results that conflict with physical reasoning and intuition, leading to a long-standing debate that has persisted for nearly a century. Based on the conditions of action quantization and determinacy relations, this study reveals that the trajectory of an elementary particle forms a virtual line. At discrete points along this line, the particle exhibits particle-like properties, whereas in the regions between these points, the particle's existence cannot be detected. The most fundamental form of matter is the matter field. Its motion is wave-like and is described by a continuous wave equation and wave function over the entire spacetime domain. At specific spacetime points defined by xn =nh px and tn = nh E⁄ for n=1,2,3,…, the quantized matter field gives rise to the excitation and annihilation of elementary particles, indicating their discontinuous nature. This creation–annihilation process constitutes the wave motion of the matter field. The time and spatial scales associated with this process correspond to one period and one wavelength of the matter field wave, respectively. The Schrödinger equation, in this context, is understood as the non-relativistic equation of motion for a low-speed matter field. The eigenfunction of an elementary particle is, in essence, a wave function that describes the dynamic behavior of the matter field, encompassing the processes of its motion, as well as the generation and annihilation of the particle.
In quantum mechanics, the probabilistic interpretation, the complementarity principle (wave-particle duality), and the uncertainty principle form the foundation of the orthodox Copenhagen interpretation. However, this interpretation occasionally produces results that conflict with physical reasoning and intuition, leading to a long-standing debate that has persisted for nearly a century. Based on the conditions of action quantization and determinacy relations, this study reveals that the trajectory of an elementary particle forms a virtual line. At discrete points along this line, the particle exhibits particle-like properties, whereas in the regions between these points, the particle's existence cannot be detected. The most fundamental form of matter is the matter field. Its motion is wave-like and is described by a continuous wave equation and wave function over the entire spacetime domain. At specific spacetime points defined by xn =nh px and tn = nh E⁄ for n=1,2,3,…, the quantized matter field gives rise to the excitation and annihilation of elementary particles, indicating their discontinuous nature. This creation–annihilation process constitutes the wave motion of the matter field. The time and spatial scales associated with this process correspond to one period and one wavelength of the matter field wave, respectively. The Schrödinger equation, in this context, is understood as the non-relativistic equation of motion for a low-speed matter field. The eigenfunction of an elementary particle is, in essence, a wave function that describes the dynamic behavior of the matter field, encompassing the processes of its motion, as well as the generation and annihilation of the particle.
Abstract The uncertainty principle is a fundamental principle of quantum mechanics, but its exact mathematical expression cannot obtain correct results when used to solve theoretical problems such as the energy levels of hydrogen atoms, one-dimensional deep potential wells, one-dimensional harmonic oscillators, and double-slit experiments. Even after approximate treatment, the results obtained are not completely consistent with those obtained by solving Schrödinger’s equation. This indicates that further research on the uncertainty principle is necessary. Therefore, using the de Broglie matter wave hypothesis, we quantize the action of an elementary particle in natural coordinates and obtain the quantization condition and a new deterministic relation. Using this quantization condition, we obtain the energy level formulas of an elementary particle in different conditions in a classical way that is completely consistent with the results obtained by solving Schrödinger’s equation. A new physical interpretation is given for the particle eigenfunction independence of probability for an elementary particle: an elementary particle is in a particle state at the space-time point where the action is quantized, and in a wave state in the rest of the space-time region. The space-time points of particle nature and the wave regions of particle motion constitute the continuous trajectory of particle motion. When an elementary particle is in a particle state, it is localized, whereas in the wave state region, it is nonlocalized.
In the particle-wave duality interpretation of matter and radiation (Topic G3) a particle moving in space can also be described as a wave in space with a wavelength related to the particle momentum by de Broglie’s equation, λ=h/p. In quantum mechanics, the notion of a particle moving in defined trajectories in a system is replaced entirely by this description of the system in terms of its wavefunctions, ψ. The wavefunction simultaneously describes all regions of space in which the particle it represents can be found. This, in turn, introduces the idea of uncertainty into quantum mechanics because the exact position of the particle at each point in time is not defined, only the region of space of all its possible positions. The exact shape of the wavefunction is important because the probability of finding the particle at each point is proportional to ψ2 at that point; a greater amplitude in the wavefunction corresponds to a greater probability density in the particle’s distribution.
Following the spirit of de Broglie and Einstein, we think the concepts of matter and radiation can be unified. We know a particle propagates like a wave; its motion is described by certain wave equations. At this point, it is not clear what the wave function represents. Besides the statistical meaning suggested by the Copenhagen interpretation, does the wave function represent any physical motion? For photon, we know it is an electro-magnetic wave. But what about particles with rest mass, such as an electron? To investigate the physical nature of matter wave, we propose that: (1) Like the photon, a particle is an excitation wave of a real physical field. (2) Different types of particles are different excitation modes of the same field. Based on this thinking, we show that the concept of quantum mechanics can be a natural extension of classical mechanics. By critically analyzing the transition from classical physics to quantum physics, we found a new physical meaning for the quantum wave function. This work suggests that various quantum wave equations, including the Klein-Gordon equation, the Dirac equation and the Schrodinger equation, could have a common base relating to the Basic Wave Equation of the matter wave. This work has some interesting implications. It suggests a possible way to explain the origin of visible matters and dark matters in our universe.
A well-known mystery in quantum mechanics is wave–particle duality: Is an electron a point mass or a physical wave? What is the physical meaning of its wave function? About a hundred years ago, there was a famous debate between Bohr and Einstein on this topic. Their question is still open today. This paper reviews a new theoretical framework to address this problem. Here, it is hypothesized that both photons and electrons are quantized excitation waves of the vacuum, the physical properties of which can be modeled based on the Maxwell theory. Using the method of Helmholtz decomposition, one can show that the wave function of the particle is associated with an electric vector potential called “ Z”, which plays the role of basic field for the excitation wave. Using this framework, the quantum wave equations can be derived based on a quantization of the Maxwell theory. This work suggests that, the quantum wave function truly represents a physical wave; the wave packet looks like a “particle” only in the macroscopic view. Because the vacuum excitation obeys the principle of all-or-none, the probability of detecting this “particle” is related to the wave function as suggested in the Copenhagen interpretation.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
Quantum mechanics describes physical systems through probabilistic wave functions, accounting for phenomena such as wave–particle duality and the uncertainty principle. Although the mathematical formalism of quantum mechanics is well established, conceptual understanding of how quantum entities transition between wave-like and particle-like behavior remains an active area of research. In this work, quantum mechanical principles are re-examined from an interaction-based perspective, emphasizing the roles of environmental coupling, measurement geometry, and energy exchange. Using the photoelectric effect, de Broglie relations, and Heisenberg’s uncertainty principle as foundational elements, a conceptual framework is proposed in which particle identity emerges dynamically from wave interaction with matter and measurement constraints. This interpretation provides an intuitive bridge between classical reasoning and quantum behavior while fully preserving the standard mathematical structure of quantum mechanics...........Please check the attachment for details