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

TL;DR

This paper develops a general framework for constructing nonspreading wave packets in one-dimensional quantum systems with time-dependent potentials, extending previous solutions and enabling arbitrary wave packet motion.

Contribution

It introduces a unified method to derive nonspreading wave packets from general potentials, including moving packets in time-dependent scenarios, expanding beyond known stationary solutions.

Findings

Derived general rules for constructing nonspreading wave packets.

Showed shape function is an eigenfunction of an effective Schrödinger equation.

Demonstrated how to generate moving nonspreading wave packets.

Abstract

We discuss nonspreading wave packets in one dimensional Schr\"{o}dinger 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 $\phi(x,t)$ 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 Schr\"{o}dinger equation with an effective potential $V_{\textmd{eff}}$ and an energy $E_{\textmd{eff}}$. We derive nonspreading wave packets found by Schr\"{o}dinger, 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.