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

Special Nature of Orbital Angular Momentum

James D. Louck

Published May 1, 1963• 12 citations
Publisher

Abstract

The properties of orbital angular momentum operators are examined within the framework of the formal theory of angular momentum. It is demonstrated that the occurrence of only integral quantum numbers in the orbital theory is a consequence of the particular form of the orbital operators. Single-valuedness of the eigenfunctions need not be postulated.

DOI
10.1119/1.1969513

Related papers

Remark Concerning the Eigenvalues of Orbital Angular Momentum

H. A. Buchdahl

That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.

American Journal of Physics•Nov 1, 1962•11 citations

Orbital Angular Momentum

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials

•Sep 7, 2023•1 citation

N-Dimensional Total Orbital Angular-Momentum Operator. II. Explicit Representations

Kenneth D. Granzow

A method of generating orthogonal polar coordinate systems in N-dimensional space is given. Commuting angular-momentum operators are easily found in the coordinate systems generated; these operators are of a single form which depends on two parameters. A short table of coordinate systems and the resulting structure of quantum numbers and eigenfunctions is given.

Journal of Mathematical Physics•Oct 1, 1964•7 citations

Orbital Angular Momentum and Central Potentials

Gordon Baym

The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.

CRC Press eBooks•Mar 5, 2018•0 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

Orbital angular momentum

Victor M. Galitski, Boris Mikhailovich Karnakov, Vladimir I Kogan +1 more

Abstract This chapter analyzes issues that deal with the general properties of angular momentum; angular momentum, l = 1; the addition of angular momenta; and tensor formalism in angular momentum theory.

•Feb 28, 2013•0 citations
The properties of orbital angular momentum operators are examined within the framework of the formal theory of angular momentum. It is demonstrated that the occurrence of only integral quantum numbers in the orbital theory is a consequence of the particular form of the orbital operators. Single-valuedness of the eigenfunctions need not be postulated.
That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.
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.