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arXiv:1010.1678

Linear Potentials, Airy Wave Packets and Airy Transform

G. Dattoli, Жуковский Константин Владимирович

Published Oct 8, 2010• 2 citations
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Abstract

The solution of the Schrodinger equation with a linear potential is considered. We use algebraic methods to obtain the explicit form of the solution for the explicitly time dependent Hamiltonian and discuss the general conditions which allow us to get solutions in terms of the Airy functions, yelding non spreading wave packets. We analyze the relevant physical meaning of these solutions and give examples of their applications. We discuss the analogy between Airy and Gauss-Weirestrass transform.

DOI
10.48550/arxiv.1010.1678

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The solution of the Schrodinger equation with a linear potential is considered. We use algebraic methods to obtain the explicit form of the solution for the explicitly time dependent Hamiltonian and discuss the general conditions which allow us to get solutions in terms of the Airy functions, yelding non spreading wave packets. We analyze the relevant physical meaning of these solutions and give examples of their applications. We discuss the analogy between Airy and Gauss-Weirestrass transform.
We show that for Hamiltonian of the form, H = c ( t )∂ x ∂ x + V ( x , t ), the general potential V ( x , t ) for which Airy packets remain nonspreading under time evolution can only be of the form V ( x , t )= p ( t ) x + q ( t ), where p ( t ) and q ( t ) are real functions. We derive rules for constructing nonspreading Airy packets from this general potential. We also consider normalizable packets obtained from Airy packet times a converging factor, such as e β x , with β> 0. We show that there is no potential which sustains nonspreading wave packets of this kind.
In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schrödinger equation, which is given by the Airy function. The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation. Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential. The method is shown to work in both one and two dimensions, and can be easily extended if required. The applicability of the method is discussed in relation to the nonlinear case.
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.