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

Journal of Mathematical Physics

N-Dimensional Total Orbital Angular-Momentum Operator

Kenneth D. Granzow

Published Jul 1, 1963• 20 citations
Publisher

Abstract

A set of generalized polar coordinate systems are defined in N-dimensional space. The total orbital angular-momentum operator as defined by Louck is found to be a tensor invariant on the (N − 1)-dimensional unit sphere; hence it is easily explicitly determined in any of the possible generalized polar coordinate systems. Commuting operators can be found and the eigenvalue problem solved in many coordinate systems. Two examples are given: (1) a coordinate system of 3M dimensions where M is an integer, and (2) a coordinate system of 8 dimensions.

DOI
10.1063/1.1704015

Related papers

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

Quantum formulation of angle and orbital angular momentum

Jörg B. Götte, Stephen Mark Barnett

The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.

Cambridge University Press eBooks•Nov 8, 2012•2 citations

SOME ASPECTS OF ANGULAR MOMENTUM IN FOUR DIMENSIONS

Mary Dale Alexander Cox

The solution of the three-dimensional angular momentum problem based on the commutation rules is well-known. 1-5 The commuting operators are the square of the total angular momentum and any one of the three components, called M₂. The matrix elements of these are found to be ²(j + 1) and m, respectively.Gallup has worked out the orbital angular momentum problem in n-dimensional space using the Mij commutation.

•Jan 1, 1962•0 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

A physically motivated derivation of the Laplacian in terms of the total angular momentum operator

Jacob Katriel

Abstract The straightforward transformation of the three-dimensional Laplacian and of the total angular momentum operator from Cartesian to spherical polar coordinates is tedious, unrewarding, and prone to errors. An alternative derivation, starting from the total angular momentum operator in Cartesian coordinates and using the generator of homogeneous scaling, readily yields the expression for the three-dimensional Laplacian in spherical polar coordinates in a manner that accounts very explicitly for the role of the total angular momentum operator. The two-dimensional case is presented as a preparatory exercise, and the N -dimensional case, which may be useful in a graduate course on the few-body problem, is shown to be as straightforward as the three-dimensional one.

European Journal of Physics•Apr 5, 2019•1 citation

Particle in a Central Potential; Orbital Angular Momentum

Rocco Schiavilla

The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.

Cambridge University Press eBooks•Dec 5, 2024•0 citations
A set of generalized polar coordinate systems are defined in N-dimensional space. The total orbital angular-momentum operator as defined by Louck is found to be a tensor invariant on the (N − 1)-dimensional unit sphere; hence it is easily explicitly determined in any of the possible generalized polar coordinate systems. Commuting operators can be found and the eigenvalue problem solved in many coordinate systems. Two examples are given: (1) a coordinate system of 3M dimensions where M is an integer, and (2) a coordinate system of 8 dimensions.
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
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
Abstract The straightforward transformation of the three-dimensional Laplacian and of the total angular momentum operator from Cartesian to spherical polar coordinates is tedious, unrewarding, and prone to errors. An alternative derivation, starting from the total angular momentum operator in Cartesian coordinates and using the generator of homogeneous scaling, readily yields the expression for the three-dimensional Laplacian in spherical polar coordinates in a manner that accounts very explicitly for the role of the total angular momentum operator. The two-dimensional case is presented as a preparatory exercise, and the N -dimensional case, which may be useful in a graduate course on the few-body problem, is shown to be as straightforward as the three-dimensional one.
The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.