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:0801.1834 · Journal of Physics A Mathematical and Theoretical

Non-dispersive wavepacket solutions of the Schrödinger equation

Shaun N Mosley

Published Jun 11, 2008• 2 citations
arXivHTMLPDF

Abstract

The free Schrödinger equation has localized constant velocity solutions ψ v of the form ψ = f ( r − v t ) e iλ t with λ being a constant. These solutions are eigenvectors of a momentum operator such that . The wavepacket, while not normalizable, is both localized and in a definite momentum state. The ψ v are orthogonal in the inner product space ⟨ϕ| r 2 |ψ⟩, and the operator is symmetric therein. We discuss whether these ψ v can act as basis states rather than the usual plane waves.

DOI
10.1088/1751-8113/41/26/265305

Related papers

NON-DISPERSIVE WAVEPACKET SOLUTIONS OF THE SCHRODINGER EQUATION

Shaun N. Mosley

The free Schrodinger equation has constant velocity wavepacket solutions ψv of the form ψ(r − vt). These solutions are eigenvectors of a velocity operator ˜v which is symmetric in a positive definite scalar product space. We discuss whether these ψv can act as basis states rather than the usual plane wave solutions.

•Nov 21, 2012•2 citations

Wavepacket Solutions of the Klein-Gordon Equation

Shaun N. Mosley

We find dispersion-free wavepacket solutions to the Klein-Gordon equation, with the only free parameter being the wavepacket velocity $ {\bf v} $. These wavefunctions are eigenvectors of a velocity operator with commuting components which is symmetric in a certain scalar product space. We show that this velocity operator corresponds to a classical generator which may be obtained by a canonical tranformation from $ {\bf x}, {\bf k} $.

arXiv:0707.3352•Jul 23, 2007•1 citation

The diffraction and spreading of a wavepacket

Guy Vandegrift

The spreading of a one-dimensional wavepacket solution of Schrödinger’s equation is related to the diffraction of light, as can be verified by considering the three-dimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrödinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to non-Gaussian wavepackets and to Fraunhoffer diffraction also is discussed.

American Journal of Physics•Feb 12, 2004•6 citations

Reformulation of the Schrödinger Equation

K. Wildermuth, Y. C. Tang

A symmetrical treatment of the incoming and outgoing channels can be done in a basically very simple way by formulating the Schrödinger equation (1.1) in the form of a projection equation (see, e.g., ref. [WI 72]) (2.1) $$ \langle \delta \psi \left| {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right|\psi \rangle = 0 $$ As usual, the Dirac bracket denotes the integration over spatial coordinates and the summation over spin and isospin coordinates. If δΨ represents a completely arbitrary variation of Ψ at a given instant t in the complete Hilbert space, then eq. (2.1) implies that the vector $$ \left( {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right)\left| {\psi \rangle } \right. $$ must be orthogonal to any arbitrary vector in this space. This evidently will be the case only if Ψ obeys eq. (1.1). Therefore, eq. (2.1) is merely another formulation of the time-dependent Schrödinger equation. However, as we shall see later, eq. (2.1) does allow us to treat the incoming and outgoing channels in a symmetrical way.

Vieweg+Teubner eBooks•Jan 1, 1977•0 citations

Airy wavepacket solutions of the Schrödinger equation

John Lekner

A Galilean boost to a frame moving with a complex velocity transforms the Berry–Balazs Airy function solution of the time-dependent Schrodinger equation into a square-integrable wavepacket. This packet has fixed momentum and energy; it contracts as time increases to zero, and then spreads again. The Berry–Balazs solution is revealed as a limiting form of a well-behaved free particle wavepacket. The topic is suitable for graduate or advanced undergraduate quantum mechanics courses.

European Journal of Physics•Apr 7, 2009•21 citations

Zero-mass fermion wavepackets

John Lekner

Abstract Localized spinor solutions of the zero-mass free-space Dirac equation, which are eigenstates of the angular momentum component along the direction of propagation, satisfy the three-dimensional wave equation. General expressions for and special compact forms of space-time localized spinors are given. An exact evaluation of the norm, energy, momentum, and angular momentum of such wavepackets is made. These are all finite. The integrals of the densities over all space, at fixed time, are shown to be equivalent to integrals over the wavenumber weight function. The localized wavepackets converge onto or diverge from a focal region, and their net momentum is smaller than their energy divided by the speed of light. They can be transformed to a (net) zero-momentum frame.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 23, 2026•0 citations
The free Schrödinger equation has localized constant velocity solutions ψ v of the form ψ = f ( r − v t ) e iλ t with λ being a constant. These solutions are eigenvectors of a momentum operator such that . The wavepacket, while not normalizable, is both localized and in a definite momentum state. The ψ v are orthogonal in the inner product space ⟨ϕ| r 2 |ψ⟩, and the operator is symmetric therein. We discuss whether these ψ v can act as basis states rather than the usual plane waves.
The spreading of a one-dimensional wavepacket solution of Schrödinger’s equation is related to the diffraction of light, as can be verified by considering the three-dimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrödinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to non-Gaussian wavepackets and to Fraunhoffer diffraction also is discussed.
A symmetrical treatment of the incoming and outgoing channels can be done in a basically very simple way by formulating the Schrödinger equation (1.1) in the form of a projection equation (see, e.g., ref. [WI 72]) (2.1) ⟨δψ∣H+ℏi∂∂t∣ψ⟩=0 \langle \delta \psi \left| {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right|\psi \rangle = 0 ⟨δψ​H+iℏ​∂t∂​​ψ⟩=0 As usual, the Dirac bracket denotes the integration over spatial coordinates and the summation over spin and isospin coordinates. If δΨ represents a completely arbitrary variation of Ψ at a given instant t in the complete Hilbert space, then eq. (2.1) implies that the vector (H+ℏi∂∂t)∣ψ⟩ \left( {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right)\left| {\psi \rangle } \right. (H+iℏ​∂t∂​)∣ψ⟩ must be orthogonal to any arbitrary vector in this space. This evidently will be the case only if Ψ obeys eq. (1.1). Therefore, eq. (2.1) is merely another formulation of the time-dependent Schrödinger equation. However, as we shall see later, eq. (2.1) does allow us to treat the incoming and outgoing channels in a symmetrical way.
Abstract Localized spinor solutions of the zero-mass free-space Dirac equation, which are eigenstates of the angular momentum component along the direction of propagation, satisfy the three-dimensional wave equation. General expressions for and special compact forms of space-time localized spinors are given. An exact evaluation of the norm, energy, momentum, and angular momentum of such wavepackets is made. These are all finite. The integrals of the densities over all space, at fixed time, are shown to be equivalent to integrals over the wavenumber weight function. The localized wavepackets converge onto or diverge from a focal region, and their net momentum is smaller than their energy divided by the speed of light. They can be transformed to a (net) zero-momentum frame.