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arXiv:quant-ph/0609092

Wave-particle duality and `bipartite' wave functions for a single particle

Chen, Zeqian

Published Sep 12, 2006• 0 citations
arXivHTMLPDF

Abstract

It is shown that `bipartite' wave functions can present a mathematical formalism of quantum theory for a single particle, in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This extension of Schrödinger's form establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. In particular, this formalism suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one. Our results shed considerable light on the basis of quantum mechanics, including quantum measurement.

DOI
10.48550/arxiv.quant-ph/0609092

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It is shown that bipartite' wave functions can present a mathematical formalism of quantum theory for a single particle, in which the associated Schrödinger's wave functions correspond to those bipartite' wave functions of product forms. This extension of Schrödinger's form establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. In particular, this formalism suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one. Our results shed considerable light on the basis of quantum mechanics, including quantum measurement.
It is shown that von Neumann-Landau equation for wave functions can present a mathematical formalism of motion of quantum mechanics. The wave functions of von Neumann-Landau equation for a single particle are `bipartite', in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This formalism establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. Furthermore, this extension of Schrödinger's form suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one.
An differential equation for wave functions is proposed, which is equivalent to Schrödinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schrödinger's wave equation, this equation is on `bipartite' wave functions. It is shown that those `bipartite' wave functions satisfy all the basic properties of Schrödinger's wave functions. Further, it is argued that `bipartite' wave functions can present a mathematical expression of wave-particle duality. This provides an alternative approach to the mathematical formalism of quantum mechanics.
It was Erwin Schrodinger who, in 1925, discovered an appropriate form of wave equation, making use of some deep formal analogies between optics and classical particle mechanics that had been evolved in the nineteenth century by W. R. Hamilton and others. There is all the accumulated evidence that the Schrodinger equations work; they provide the basis for a correct analysis of all kinds of molecular, atomic, and nuclear systems. The theoretical basis for the existence of states of discrete energies emerges more or less directly from the wave-particle duality; one of the most familiar features of classical wave behavior is that of standing waves. The quantum amplitudes for a particle in a bound state can be normalized. In order to make the statistical interpretation of the wave function more precise, we need to use the quantitative ideas of probability distributions.
The problem of measurement in quantum mechanics is that the quantum particle in the course of evolution, as described by the linear Schrodinger equation, exists in all of its possible states, but in measuring, the particle is always detected in only one of its states. This property is called the "collapse of the wave function" and was formulated by Von Neumann as one of the postulates of quantum mechanics. However, it remains unclear at what point in time and under what laws this transition occurs. This article demonstrates that the collapse of the wave function may be due to the creation or annihilation of particles (quasi-particles). The processes of the creation or annihilation of particles play a key role in the measurements and are described on the basis of quantum field theory. The system of equations of quantum field theory of particles and fields is non-linear; as a result, the principle of superposition does not hold for the theory. The collapse of the wave function is a consequence of this non-linearity and occurs at the moment of creation (annihilation) of a particle. This result demonstrates that the wave function collapse can occur in both microscopic and macroscopic systems. Understanding the mechanisms of the collapse of the wave function can lead to the creation of microscopic devices involved in the calculations based on quantum computing.
Abstract The double-slit experiment is a classic experiment that presents the clearest manifestation of wave-particle duality, one of the fundamental principles of quantum mechanics. The double-slit experiment still remains the phenomenon that demonstrates wave-particle duality at the most fundamental level. This phenomenon quantitatively and conceptually demonstrates that microscopic objects can behave like both waves and particles at the same time. This article provides an in-depth analysis of the physical nature of dualism, the mechanism of formation of the interference pattern, and the impact of the act of observation on the dynamics of quantum systems. The need for a new perspective on this topic is due to the fact that the current uncertainty in the interpretation of quantum reality has not yet been completely eliminated, and the boundaries between classical and quantum behavior in microscopic objects are still being explored. The results presented here restructure the deep mechanisms of the interference picture, more clearly frame the relationship between observation and wavefunction collapse, and provide a quantum. Since the article attempts to provide a complete answer to the mechanism of the formation of wave-particle duality, extensive space is given to sections such as historical development, theoretical models, modern research, etc. New approaches to this phenomenon have been developed using modern research and string theory. As a result, a new mathematical expression has been proposed that explains the reasons for the initial emergence of wave-particle duality, and is also important for string theory. The article explains the measurement problem with a more effective physical structure and ontological model.
We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a ``triality'' relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell's inequality, arises between single- and bipartite properties.