The Changes of States and Properties of Microscopic Particles Under Influences of Different External Potential Fields

Abstract

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.

The Properties of the Solutions of Nonlinear Schrödinger Equation with Center Potential

“…Meanwhile, Pang used the nonlinear Schrödinger equation in Eq. (3) to depict the states and motions of microscopic particles in nonlinear quantum mechanics [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], where ( , ) r t f  is a wave function of state of microscopic particle, m is the mass of the particle, 2 2 / 2m ∇  is its kinetic energy operator, ( ) , V r t  is the externally applied potential, which is related to the positions of particles or a constant, r  is its position,…”

“…This equation has, in general, a soliton solution [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. This means that the microscopic particles move in a soliton states in microscopic systems.…”