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 A

Exact classical limit of quantum mechanics: Noncentral potentials and Ermakov-type invariants

Adam J. Makowski

Published Aug 6, 2003• 9 citations
Publisher

Abstract

Two-dimensional noncentral potentials are derived in the exact classical limit of quantum mechanics where the quantum correction to the classical Hamilton-Jacobi equation is canceled. It is then shown explicitly that there exist Ermakov-type invariants I for a large class of such potentials. Some of the invariants are found to be positive definite, $I>0,$ and for the other, we have found $I=0.$

DOI
10.1103/physreva.68.022102

Related papers

Exact classical limit of quantum mechanics: Central potentials and specific states

Adam J. Makowski

A derivation has been performed of some central potentials with no quantum correction to the Hamilton-Jacobi equation. In this exact limit of quantum mechanics, quantum trajectories identical to the classical ones are obtained. Interestingly, some of them are closed orbits. Applications of the found potentials in many areas of physics are also discussed.

Physical Review A•Jan 30, 2002•17 citations

Bohr’s correspondence principle: The cases for which it is exact

Adam J. Makowski, K. Górska

Two-dimensional central potentials leading to the identical classical and quantum motions are derived and their properties are discussed. Some of zero-energy states in the potentials are shown to cancel the quantum correction $Q=(\ensuremath{-}{\ensuremath{\Elzxh}}^{2}/2m)\ensuremath{\Delta}R/R$ to the classical Hamilton-Jacobi equation. The Bohr's correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0,$ or of the like. In this exact limit of $Q=0,$ classical trajectories are found and classified. Interestingly, many of them are represented by closed curves. Applications of the found potentials in many areas of physics are briefly commented.

Physical Review A•Dec 6, 2002•18 citations

On the Classical Limit of the Quantum Mechanics Based on the Hamilton-Jacobi Equation

Li-Feng Wang

Based on the Hamilton-Jacobi equation and the Bohm's quantum potential,it is pointed out that the classical limit of the quantum mechanics is the classical statistical mechanics.Through analyzing the Gaussian packet from two approaches,the perspective of the classical limit via the H-J equation is shown to be more profound than that via the Ehrenfest theorem.

Journal of Anyang Normal University•Jan 1, 2009•0 citations

Exactly solvable noncentral potentials in two and three dimensions

Avinash Khare, Rajat K. Bhaduri

We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.

arXiv:hep-th/9310104•American Journal of Physics•Nov 1, 1994•57 citations

Electromagnetic potentials and Aharonov-Bohm effect

A. I. Ershkovich

Hamilton-Jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials (A,ϕ), similar to Schroedinger equation. We derived the Aharonov-Bohm effect from Hamilton-Jacobi equation thereby having proved that this effect is of classical origin. These facts enable us to arrive at the following conclusions: a) the very idea of special role of potentials (A,ϕ) in quantum mechanics (different from their role in classical physics) lost the ground, and becomes dubious, as this idea is based on the Aharonov-Bohm effect, b) failure to find any signs of a special role of these potentials in the appropriate experiments (Feinberg, 1963) is thereby explained, and c) discovery of classical analogues of the Aharonov-Bohm effect (Berry et al., 1980) is also explained by a classical nature of this effect. Elucidation of the "unlocal" interaction problem was made.

arXiv:1209.1078•Sep 4, 2012•2 citations

The classical limit of entropic quantum dynamics

Anthony Demme, Ariel Caticha

The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ħ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.

arXiv:1612.01905•AIP conference proceedings•Jan 1, 2017•10 citations
Two-dimensional noncentral potentials are derived in the exact classical limit of quantum mechanics where the quantum correction to the classical Hamilton-Jacobi equation is canceled. It is then shown explicitly that there exist Ermakov-type invariants I for a large class of such potentials. Some of the invariants are found to be positive definite, I>0,I>0,I>0, and for the other, we have found I=0.I=0.I=0.
Two-dimensional central potentials leading to the identical classical and quantum motions are derived and their properties are discussed. Some of zero-energy states in the potentials are shown to cancel the quantum correction $Q=(\ensuremath{-}{\ensuremath{\Elzxh}}^{2}/2m)\ensuremath{\Delta}R/R$ to the classical Hamilton-Jacobi equation. The Bohr's correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0,$ or of the like. In this exact limit of Q=0,Q=0,Q=0, classical trajectories are found and classified. Interestingly, many of them are represented by closed curves. Applications of the found potentials in many areas of physics are briefly commented.
We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.
Hamilton-Jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials (A,ϕ), similar to Schroedinger equation. We derived the Aharonov-Bohm effect from Hamilton-Jacobi equation thereby having proved that this effect is of classical origin. These facts enable us to arrive at the following conclusions: a) the very idea of special role of potentials (A,ϕ) in quantum mechanics (different from their role in classical physics) lost the ground, and becomes dubious, as this idea is based on the Aharonov-Bohm effect, b) failure to find any signs of a special role of these potentials in the appropriate experiments (Feinberg, 1963) is thereby explained, and c) discovery of classical analogues of the Aharonov-Bohm effect (Berry et al., 1980) is also explained by a classical nature of this effect. Elucidation of the "unlocal" interaction problem was made.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ħ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.