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

The motion of wave packets through their expectation values and uncertainties

Daniel F. Styer

Published Aug 1, 1990• 40 citations
Publisher

Abstract

The Ehrenfest equations govern the time evolution of quantal expectation values, and they have a widely known generalization that governs the time evolution of quantal uncertainties. Unfortunately, these results are usually applied only to free particles. Here, these methods are reviewed and applied to the simple harmonic oscillator, revealing a simple and beautiful picture of wave-packet motion with an obvious classical limit.

DOI
10.1119/1.16396

Related papers

The Time-Dependent Schrödinger Equation

Reinhold A. Bertlmann, Nicolai Friis

Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description

•Sep 7, 2023•0 citations

A new look at quantal time evolution

Aparajita Bhattacharyya, Jayanta K. Bhattacharjee, Debabrata Sinha

Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.

American Journal of Physics•May 21, 2021•3 citations

Evolution of the wave function's shape in a time-dependent harmonic potential

Etera R. Livine

Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.

arXiv:2305.03847•Europhysics Letters (EPL)•Aug 1, 2023•2 citations

Wave Functions of a Harmonic Oscillator

Jerzy F Plebanski

By means of some special unitary operators, theorems are obtained extending classical properties of of Kennard's packets (classical motion of the center, and periodic time-depencence of $\ensuremath{\Delta}x$) over all solutions of the oscillator wave equation. These results are also generalized for the quantum field theory. Using a special "functional representation," nonspreading wave packets with classical properties are given for a general free field.

Physical Review•Mar 15, 1956•52 citations

Wave function of a Brownian particle

R. M. Cavalcanti

Using the Hamiltonian of Caldirola [Nuovo Cimento 18, 393 (1941)] and Kanai [Prog. Theor. Phys. 3, 440 (1948)], we study the time evolution of the wave function of a particle whose classical motion is governed by the Langevin equation $m\mathrm{x\ifmmode\ddot\else\textasciidieresis\fi{}}+\ensuremath{\eta}\mathrm{x\ifmmode \dot{}\else \.{}\fi{}}=F(t).$ We show in particular that if the initial wave function is Gaussian, then (i) it remains Gaussian for all times, (ii) its width grows, approaching a finite value when $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ and (iii) its center describes a Brownian motion and so the uncertainty in the position of the particle grows without limit.

arXiv:quant-ph/9805005•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 1, 1998•8 citations

On the Ehrenfest theorem of quantum mechanics

Gero Friesecke, Mario Koppen

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions.

arXiv:0907.1877•Journal of Mathematical Physics•Aug 1, 2009•17 citations
The Ehrenfest equations govern the time evolution of quantal expectation values, and they have a widely known generalization that governs the time evolution of quantal uncertainties. Unfortunately, these results are usually applied only to free particles. Here, these methods are reviewed and applied to the simple harmonic oscillator, revealing a simple and beautiful picture of wave-packet motion with an obvious classical limit.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.
Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.
Using the Hamiltonian of Caldirola [Nuovo Cimento 18, 393 (1941)] and Kanai [Prog. Theor. Phys. 3, 440 (1948)], we study the time evolution of the wave function of a particle whose classical motion is governed by the Langevin equation $m\mathrm{x\ifmmode\ddot\else\textasciidieresis\fi{}}+\ensuremath{\eta}\mathrm{x\ifmmode \dot{}\else \.{}\fi{}}=F(t).$ We show in particular that if the initial wave function is Gaussian, then (i) it remains Gaussian for all times, (ii) its width grows, approaching a finite value when $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ and (iii) its center describes a Brownian motion and so the uncertainty in the position of the particle grows without limit.