Nonlinearity of Quantum Mechanics and Solution of the Problem of Wave Function Collapse

The problem of the wave function collapse (a problem of measurement in quantum mechanics) is considered. It is shown that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The particle creation and annihilation processes, which are described based on quantum field theory, play a key role in the measurement processes. Superposition principle is not valid for the system of equations of quantum field theory for particles and fields, because this system is a non-linear. As a result of the creation (annihilation) of a particle, an additional uncertainty arises, which "smears" the interference pattern. The imposition of such a large number of uncertainties in the repetitive measurements leads to the classical behavior of particles. The decoherence theory also implies the creation and annihilation of particles, and this processes are the consequence of non-linearity of quantum mechanics. In this case, the term "collapse of the wave function" becomes a consequence of the other statements of quantum mechanics instead of a separate postulate of quantum mechanics.     

Nonlinearity of Quantum Mechanics and Solution of the Problem of Wave Function Collapse

The problem of the wave function collapse(a problem of measurement in quantum mechanics) is considered.It is shown that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The particle creation and annihilation processes, which are described based on quantum field theory, play a key role in the measurement processes. Superposition principle is not valid for the system of equations of quantum field theory for particles and fields, because this system is a non-linear. As a result of the creation(annihilation) of a particle,an additional uncertainty arises, which "smears" the interference pattern. The imposition of such a large number of uncertainties in the repetitive measurements leads to the classical behavior of particles. The decoherence theory also implies the creation and annihilation of particles, and this processes are the consequence of non-linearity of quantum mechanics. In this case, the term "collapse of the wave function" becomes a consequence of the other statements of quantum mechanics instead of a separate postulate of quantum mechanics.     

Particle Trajectories for Quantum Field Theory

The formulation of quantum mechanics developed by Bohm, which can generate well-defined trajectories for the underlying particles in the theory, can equally well be applied to relativistic quantum field theories to generate dynamics for the underlying fields. However, it does not produce trajectories for the particles associated with these fields. Bell has shown that an extension of Bohm’s approach can be used to provide dynamics for the fermionic occupation numbers in a relativistic quantum field theory. In the present paper, Bell’s formulation is adopted and elaborated on, with a full account of all technical detail required to apply his approach to a bosonic quantum field theory on a lattice. This allows an explicit computation of (stochastic) trajectories for massive and massless particles in this theory. Also particle creation and annihilation, and their impact on particle propagation, is illustrated using this model.     

量子力学的整体性概念和概率诠释的物理内涵

本文回答了文献1～2对量子力学提出的疑问.基于量子力学的整体性概念指出,Landau和Lifshitz给出的一维无限深方势阱中粒子的动能概率分布函数是正确的,Pauli等人给出的概率分布函数是不正确的.从量子测量理论的角度讨论了一维谐振子的动量概率分布问题,并且指出势能大于本征能量的概率不为零并不表示存在负动能的概率分布区域。     

Quantum Potential in Relativistic Dynamics

The experimental confirmation of nonlocality has renewed interest in Bohm's quantum potential. The construction of quantum potentials for relativistic systems has encountered difficulties which do not arise in a parametrized formulation of relativistic quantum mechanics known as Relativistic Dynamics. The purpose of this paper is to show how to construct a quantum potential in the relativistic domain by deriving a relativistically invariant quantum potential using Relativistic Dynamics. The formalism is applied to three relativistic scalar particle models: a single particle interacting with a scalar potential; N particles interacting with a scalar potential; and a single particle interacting with an electromagnetic 4-vector potential.     

Bohmian mechanics and quantum field theory

We discuss a recently proposed extension of Bohmian mechanics to quantum field theory. For more or less any regularized quantum field theory there is a corresponding theory of particle motion, which, in particular, ascribes trajectories to the electrons or whatever sort of particles the quantum field theory is about. Corresponding to the nonconservation of the particle number operator in the quantum field theory, the theory describes explicit creation and annihilation events: the world lines for the particles can begin and end.     

Generalized relativistic Fock space

We consider a generalized Fock space obtained by eliminating the restriction to symmetric components for bosons or antisymmetric ones for fermions. In this space we can extend the many times formalism of relativistic quantum mechanics to quantum field theory, in which each particle has a time parameter that has to be included in any exchange of variables. Physical states in which all particle times, or all antiparticle times, are equal, still have the right symmetry. We define creation and annihilation operators for numbered particles in this space, and relate them to the usual operators.     

On the geometrization of quantum mechanics

Nonrelativistic quantum mechanics is commonly formulated in terms of wavefunctions (probability amplitudes) obeying the static and the time-dependent Schrödinger equations (SE). Despite the success of this representation of the quantum world a wave–particle duality concept is required to reconcile the theory with observations (experimental measurements). A first solution to this dichotomy was introduced in the de Broglie–Bohm theory according to which a pilot-wave (solution of the SE) is guiding the evolution of particle trajectories. Here, I propose a geometrization of quantum mechanics that describes the time evolution of particles as geodesic lines in a curved space, whose curvature is induced by the quantum potential. This formulation allows therefore the incorporation of all quantum effects into the geometry of space–time, as it is the case for gravitation in the general relativity.     

