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AIP conference proceedings

Wavelets and the Cauchy problem for the Schrodinger equation on analytic ultrametric space

Andrei Yu. Khrennikov

Published Jan 1, 2006• 7 citations
Publisher

Abstract

Free evolution for quantum particle in general ultrametric space is considered. We find that if mean zero wave packet is localized in some ball in the ultrametric space then its evolution remains localized in the same ball.
DOI
10.1063/1.2205817

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Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.
The behavior of a particle moving according to the laws of quantum mechanics in a field, which potential changes over time, is studied. The method of unitary transformations for solving the temporal Schrödinger equation is considered. The reduced Hamiltonian of the system of a quantum­-mechanical particle in time­-dependent potential is obtained, as well as the total operator of the evolution of such a system. We find a new version of the unitary transformation, which, compared to the known ones, simplifies solving the problem in analytical form. This transformation associates the linear potential quantum model with time-­dependent parameters and free­particle system. Based on the constructed evolution operator, we consider the wave packet dispersion. It is shown that the wave packet dispersion for this model is the same as that for a free particle. This conclusion is the general result of the behavior of a quantum­mechanical particle moving in the field of a time­dependent linear potential.