Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Do free quantum-mechanical wave packets always spread?

James R. Klein

Published Dec 1, 1980• 20 citations
Publisher

Abstract

The spreading or shrinking of free three-dimensional quantum-mechanical wave packets is studied, and several results are presented. A seeming paradox concerning the time evolution operator and nonspreading wave packets is discussed, and the necessity of taking into account the appropriate mathematical structure of quantum mechanics is emphasized. Implications for the teaching of quantum mechanics are given.

DOI
10.1119/1.12300

Related papers

Analyzing and constructing general nonspreading wave packets

Chyi-Lung Lin

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.

arXiv:1708.09589•Aug 31, 2017•0 citations

Uniqueness of the Airy packet in quantum mechanics

K. Unnikrishnan, A. R. P. Rau

The general form of a nonspreading wave packet in one-dimensional free space is derived from first principles, employing a decomposition of the quantum mechanical evolution operator of the free particle. In agreement with the classical analysis of Berry and Balazs, the corresponding probability density is proportional to the square of an Airy function, and the packet propagates with constant acceleration.

American Journal of Physics•Aug 1, 1996•75 citations

Is the Spreading of Quantum Mechanical Wave Packets Indeed Inevitable?

I. Mayer

It is demonstrated that -- contrary to the common belief -- it is possible to construct solutions of the non-relativistic Schrödinger equation of a free particle, that do not exhibit dispersion. However, it seems that no normalizable wave packets can be built up by their use, so the spreading of the wave packets is indeed inevitable.

arXiv:1209.0711•Sep 3, 2012•0 citations

The evolution and revival structure of localized quantum wave packets

Robert Bluhm, V. Alan Kostelecký, James A. Porter

Localized quantum wave packets can be produced in a variety of physical systems and are the subject of much current research in atomic, molecular, chemical, and condensed-matter physics. They are particularly well suited for studying the classical limit of a quantum-mechanical system. The motion of a localized quantum wave packet initially follows the corresponding classical motion. However, in most cases the quantum wave packet spreads and undergoes a series of collapses and revivals. We present a generic treatment of wave-packet evolution, and we provide conditions under which various types of revivals occur in ideal form. The discussion is at a level appropriate for an advanced undergraduate or first-year graduate course in quantum mechanics. Explicit examples of different types of revival structure are provided, and physical applications are discussed.

arXiv:quant-ph/9510029•American Journal of Physics•Jul 1, 1996•145 citations

Wave-Packet Evolution and Quantization

Robert G. Littlejohn

A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave packet. Instead, one must propagate the wave packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.

Physical Review Letters•May 12, 1986•20 citations

The effective Hamiltonian which governs the propagation dynamics of nonspreading wave packets

Chyi-Lung Lin

We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets.

arXiv:1306.1311•Jun 6, 2013•0 citations
The spreading or shrinking of free three-dimensional quantum-mechanical wave packets is studied, and several results are presented. A seeming paradox concerning the time evolution operator and nonspreading wave packets is discussed, and the necessity of taking into account the appropriate mathematical structure of quantum mechanics is emphasized. Implications for the teaching of quantum mechanics are given.
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.
Localized quantum wave packets can be produced in a variety of physical systems and are the subject of much current research in atomic, molecular, chemical, and condensed-matter physics. They are particularly well suited for studying the classical limit of a quantum-mechanical system. The motion of a localized quantum wave packet initially follows the corresponding classical motion. However, in most cases the quantum wave packet spreads and undergoes a series of collapses and revivals. We present a generic treatment of wave-packet evolution, and we provide conditions under which various types of revivals occur in ideal form. The discussion is at a level appropriate for an advanced undergraduate or first-year graduate course in quantum mechanics. Explicit examples of different types of revival structure are provided, and physical applications are discussed.
A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave packet. Instead, one must propagate the wave packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.
We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets.