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

The European Physical Journal Plus

Evolution and invariants of oscillator moments

Mark Andrews

Published Apr 1, 2022• 1 citation
Publisher

Abstract

Abstract Moments are expectation values over wave functions (or averages over a set of classical particles) of products of powers of position and momentum. For the harmonic oscillator, the evolution in the quantum case is very closely related to that of the classical case. Here we consider the non-relativistic evolution of moments of all orders for the oscillator in one dimension and investigate invariant combinations of the moments. In particular, we find an infinite set of invariants that enable us to express the evolution of any moment in terms of sinusoids. We also find explicit expressions for the inverse of these relations, thus enabling the expression of the evolution of any moment in terms of the initial set of moments. More detailed attention is given to moments of the third and fourth order in terms of the invariant combinations.

DOI
10.1140/epjp/s13360-022-02656-0

Related papers

Evolution and invariants of free-particle moments

Mark Andrews

Abstract Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a set of classical particles. Here we consider the evolution of symmetrized moments for free particles in one dimension, first examining the geometric properties of the evolution for moments up to the fourth order, as determined by their extrema and inflections. These properties are specified by combinations of the moments that are invariant in that they remain constant under free evolution. An inequality constrains the fourth-order moments and shows that some geometric types of evolution are possible for a quantum particle but not possible classically, and some examples are examined. Explicit expressions are found for the moments of any order in terms of their initial values, for the invariant combinations, and for the moments in terms of these invariants.

arXiv:2012.06699•Journal of Physics A Mathematical and Theoretical•Mar 26, 2021•2 citations

Classical and quantum behavior of the harmonic and the quartic oscillators

David Brizuela

In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.

arXiv:1411.1522•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 18, 2014•26 citations

Invariant operators for quadratic Hamiltonians

Mark Andrews

The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.

American Journal of Physics•Apr 1, 1999•23 citations

The evolution of oscillator wave functions

Mark Andrews

We investigate how wave functions evolve with time in the harmonic oscillator. We first review the periodicity properties over each multiple of a quarter of the classical oscillation period. Then we show that any wave function can be simply transformed so that its centroid, defined by the expectation values of position and momentum, remains at rest at the center of the oscillator. This implies that we need only consider the evolution of this restricted class of wave functions; the evolution of all others can be reduced to these. The evolution of the spread in position Δx and momentum Δp throws light on energy and uncertainty and on squeezed and coherent states. Finally, we show that any wave function can be transformed so that Δx and Δp do not change with time and that the evolution of all wave functions can easily be found from the evolution of those at rest at the origin with unchanging Δx and Δp.

arXiv:1509.05968•American Journal of Physics•Mar 21, 2016•5 citations

Position moments linearly averaged over the wave function. II

Toshikatsu Koga

Based on the Fourier transform of the position–space Schrödinger equation in a finite domain, a set of relations is derived among position moments linearly averaged over the wave function in that domain. The present results add nontrivial information for the structure of the Schrödinger equation, including the known relations for an infinite domain. A special case provides a regional form of the zero momentum energy expression, which measures the accuracy of wave functions in some specific region.

The Journal of Chemical Physics•May 15, 1989•3 citations

Some simple solutions of Schrödinger's equation for a free particle or for an oscillator

Mark Andrews

Abstract For a non-relativistic free particle, we show that the evolution of some simple initial wave functions made up of linear segments can be expressed in terms of Fresnel integrals. Examples include the square wave function and the triangular wave function. The method is then extended to wave functions made from quadratic elements. The evolution of all these initial wave functions can also be found for the harmonic oscillator by a transformation of the free evolutions.

European Journal of Physics•Jan 26, 2018•0 citations
Abstract Moments are expectation values over wave functions (or averages over a set of classical particles) of products of powers of position and momentum. For the harmonic oscillator, the evolution in the quantum case is very closely related to that of the classical case. Here we consider the non-relativistic evolution of moments of all orders for the oscillator in one dimension and investigate invariant combinations of the moments. In particular, we find an infinite set of invariants that enable us to express the evolution of any moment in terms of sinusoids. We also find explicit expressions for the inverse of these relations, thus enabling the expression of the evolution of any moment in terms of the initial set of moments. More detailed attention is given to moments of the third and fourth order in terms of the invariant combinations.
Abstract Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a set of classical particles. Here we consider the evolution of symmetrized moments for free particles in one dimension, first examining the geometric properties of the evolution for moments up to the fourth order, as determined by their extrema and inflections. These properties are specified by combinations of the moments that are invariant in that they remain constant under free evolution. An inequality constrains the fourth-order moments and shows that some geometric types of evolution are possible for a quantum particle but not possible classically, and some examples are examined. Explicit expressions are found for the moments of any order in terms of their initial values, for the invariant combinations, and for the moments in terms of these invariants.
In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.
The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.
We investigate how wave functions evolve with time in the harmonic oscillator. We first review the periodicity properties over each multiple of a quarter of the classical oscillation period. Then we show that any wave function can be simply transformed so that its centroid, defined by the expectation values of position and momentum, remains at rest at the center of the oscillator. This implies that we need only consider the evolution of this restricted class of wave functions; the evolution of all others can be reduced to these. The evolution of the spread in position Δx and momentum Δp throws light on energy and uncertainty and on squeezed and coherent states. Finally, we show that any wave function can be transformed so that Δx and Δp do not change with time and that the evolution of all wave functions can easily be found from the evolution of those at rest at the origin with unchanging Δx and Δp.
Based on the Fourier transform of the position–space Schrödinger equation in a finite domain, a set of relations is derived among position moments linearly averaged over the wave function in that domain. The present results add nontrivial information for the structure of the Schrödinger equation, including the known relations for an infinite domain. A special case provides a regional form of the zero momentum energy expression, which measures the accuracy of wave functions in some specific region.