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arXiv:0809.4105

Nonspreading wave packets in a general potential V(x,t) in one dimension

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

Published Sep 24, 2008• 1 citation
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Abstract

We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential $\textmd{V}(x,t)$. The essential ingredients of a nonspreading wave packet, the shape function $f(x)$, the motion $d(t)$, the phase function $ϕ(x,t)$ 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 $f(x)$ is the eigenfunction of the time independent Schrödinger equation with an effective potential $V_{\textmd{eff}}$ and an energy $E_{\textmd{eff}}$. 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.

DOI
10.48550/arxiv.0809.4105

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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.
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