Journal of Mathematical Physics

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

Kenneth D. Granzow

Published Oct 1, 1964• 7 citations

Abstract

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.

DOI: 10.1063/1.1704084

Related papers

N-Dimensional Total Orbital Angular-Momentum Operator

Kenneth D. Granzow

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.

Journal of Mathematical PhysicsJul 1, 196320 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, 20231 citation

Construction of Orthonormal Angular-Momentum Operators

Paul L. Corio

The problem of constructing an orthonormal basis for the (2j + 1)2-dimensional space of angular-momentum operators is reduced to the problem of constructing a sequence of orthogonal polynomials of a discrete real variable. A number of properties of the polynomials are obtained, including a pure recurrence formula and several mixed recurrence relations. Three explicit representations of the polynomials are given, together with a table of orthonormal angular-momentum operators.

Journal of Mathematical PhysicsJul 1, 196812 citations

Orbital angular momentum eigenfunctions for many-particle systems

John Avery

Methods are presented for constructing eigenfunctions of the total orbital angular momentum operator of a many-particle system without the use of the Clebsch–Gordan coefficients. One of the equations derived in this paper is analogous to Dirac's identity for total spin; and through this equation, a connection is established between eigenfunctions of L2 and irreducible representations of the symmetric group Sn.

Canadian Journal of ChemistryJul 1, 19852 citations

Angular Momentum Operators in Orthogonal Curvilinear Coordinate Systems

Sun Shu-sheng

Using the transformation matrix between the unit basis vectors in deferent orthogonal curvilinear coordinate systems,we deduce the general form of the transformation between the angular momentum operators in these coordinate systems.We give the representations of Hamilton operator  and angular momentum operators in the orthogonal curvilinear coordinate system.We discuss the transformation matrix of unit basis vectors among the spherical coordinate system,cylindrical coordinates system and rectangular coordinates system,and the transformation between the angular momentum operators.Furthermore,we discuss the treatment of 2operator in different coordinate systems.

Journal of Shenyang University of Chemical TechnologyJan 1, 20100 citations

Angular momentum wave functions constructed by projection operators. Technical Note No. 12

Per‐Olov Löwdin

The conventional method of constructing wave functions of pure angular momentum for a composite system is based on the idea of coupling the angular momenta of the constituents. Instead of this synthetic method, an analytical approach is here introduced. It is pointed out that an arbitrary trial wave function for the total system in a unique way must be resolvable into orthogonal components of pure angular momentum associated with different quantum numbers. A particular component can be selected by means of certain 'projection operators', which annihilate all components except the one desired. Such a projection operator is simply a product of commuting factors, each one of which annihilates a specific eigenfunction to the angular momentum under consideration. Physically this idea is of importance, since one can now start out from a rough model wave function based essentially on qualitative arguments, for instance of the independent-particle type, and then obtain a mathematically and physically much better trial function by selecting the particular component of the original function which has the correct symmetry type desired. The projection operators are studied in some detail both as products and in expanded form. The case of degenerate subspaces of the same angular momentum quantum numbers is discussed, and the problem of constructing an orthogonal subset of functions is solved by a simple elimination procedure; the connection with the conventional seniority idea is briefly discussed.

May 10, 19580 citations