Abstract
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.
Highlights
We debate first the properties of quantum mechanics and its difficulties and the reasons resulting in these diffuculties and its direction of development
It is well known that several great scientists, such as Bohr, Born, Schrödinger and Heisenberg, etc. established quantum mechanics in the early 1900s [1,2,3,4,5,6,7,8,9], which is the foundation and pillar of modern science and provides an unique way of describing the properties and rules of motion of microscopic particles (MIP) in microscopic systems
The simplest system in nature is the hydrogen atom, but it consists of two particles. In such a case, when we study the states of particles in realistic systems composed of many particles and many bodies using quantum mechanics, we have to use a simplified and uniform average-potential unassociated with the states of particles to replace the complicated and nonlinear interaction among these particles [19,20,21,22,23,24,25]
Summary
Pang worked out the NLQM describing the properties of motion of MIPs in nonlinear systems [17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The elementary principles, theory, calculated rules and applications of NLQM are proposed and established based on the relations among the nonlinear interaction and soliton motions and macroscopic quantum effect through incorporating modern theories of superconductors, superfluids and solitons [23,24,25,26,27]. Based on the elementary principle Pang [23,24,25,26,27] established the theory of nonlinear quantum mechanics, which includes the superposition theorem of state of the particles, relation of nonlinear Fourier transformation, nonlinear perturbation theory, theory of nonlinear quantization, eigenvalue theory of nonlinear Schrödinger equation, calculated method of eigenenergy of Hamiltonian operator and relativistic theory of nonlinear quantum mechanics, collision and scattering theory of microscopic particles, and so on [25,26,27]. We can investigate the rules and properties of motion of microscopic particles in any physical systems using these principle and theories of nonlinear quantum mechanics
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