Causal stochastic interpretation of quantum statistics

The differences between Einstein and Bohr on the interpretation of quantum mechanics revolved around the question of completeness of the Copenhagen Interpretation. This fundamental problem is examined here in the light of recent neutron interference experiments which allow for novel experimental situations. Exploiting the possibility of neutron spin flip in these experiments, the inadequacy of the Copenhagen interpretation to fully understand the experimental results is brought out. Instead a causal interpretation of quantum mechanics is advocated, in which the neutron, as a particle, does always have a definite space time trajectory but also involves a wave which creates a potential affecting the particle neutron. The reestablishment of definite particle trajectories in the microscopic domain obliges us to reexamine the statistical treatment of ‘identical’ particles, as well as the problem of negative energies and probabilities in relativistic quantum mechanics.     

Quantum electrodynamics of one scalar particle

The theory of the interaction between a complex scalar field and the electromagnetic field is presented with initial and final conditions that allow an interpretation in the context of the relativistic quantum mechanics of a single charged scalar particle. Included are particle scattering, antiparticle scattering, pair creation, and pair annihilation due to a classical dynamical electromagnetic field. The equations of motion are solved by a perturbation expansion, which does not lead to the troublesome divergent terms of quantum field theory.     

The Madelung Picture as a Foundation of Geometric Quantum Theory

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.     

Quantum Chaos and Dynamical Entropy

We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion. Received: 26 June 1997 / Accepted: 13 April 1998     

无反射势阱中粒子运动的双波函数描述

应用双波函数量子理论,得到了描述无反射势阱中单粒子运动状态的力学量的时间演化方程,而通常量子力学中的平均值公式可描述为双波函数对某类系综的平均结果。 关键词：     

Fractals and quantum mechanics

A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.     

势场中粒子行为的量子计算与模拟

从量子力学的基本原理出发,采用指数差分方法,利用Matlab,模拟了入射不同势场后的粒子密度分布情况.结果表明,改变势场分布及粒子入射方式,粒子密度分布会产生显著变化.该模拟适用面广,可应用于不同的物理问题.     

Some Clarifications on the Relation Between Bohmian Quantum Potential and Mach’s Principle

Mach’s principle asserts that the inertial mass of a body is related to the distribution of other distant bodies. This means that in the absence of other bodies, a single body has no mass. In this case, talking about motion is not possible, because the detection of motion is possible only relative to other bodies. But in physics we are faced with situations that are not fully Machian. As in the case of general theory of relativity where geodesics exist in the absence of any matter, the motion has meaning. Another example which is the main topic of our discussion, refers to Bohmian quantum mechanics, where the inertial mass of a single particle does not vanish, but is modified. We can call such situations in which motion or mass of a single particle has meaning, pseudo-Machian situations. In this paper, we use the Machian or pseudo-Machian considerations to clarify under what circumstances and how a Machian effect leads us to Bohmian quantum mechanics. Then, we shall get the Bohmian quantum potential and its higher order terms for the Klein-Gordon particle through Machian considerations, without using any quantum mechanical postulate or operator formalism.     

瞬时响应的粒子探测器模型

本文中将Unruh-DeWitt型粒子探测器改造成瞬时响应的粒子探测器。经过改进之后,所有理想的探测器应是等价的,尽管它们对同一物理状态的响应可能不一样。本文给出一个可行的方法用以计算在Minkowski真空中作任意运动的探测器的响应函数。 关键词：     

一维无限深方势阱中粒子动量概率分布引出的问题

本文强调泡利关于一维无限深方势阱中粒子动量的结论与标准量子力学的逻辑推论不一致,而标准量子力学是自洽的。指出,当我们在一个量子态上掺入某种直观的经典力学内容时要很谨慎。至于对量子力学本身,至今尚无一种公认的诠释。     

Quantum information theory

A conceptual analysis of the classical information theory of Shannon (1948) shows that this theory cannot be directly generalized to the usual quantum case. The reason is that in the usual quantum mechanics of closed systems there is no general concept of joint and conditional probability. Using, however, the generalized quantum mechanics of open systems (A. Kossakowski 1972) and the generalized concept of observable (“semiobservable”, E.B. Davies and J.T. Lewis 1970) it is possible to construct a quantum information theory being then a straightforward generalization of Shannon's theory.     

Finite-dimensional quantum mechanics of a particle. II

The finite-dimensional quantum mechanics (FDQM) based on Weyl’s form of Heisenberg’s canonical commutation relations, developed for the case of one-dimensional space, is extended to three-dimensional space. This FDQM is applicable to the physics of particles confined to move within finite regions of space and is significantly different from the current quantum mechanics in the case of atomic and subatomic particles only when the region of confinement is extremely small—of the order of nuclear or even smaller dimensions. The configuration space of such a particle has a quantized eigenstructure with a characteristic dependence on its rest mass and dimension of the region of confinement, and the current Schrödinger-Heisenberg formalism of quantum mechanics becomes an asymptotic approximation of this FDQM. As an example, a spherical harmonic oscillator with a particular radius of confinement is considered.