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

American Journal of Physics

Quantum pendulum

R. Aldrovandi, Paulo Leal Ferreira

Published Aug 1, 1980• 27 citations
Publisher

Abstract

The problem of the mathematical pendulum is discussed in its classical, semiclassical, and quantum aspects. The energy spectrum and the eigenfunctions are presented under the usual requirement of single valuedness of the solutions.
DOI
10.1119/1.12332

Related papers

The quantum point-mass pendulum

Gayle Cook, Clyde S. Zaidins

The question of how long it takes an inverted point-mass pendulum to fall according to classical, semiclassical, and quantum theories is examined. The semiclassical analysis is based on the uncertainty introduced by measurement.

American Journal of Physics•Mar 1, 1986•26 citations

The quantum pendulum

Gregory L. Baker, James A. Blackburn

Abstract The classical pendulum has a long and rich history in physics. On the other hand, the history of the quantum pendulum can only stretch back as far as the appearance of quantum mechanics itself in 1925–1926. Yet within three years of that exciting time, a paper on the quantum pendulum appeared in Physical Review authored by one of the pioneers of atomic physics, Edward Condon (Condon 1928). Condon’s paper laid out the basic problem and the means to its solution. Contributions to the study of the quantum pendulum continue to be made even in recent times. See, for example, (Pradhan and Khare 1973; Aldrovandi and Ferreira 1980; Cook and Zaidins 1986; Baker et al. 2002). In this tale we describe some of the novel and surprising features of the pendulum when it is analyzed according to the rules of quantum physics.

•Jun 2, 2005•0 citations

The Physical Pendulum in Quantum Mechanics

E. U. Condon

It is pointed out that the Mathieu functions of even order are the characteristic functions of the physical pendulum in the sense of Schr\"odinger's wave mechanics. The relation of various properties of the functions, as known from purely analytical investigations of them, to the pendulum problem is discussed.

Physical Review•May 1, 1928•79 citations

Critical comments on the quantization of the angular momentum: II. Analysis based on the requirement that the eigenfunction of the third component of the operator of the angular momentum must be a single valued periodic function

G. S. Japaridze, Anzor Khelashvili, K. Sh. Turashvili

We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of angular momentum could be only integer. We reexamine the arguments based on this requirement and alternate condition imposed by Pauli and show that they do not follow from the first principles and therefore these constraints can dropped. Consequently, we arrive to the same conclusion as in [1]: there exist regular, normalizable eigenfunctions with the non-integer eigenvalues thus a non-integer angular momentum is perfectly admissible from the theoretical viewpoint. The issue of the nature of eigenvalues forming the spectrum of the angular momentum remains open. What can be derived from the first principles is that to a fixed value of the angular momentum L corresponds a discrete spectrum of eigenvalues of the third component of the angular momentum, m, defined by the relation |m|=L-k, k=0,1,...,[L], where [L] is an integer part of L. As a mathematical byproduct of our analysis of eigenfunctions, we present an alternate definition of a power of a complex number allowing to retain initial translational invariance of a base.

arXiv:2004.10673•Apr 17, 2020•1 citation

The Harmonic Oscillator, a Review of Classicaland Elementary Quantum Mechanics

Carl W David

The classical harmonic oscillator and an elementary discussion of the quantum mechanical solutions for the harmonic oscillator are discussed.

•Jan 1, 2006•0 citations

Plane Pendulum in Quantum Mechanics

T. Pradhan, Avinash Khare

A detailed quantum mechanical treatment of the motion of a plane pendulum is given. Exact solution of the Schrödinger equation shows that the energy eigenfunctions and eigenvalues go over to those of a plane rotator for vanishing acceleration due to gravity g and to those of a harmonic oscillator in the limit of infinitely massive bob. In view of this, perturbation calculations have been done using rotator and oscillator basis, and conditions for convergence of the perturbation series in both cases have been discussed.

American Journal of Physics•Jan 1, 1973•36 citations
Abstract The classical pendulum has a long and rich history in physics. On the other hand, the history of the quantum pendulum can only stretch back as far as the appearance of quantum mechanics itself in 1925–1926. Yet within three years of that exciting time, a paper on the quantum pendulum appeared in Physical Review authored by one of the pioneers of atomic physics, Edward Condon (Condon 1928). Condon’s paper laid out the basic problem and the means to its solution. Contributions to the study of the quantum pendulum continue to be made even in recent times. See, for example, (Pradhan and Khare 1973; Aldrovandi and Ferreira 1980; Cook and Zaidins 1986; Baker et al. 2002). In this tale we describe some of the novel and surprising features of the pendulum when it is analyzed according to the rules of quantum physics.
We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of angular momentum could be only integer. We reexamine the arguments based on this requirement and alternate condition imposed by Pauli and show that they do not follow from the first principles and therefore these constraints can dropped. Consequently, we arrive to the same conclusion as in [1]: there exist regular, normalizable eigenfunctions with the non-integer eigenvalues thus a non-integer angular momentum is perfectly admissible from the theoretical viewpoint. The issue of the nature of eigenvalues forming the spectrum of the angular momentum remains open. What can be derived from the first principles is that to a fixed value of the angular momentum L corresponds a discrete spectrum of eigenvalues of the third component of the angular momentum, m, defined by the relation |m|=L-k, k=0,1,...,[L], where [L] is an integer part of L. As a mathematical byproduct of our analysis of eigenfunctions, we present an alternate definition of a power of a complex number allowing to retain initial translational invariance of a base.
A detailed quantum mechanical treatment of the motion of a plane pendulum is given. Exact solution of the Schrödinger equation shows that the energy eigenfunctions and eigenvalues go over to those of a plane rotator for vanishing acceleration due to gravity g and to those of a harmonic oscillator in the limit of infinitely massive bob. In view of this, perturbation calculations have been done using rotator and oscillator basis, and conditions for convergence of the perturbation series in both cases have been discussed.