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

Inequivalence of the classes of classical and quantum harmonic potentials: Proof by example

Michael Martin Nieto, Vincent P. Gutschick

Published Feb 15, 1981• 26 citations
Publisher

Abstract

There are an infinite number of potentials whose classical angular velocity ${\ensuremath{\omega}}_{c}$ is independent of energy. There are also an infinite number of potentials whose quantum eigenenergies are equally spaced, $\ensuremath{\hbar}{\ensuremath{\omega}}_{q}$. If ${\ensuremath{\omega}}_{c}={\ensuremath{\omega}}_{q}$, then in the WKB approximation these classes are the same. (Certain specific potentials belong to both of these classes exactly.) However, we show by a different specific example that in general these classes are not equivalent.

DOI
10.1103/physrevd.23.922

Related papers

Inequivalence of the classes of quantum and classical harmonic potentials: Proof by example

Gautam Kumar Ghosh, Rainer W. Hasse

It has been proven by explicit construction that there exists a quantum harmonic potential (equally spaced energy levels) which does not belong to the class of classical harmonic potentials (frequency independent of energy). In this paper we prove the converse, also by example, that all classical harmonic potentials are not necessarily quantum harmonic potentials. Thus all classical harmonic potentials are not quantum harmonic potentials and vice versa.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Aug 15, 1981•18 citations

Comment on “Identical motion in classical and quantum mechanics”

Ali Mostafazadeh

Makowski and Konkel [Phys. Rev. A 58, 4975 (1998)] have obtained certain classes of potentials which lead to identical classical and quantum Hamilton-Jacobi equations. We obtain the most general form of these potentials.

arXiv:quant-ph/9909092•Physical Review A•Dec 1, 1999•2 citations

Relative convergences of the WKB and SWKB approximations

Y. P. Varshni

The relative convergences of the WKB and SWKB approximations are examined by calculating the eigenenergies for four potentials by one- and two-term WKB and SWKB approximations. Exact eigenenergies for these four potentials are also calculated by a numerical integration of the Schrodinger equation. Varied results are found for the four potentials. It is found that in general the effect of the second term in the WKB and SWKB approximations depends on the potential, the parameters involved and the quantum number of the state. No simple generalizations are possible.

Journal of Physics A Mathematical and General•Nov 7, 1992•22 citations

Classical and Quantum Dispersion Relations

Sergio A. Hojman, Felipe A. Asenjo

Abstract It is showed that, in general, classical and quantum dispersion relations are different due to the presence of the Bohm potential. There are exact particular solutions of the quantum (wave) theory which obey the classical dispersion relation, but they differ in the general case. The dispersion relations may also coincide when additional assumptions are made, such as WKB or eikonal approximations, for instance. This general result also holds for non–quantum wave equations derived from classical counterparts, such as in ray and wave optics, for instance. Explicit examples are given for covariant scalar, vectorial and tensorial fields in flat and curved spacetimes.

arXiv:2003.13559•Physica Scripta•Jun 1, 2020•11 citations

Classical Concepts and Quantal Inequivalences

F. S. Levin

Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts. This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.

Cambridge University Press eBooks•Dec 3, 2001•0 citations

On classical and quantum harmonic potentials

Pirooz Mohazzabi

To date, the only potential energy function that has been demonstrated to be classical harmonic but not quantum harmonic is that of the asymmetrically matched harmonic oscillator. By investigating the accurate quantum mechanical energy levels of the potential V = V0 [Formula: see text], we demonstrate that this is the second member of the class. PACS No.: 03.65Ge

Canadian Journal of Physics•Jan 1, 2000•5 citations
There are an infinite number of potentials whose classical angular velocity \ensuremathωc{\ensuremath{\omega}}_{c}\ensuremathωc​ is independent of energy. There are also an infinite number of potentials whose quantum eigenenergies are equally spaced, \ensuremathℏ\ensuremathωq\ensuremath{\hbar}{\ensuremath{\omega}}_{q}\ensuremathℏ\ensuremathωq​. If \ensuremathωc=\ensuremathωq{\ensuremath{\omega}}_{c}={\ensuremath{\omega}}_{q}\ensuremathωc​=\ensuremathωq​, then in the WKB approximation these classes are the same. (Certain specific potentials belong to both of these classes exactly.) However, we show by a different specific example that in general these classes are not equivalent.
The relative convergences of the WKB and SWKB approximations are examined by calculating the eigenenergies for four potentials by one- and two-term WKB and SWKB approximations. Exact eigenenergies for these four potentials are also calculated by a numerical integration of the Schrodinger equation. Varied results are found for the four potentials. It is found that in general the effect of the second term in the WKB and SWKB approximations depends on the potential, the parameters involved and the quantum number of the state. No simple generalizations are possible.
Abstract It is showed that, in general, classical and quantum dispersion relations are different due to the presence of the Bohm potential. There are exact particular solutions of the quantum (wave) theory which obey the classical dispersion relation, but they differ in the general case. The dispersion relations may also coincide when additional assumptions are made, such as WKB or eikonal approximations, for instance. This general result also holds for non–quantum wave equations derived from classical counterparts, such as in ray and wave optics, for instance. Explicit examples are given for covariant scalar, vectorial and tensorial fields in flat and curved spacetimes.
Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts. This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.