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

arXiv:quant-ph/0208106

Wave packets of a harmonic oscillator with various degrees of rigidity

Qiong-Gui Lin

Published Aug 15, 2002• 0 citations
arXivHTMLPDF

Abstract

The time evolution of wave packets in a harmonic oscillator potential is studied. Some new results for the most general case are obtained. A natural number, called ``degree of rigidity'', is introduced to describe qualitatively how much the shape of a wave packet is changed with time. Two classes of wave packets with an arbitrarily given degree of rigidity are presented.

DOI
10.48550/arxiv.quant-ph/0208106

Related papers

Harmonic Oscillator Wave Functions

I. R. Senitzky

A harmonic oscillator wave packet which oscillates sinusoidally without change in shape has been described by Schr\"odinger and others. It is shown in the present note that there are an infinite number of such wave packets---of which the above is a special one-having the shapes of the various eigenfunctions of the harmonic oscillator Hamiltonian. The relation of these wave packets to the classical oscillator is discussed.

Physical Review•Sep 1, 1954•45 citations

Shape-invariant wave packets for three-dimensional harmonic oscillators

Frans Arickx, J. Broeckhove, Piet Van Leuven

It is shown that for a three-dimensional harmonic oscillator there exist families of Gaussian wave packets that retain their shape during the time evolution, i.e., the wave packet rotates around its center without spreading or deforming. This generalizes the concept of coherent vibrational motion to coherent rotational motion.

Physical Review A•Feb 1, 1991•8 citations

Almost-periodic wave packets and wave packets of invariant shape

Detlef Mentrup, Marshall Luban

Wave packets of one-dimensional, single-particle quantum systems exhibit a rich diversity of dynamical behavior. We focus on confining potentials where the entire energy spectrum is discrete so that wave packets do not spread indefinitely with time and instead exhibit characteristics of almost-periodic functions. For the harmonic oscillator, for which all wave packets are necessarily strictly periodic, Senitzky has shown that is possible to construct an infinite class of normalized wave packets that maintain a fixed shape while oscillating sinusoidally with time. We present two alternate and instructive methods for deriving this result using coherent states and the displacement operator.

American Journal of Physics•May 12, 2003•3 citations

Harmonic Oscillator Wave Packets

Saul T. Epstein

Time-dependent operators are used to find harmonic oscillator wave packets which oscillate without change in shape. The results are in agreement with those of Senitzky and others. Some useful averages are computed for these packets and some comments are made on the classical limit.

American Journal of Physics•May 1, 1959•11 citations

Simple approach to the time evolution of wave packets in a harmonic oscillator potential

Qiong-Gui Lin

A simple approach is presented to calculate the explicit time dependent wave functions of some wave packets in a harmonic oscillator potential. In this approach no knowledge on special functions is needed.

College Physics•Jan 1, 2003•0 citations

Soliton like solutions of the Schrödinger equation for simple harmonic oscillator

Chai Yan

By solving exactly with a simple method the time dependent Schrödinger equation for simple harmonic oscillator, we demonstrate that the probability density associated with any displaced eigenstate has the following properties: (a) Its centroid oscillates according to the classical law. (b) Its wave packet form remains rigid. The same method is also applied to solve exactly the problem of simple harmonic oscillator with the force constant depending on time. In this case, probability densities with the same wave packet structures as that of the usual simple harmonic oscillator can be found to move in such a way that their centroids follow the classical law but their wave packets deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.

American Journal of Physics•Feb 1, 1994•32 citations
The time evolution of wave packets in a harmonic oscillator potential is studied. Some new results for the most general case are obtained. A natural number, called ``degree of rigidity'', is introduced to describe qualitatively how much the shape of a wave packet is changed with time. Two classes of wave packets with an arbitrarily given degree of rigidity are presented.
Wave packets of one-dimensional, single-particle quantum systems exhibit a rich diversity of dynamical behavior. We focus on confining potentials where the entire energy spectrum is discrete so that wave packets do not spread indefinitely with time and instead exhibit characteristics of almost-periodic functions. For the harmonic oscillator, for which all wave packets are necessarily strictly periodic, Senitzky has shown that is possible to construct an infinite class of normalized wave packets that maintain a fixed shape while oscillating sinusoidally with time. We present two alternate and instructive methods for deriving this result using coherent states and the displacement operator.
By solving exactly with a simple method the time dependent Schrödinger equation for simple harmonic oscillator, we demonstrate that the probability density associated with any displaced eigenstate has the following properties: (a) Its centroid oscillates according to the classical law. (b) Its wave packet form remains rigid. The same method is also applied to solve exactly the problem of simple harmonic oscillator with the force constant depending on time. In this case, probability densities with the same wave packet structures as that of the usual simple harmonic oscillator can be found to move in such a way that their centroids follow the classical law but their wave packets deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.