We debate first the properties of quantum mechanics and its difficulties and the reasons resulting in these diffuculties and its direction of development. The fundamental principles of nonlinear quantum mechanics are proposed and established based on these shortcomings of quantum mechanics and real motions and interactions of microscopic particles and backgound field in physical systems. Subsequently, the motion laws and wave-corpuscle duality of microscopic particles described by nonlinear Schr?dinger equation are studied completely in detail using these elementary principles and theories. Concretely speaking, we investigate the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the mechanism and rules of particle collision and the uncertainty relation of particle’s momentum and position, and so on. We obtained that the microscopic particles obey the classical rules of collision of motion and satisfy the minimum uncertainty relation of position and momentum, etc. From these studies we see clearly that the moved rules and features of microscopic particle in nonlinear quantum mechanics is different from those in linear quantum mechanics. Therefore, nolinear quantum mechanics is a necessary result of development of quantum mechanics and represents correctly the properties of microscopic particles in nonlinear systems, which can solve difficulties and problems disputed for about a century by scientists in linear quantum mechanics field.
We debate first the properties of quantum mechanics and its difficulties and the reasons resulting in these diffuculties and its direction of development. The fundamental principles of nonlinear quantum mechanics are proposed and established based on these shortcomings of quantum mechanics and real motions and interactions of microscopic particles and backgound field in physical systems. Subsequently, the motion laws and wave-corpuscle duality of microscopic particles described by nonlinear Schr?dinger equation are studied completely in detail using these elementary principles and theories. Concretely speaking, we investigate the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the mechanism and rules of particle collision and the uncertainty relation of particle’s momentum and position, and so on. We obtained that the microscopic particles obey the classical rules of collision of motion and satisfy the minimum uncertainty relation of position and momentum, etc. From these studies we see clearly that the moved rules and features of microscopic particle in nonlinear quantum mechanics is different from those in linear quantum mechanics. Therefore, nolinear quantum mechanics is a necessary result of development of quantum mechanics and represents correctly the properties of microscopic particles in nonlinear systems, which can solve difficulties and problems disputed for about a century by scientists in linear quantum mechanics field.
We establish the nonlinear quantum mechanics due to difficulties and problems of original quantum mechanics, in which microscopic particles have only a wave feature, not corpuscle feature, which are completely not consistent with experimental results and traditional concept of particle. In this theory the microscopic particles are no longer a wave, but localized and have a wave-corpuscle duality, which are represented by the following facts, the solutions of dynamic equation describing the particles have a wave-corpuscle duality, namely it consists of a mass center with constant size and carrier wave, is localized and stable and has a determinant mass, momentum and energy, which obey also generally conservation laws of motion, their motions meet both the Hamilton equation, Euler-Lagrange equation and Newton-type equation, their collision satisfies also the classical rule of collision of macroscopic particles, the uncertainty of their position and momentum is denoted by the minimum principle of uncertainty. Meanwhile the microscopic particles in this theory can both propagate in solitary wave with certain frequency and amplitude and generate reflection and transmission at the interfaces, thus they have also a wave feature, which but are different from linear and KdV solitary waves. Therefore the nonlinear quantum mechanics changes thoroughly the natures of microscopic particles due to the nonlinear interactions. In this investigation we gave systematically and completely the distinctions and variations between linear and nonlinear quantum mechanics, including the significances and representations of wave function and mechanical quantities, superposition principle of wave function, property of microscopic particle, eigenvalue problem, uncertainty relation and the methods solving the dynamic equations, from which we found nonlinear quantum mechanics is fully new and different from linear quantum mechanics. Finally, we verify further the correctness of properties of microscopic particles described by nonlinear quantum mechanics using the experimental results of light soliton in fiber and water soliton, which are described by same nonlinear Schr¶dinger equation. Thus we affirm that nonlinear quantum mechanics is correct and useful, it can be used to study the real properties of microscopic particles in physical systems.
