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Europhysics Letters (EPL)

The general potential V(x,t) in which Airy wave packets remain nonspreading

Chyi-Lung Lin, Te-Chih Hsiung, Meng-Jie Huang

Published Jul 1, 2008• 13 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/83/30002

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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.
Berry and Balazs showed that an initial Airy packet Ai(b x) under time evolution is nonspreading in free space and also in a homogeneous time-varying linear potential V(x,t)=-F(t) x. We find both results can be derived from the time evolution operator U(t). We show that U(t) can be decomposed into ordered product of operators and is essentially a shift operator in x; hence, Airy packets evolve without distortion. By writing the Hamiltonian H as H=H_b+H_i, where H_b is the Hamiltonian such that Ai(b x) is its eigenfunction. Then, H_i is shown to be as an interacting Hamiltonian that causes the Airy packet into an accelerated motion of which the acceleration a=(-H_i/( x))/m. Nonspreading Airy packet then acts as a classical particle of mass m, and the motion of it can be described classically by H_i.
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.
We show the method for constructing nonspreading wave packets whose shape and motion can be general. We analyze the time evolution of nonspreading wave packets by decomposing the Hamiltonian into two parts. Of the two, one changes the instantaneous state, the other does not. Through this decomposition, the time evolution operator is shown to be effectively a spatial shifting operator. This explains why nonspreading wave packets can be nonspreading. And we show that the part of the Hamiltonian which changes the instantaneous state governs the motion of the nonspreading wave packets.
We present the solution of Schrödinger’s equation for time-dependent quadratic potentials in terms of an Airy packet. We demonstrate new features of the Airy packet in light of its apparent unfamiliar properties discovered by Berry and Balazs (who found a nonspreading and accelerating Airy packet to be a unique solution to the free-particle Schrödinger equation). Our solution, found using space-time scale transformations, not only overcomes the packet’s tendency to accelerate, but also make it possible to spread or contract in time.