RESONANT SPREAD WAVE FUNCTION IN PARABOLIC POTENTIAL

In this paper, we consider the parabolic potential, which as a whole is subject to dipole or quadrupole action (parametric resonance), which periodically changes with time, and the dynamics of the wave function of a particle. Based on the solutions found for the nonstationary Schrödinger equation, algorithms for calculating the dynamics of the wave function are constructed. The evolution of the wave function of a particle is analyzed. Asymptotic solutions of the equation of motion are given, using which the main characteristics of the wave packet are obtained. For selected types of potential perturbations, examples of the evolution of the wave function are given.

Asymptotics of solution to the nonstationary Schrödinger equation

The Cauchy problem with a rapidly oscillating initial condition for the homogeneous Schr?dinger equation was studied in [5]. Continuing the research ideas of this work and [3], in this paper we construct the asymptotic solution to the following mixed problem for the nonstationary Schr?dinger equation: Lhu ? ih?tu + h2?2xu-b(x,t)u = f(x,t), (x,t) ? ??= (0,1) x (0,T], u|t=0 = g(x), u|x=0 = u|x=1 = 0, (1) where h > 0 is a Planck constant, u = u(x,t,h). b(x,t), f(x,t) ? C??(??), g(x) ? C? [0,1] are given functions. The similar problem was studied in [7, 8] when the Plank constant is absent in the first term of the equation and asymptotics of solution of any order with respect to a parameter was constructed. In this paper, we use a generalization of the method used in [7].