The properties of quantum mechanics and its difficulties and problems are discussed, the reasons resulting in these diffuculties and direction of development of quantum mechanics are carefully studied. In practice, quantum mechanics neglects the real motions of the microscopic particles and backgound field as well as the real interactions between them. Therefore it can describe appromately the states of microsco[ic particles. For solving these difficulties of quantum mechanics, we proposed and established a nonlinear quantum mechanics based on the properties of macroscopic quantum effect and theories of superconductivity and superfluidity. In the new theory the elementary principles are proposed, its properties are also investigated. Subsequently, the motion laws and some main properties of microscopic particles in nonlinear quantum systems, includong the wave-particle duality of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles described by nonlinear quantum mechanics and the classical rule of microscopic particle motion, are studied using these elementary principles and theories. We obtained a lot of new and interesting results from this investigation, for example, the microscopic particles have really a wave-corpuscle duality in nonlinear quantum mechabics. The microscopic particles are localized due to the nonlinear interaction, have a determinant size, mass, energy and momentum. Thus the microscopic particles have not only a corpuscle but also a wave features, which are completely different from those in quantum mechanics. Hence we conclude that nolinear quantum mechanics is a new physical theory, a correct representation of describing microscopic particles and a necessary result of the development of quantum mechanics, but traditional quantum mechanics is only its an approximation and especial case at the nonlinear interaction to be equal to zero, it can solve the difficulties and problems disputed for about a century by scientists in quantum mechanics field. In the meanwhile, we use the new theory of nonlinear quantum mechanics to study properties of the excitons and phonons in an organic crystry (acetanilide). The results obtained show that the excitons and phonons become as a soliton having a wave-corpuscle duality in this system.Thus the correctness and avialablity of nonlinear quantum mechanics are demontrated using these experiments. We can predict that it has extensive applications in physics, chemistry, biology and polymers, etc.. Finally we state how the difficulties and problems in quantum mechanics are solved by the new theory of nonlinear quantum mechanics. Thus we can affirm from these investigations that the nonlinear quantum mechanics established by us is correct, quantum mechanics is only an appromate theory.
When the states and properties of microscopic particles were described by linear SchrÖdinger equation the quantum mechanics had a lot of difficulties, which cause a centenary controversy in physics and have not a determinate conclusions till now. Thus we used a nonlinear SchrÖdinger equation to replace the linear SchrÖdinger equation and to depict and study in detail the states, properties and rules of motion of microscopic particles. Concretely speaking, we here investigated the properties and rules of wave-particle duality of microscopic particles and its stability, the invariances and conservation laws of motion of particles, the classical rule of motion, Hamiltonian principle of particle motion, corresponding Lagrangian and Hamilton equations for the microscopic particle, the mechanism and rules of particle collision in different nonlinear systems, the uncertainty relation of position and momentum of particles, the features of reflection and transmission of the particles at interfaces as well as the eigenvalue and eigenequations of the particles, and so on.
The dynamic and collision features of microscopic particles described by nonlinear Schrodinger equation are investigated deeply using the analytic and the Runge-Kutta method of numerical simulation. The results show that the microscopic particles have a wave-corpuscle duality and are stable in propagation. When the two microscopic particles are collided, they can go through each other and retain their form after their collision of head-on from opposite directions, This feature is the same with that of the classical particles. However, a wave peak of large amplitude, which is a result of complicated superposition of two solitary waves, occurs in the colliding process. This displays the wave feature of microscopic particles. Therefore, the collision process shows clearly that the solutions of the nonlinear Schrodinger equation have a both corpuscle and wave feature, then the microscopic particles represented by the solutions have a wave-corpuscle duality. Obviously, this is due to the nonlinear interaction b||2. Thus we can determine the nonlinear Schrodinger equation can describe correctly the natures and properties of microscopic particles in quantum systems. Key words: Microscopic particle Schrodinger equation; Wave-corpuscle duality; Nonlinear interaction; Collision; Propagation; Quantum mechanics
The properties of microscopic particles are studied using the linear Schrödinger equation in quantum mechanics and nonlinear Schrödinger equation, respectively. The results obtained show that the microscopic particles have only a wave nature in quantum mechanics, but a wave-corpuscle duality in nonlinear systems depicted by the nonlinear Schrödinger equation, no matter the form of external potentials. Thus we know that the kinetic energy term in dynamic equations determines the wave feature of the particles; the nonlinear interaction term determines the corpuscle feature; their combination makes the microscopic particles have a wave-corpuscle duality. However the external potential term can change the phase and group velocities of motion, phase, amplitude, frequency and form of wave for the particles in both quantum mechanics and the nonlinear quantum systems, although it cannot change these fundamental natures of particles, no matter the forms. Meanwhile, we find that the changes of positions of the microscopic particles by increasing the time under action of an external potential satisfy the Newton-type equation of motion in nonlinear quantum systems. Thus the investigations make us not only see the limits and approximations of quantum mechanics but also know the necessity and importance of developing nonlinear quantum mechanics on the basis of the nonlinear Schrödinger equation.
