Collisions and stability of quantum wave packets

Time evolution of the quantum wave packets is discussed in context of the non-linear cubic Shrцdinger equation and equivalent hydrodynamical description. In hydrodynamical description, the quantum Hamilton-Jacobi equation for action is rewritten in variables: probability density and probability flow density. These variables are smooth at the node points. The studied dynamical systems have finite dimensions and impenetrable walls, they have been analyzed at the different initial conditions including the Gaussian-like form. Our interest in investigation of the properties of dynamical non-linear equations is caused by existence of stable solutions which correspond to the quantum non-spreading wave packets. Behavior of the localized wave packet in one-dimensional system is characterized by classic-like trajectory and collisions against walls. The packet keeps localized form during some time interval and can oscillate around some “stable” profile. To describe the time evolution of two wave packets on plane we have to integrate the non-stationary two dimensional Shrцdinger equation for the two-particle wave function taking into account the symmetry properties. But, as first step, in present investigation the problem was essentially simplified. The particles were considered as spinless, and wave function was presented in the product form of two functions. Now, the collisions between quantum wave packets of two particles will also occur. During some time interval, fragmentations of packets are generated. Then they return to its original shape and move as classical particles. In both cases, non-linearity plays self-organizing role in comparison to the regimes when non-linear cubic term is absent