The wave-corpuscle duality of microscopic particles depicted by nonlinear Schrödinger equation

We used a nonlinear Schrödinger equation to replace the linear Schrödinger equation and to study the states of microscopic particles due to plenty of difficulties of quantum mechanics. From this investigation we find that the properties of microscopic particles are considerably changed relative to those in quantum mechanics. An unusual change is that the microscopic particles have a wave-corpuscle duality. The wave feature is followed from the solution of nonlinear Schrödinger equation, which is composed of an envelope and carrier waves, which propagate with determined frequency and velocity. The corpuscle feature is verified by the following results, i.e., the solutions have a mass centre and determinant size, mass, momentum and energy, which satisfy conservation laws of mass, momentum and energy, their collision obeys the collision law of classical particles, etc. The roots generating these changes are due to nonlinear interactions among the particles or between the particles and background field in the equation. The nonlinear interactions provide a double-well potential to make the microscopic particle localized as a soliton. Thus the Hamiltonian operator of the system breaks through the fundamental hypothesis of independence of wave function of states in quantum mechanics. This investigation indicates that the microscopic particle should be described by the nonlinear Schrödinger equation, instead of the linear Schrödinger equation, and the quantum mechanics should develop towards the direction of nonlinear domain.