原有量子力学的由Schrodinger方程不能很好地描述微观粒子的局域特性和波-粒二象性，因此，量子力学必须改进和发展。在非线性系统中我们考虑了粒子之间或它与背景场之间的非线性相互作用，于是必须用非线性Schrodinger方程去描述的微观粒子的状态。至此，微观粒子的状态和特征相对于线性系统发生了根本性的变化。在这种情况下不论外势场如何变化，但非线性Schrodinger方程都能给出具有波-粒二象特性的解，它由包络孤子和一个载波组成，有一个具有确定的位置的质心；同时，微观粒子此时也具有一个确定的质量，能量和动量，和它们遵守的质量，能量和动量守恒定律，在外场作用时粒子是稳定的，并满足经典运动方程和遵守哈密顿方程和拉格朗日方程。于是用非线性Schrodinger方程去描述的微观粒子具有一个明显的的局域特性和波-粒二象性， 从而彻底消除了原有量子力学存在近一个世纪的困难和问题。基于这些新的结果我们可建立起完整和系统的非线性量子力学理论体系，推动量子力学的发展。因此这一研究具有重要意义。 In original quantum mechanics the Schrodinger equation is, in essence, a wave equation, wherein the microscopic particles depicted have only a wave feature, and not a corpuscle nature. These descriptors do not agree only with the Broglie relation of wave-corpuscle duality, but also with experimental results and the traditional knowledge of particle concept. Meanwhile, the theory gives only some approximate solutions. Thus a series of contradictory representations and problems occur in quantum mechanics, which have re-sulted in durative disputations focused on the area of physics and have not led to any united conclusions until now. The only way to solve these problems and difficulties is to develop the quantum mechanics. We inves-tigate in detail wave-corpuscle duality of microscopic particles described by a nonlinear Schrodinger equa-tion in nonlinear quantum systems. Concretely speaking, we here study the properties of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles, invariance and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, and so on. Studied results show that the solution of the nonlinear Schrodinger equation depicting microscopic particles is a soliton and have a wave-corpuscle duality, microscopic particles have always a mass center and possess determinant positions, sizes, mass, mo-mentum, energy and form, their mass, momentum and energy satisfy corresponding conservation laws, their dynamic states can be described by both nonlinear Schrodinger equation and classical Lagrangian and Ham-ilton equations, their motions obey the classical Newton-type law of motion. These properties indicate that the microscopic particles described by a nonlinear Schrodinger equation have a corpuscle nature. However, the solutions of dynamic equation are some solitary waves which are accompanied by a carrier wave to propagate in space-time, and possess certain frequency and wave speed. Thus microscopic particles described by nonlinear quantum mechanics have also wave feature. Thus we can affirm that the microscopic particles depicted by a nonlinear Schrodinger equation have a perfect wave-corpuscle duality, which are in essence different from those in linear quantum mechanics. Based on these results we can establish a new nonlinear quantum mechanics, which can solve these difficulties and problems disputed for about a century in linear quantum mechanics.