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

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Relation between the solutions of Newton's and Heisenberg's equations of motion

Francis R. Halpern

Published Sep 15, 1978• 0 citations
Publisher

Abstract

For a class of problems it is shown that a solution to Newton's equation of motion may be interpreted as the generating function for the matrix elements of the coordinate operators in the Heisenberg representation. For three one-dimensional problems, the harmonic oscillator, the particle in a box, and the ${x}^{4}$ anharmonic oscillator, solutions to the quantum-mechanical problem are given in terms of the classical solution. A brief examination is made of a two-dimensional problem in which the angular momentum is conserved.

DOI
10.1103/physrevd.18.1894

Related papers

The classical Lenz vector and the two-dimensional quantum harmonic oscillator

Stefano Palmisano, Antonio Davide Polosa, R. Senese +1 more

Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.

Physics Open•Apr 8, 2020•0 citations

Quantum mechanics from Newton's second law and the canonical commutation relation [ X , P ] = i

Mark C. Palenik

Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations $F=\frac{dP}{dt}$, $P=m\frac{dV}{dt}$, and $\left[X,P\right]=i$. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.

arXiv:1402.0396•European Journal of Physics•May 9, 2014•4 citations

Quantum Mechanics in the Heisenberg (Matrix) Picture

Ana Marı́a Cetto, Luis de la Peña

Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.

Cambridge University Press eBooks•Dec 18, 2025•0 citations

Matrix elements in harmonic oscillator representations I. General solution and application to one dimension

Stephen G. Bell

General expressions have been obtained for the matrix elements of linear or radial perturbations in the representations of the one-, two-, and three-dimensional harmonic oscillators. The solutions are also general for the two types of matrix element where the two wave functions belong to the same basis set or to two different basis sets. In this paper these expressions are applied to one- dimensional problems.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1970•13 citations

Heisenberg's Quantum Mechanics

Mohsen Razavy

A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities

WORLD SCIENTIFIC eBooks•Jan 1, 2011•16 citations

STUDY ON THE EQUIVALENCE BETWEEN COORDINATE REPRESENTATION AND MOMENTUM REPRESENTATION FOR ONE-DIMENSIONAL HARMONIC OSCILLATOR

Huaiyu JI, Zijun Zhang, Hua Zheng +1 more

The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.

Wuli yu gongcheng.•Jun 1, 2025•0 citations
For a class of problems it is shown that a solution to Newton's equation of motion may be interpreted as the generating function for the matrix elements of the coordinate operators in the Heisenberg representation. For three one-dimensional problems, the harmonic oscillator, the particle in a box, and the x4{x}^{4}x4 anharmonic oscillator, solutions to the quantum-mechanical problem are given in terms of the classical solution. A brief examination is made of a two-dimensional problem in which the angular momentum is conserved.
Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities
The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.