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

Time evolution and wavepackets

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

Abstract Here, the chapter discusses time evolution in quantum mechanics and in particular of a free-particle state which is initially prepared as a wavepacket.
DOI
10.1093/acprof:oso/9780198749349.003.0005

Related papers

Evolution of a Gaussian Wavepacket

S. M. Blinder

An alternative derivation is given for the evolution in time of a Gaussian wavepacket representing a free particle. This is based on explicit evaluation of the action of the evolution operator on the initial wavefunction, rather than on expansions in free-particle eigenstates. Both the cases of a free particle initially at rest and in uniform linear motion are considered.

American Journal of Physics•Jun 1, 1968•22 citations

Wavepackets

Mark Sheard Child

Abstract The discussion in previous chapters has centred on the use of trajectorybased information to add quantum corrections to the classical picture, in a mainly time-independent picture. This chapter takes an explicitly timedependent standpoint by examining the relevance of various types of wavepacket to classical mechanics and vice versa; Littlejohn (1986) gives a much fuller mathematical account. The intuitive connection is obvious, in the sense that a unit normalized wavepacket corresponds most nearly to a single quantum mechanical particle, but the associated uncertainty relations imply a closer connection with a classical ensemble. Analysis of this connection is one of the main concerns of the first half of the chapter.

•Jan 3, 1991•0 citations

Wavefunctions

Gary E. Bowman

This chapter focuses on a basic but systematic treatment of wavefunctions. By comparison with a spin 1/2 system, the concept of a wavefunction as a collection of coefficients is developed, followed by discussions of eigenstates, operators, and representations in the context of wavefunctions. Transformation from position to momentum representation is effected by a Fourier transform. This is developed next, and illustrated for a Gaussian. The discussion of phases and time evolution in earlier chapters is now applied to wavefunctions, and the time evolution of a free particle discussed. The concept of a wavepacket and its suitability for representing particles follows. A topic of some subtlety — and one which is discussed in detail — is the use of bra-ket (Dirac) notation for wavefunctions. The chapter closes with a short epilogue.

Oxford University Press eBooks•Nov 1, 2007•0 citations

Chapter 5. Aspects of Time-Dependent Quantum Mechanics

Eric Heller

Princeton University Press eBooks•Aug 13, 2018•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

Free expansion of a Gaussian wavepacket using operator manipulations

Alessandro M. Orjuela, James K. Freericks

The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.

arXiv:2305.00059•American Journal of Physics•May 23, 2023•3 citations
Abstract The discussion in previous chapters has centred on the use of trajectorybased information to add quantum corrections to the classical picture, in a mainly time-independent picture. This chapter takes an explicitly timedependent standpoint by examining the relevance of various types of wavepacket to classical mechanics and vice versa; Littlejohn (1986) gives a much fuller mathematical account. The intuitive connection is obvious, in the sense that a unit normalized wavepacket corresponds most nearly to a single quantum mechanical particle, but the associated uncertainty relations imply a closer connection with a classical ensemble. Analysis of this connection is one of the main concerns of the first half of the chapter.
This chapter focuses on a basic but systematic treatment of wavefunctions. By comparison with a spin 1/2 system, the concept of a wavefunction as a collection of coefficients is developed, followed by discussions of eigenstates, operators, and representations in the context of wavefunctions. Transformation from position to momentum representation is effected by a Fourier transform. This is developed next, and illustrated for a Gaussian. The discussion of phases and time evolution in earlier chapters is now applied to wavefunctions, and the time evolution of a free particle discussed. The concept of a wavepacket and its suitability for representing particles follows. A topic of some subtlety — and one which is discussed in detail — is the use of bra-ket (Dirac) notation for wavefunctions. The chapter closes with a short epilogue.
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.
The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.