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SOME ASPECTS OF ANGULAR MOMENTUM IN FOUR DIMENSIONS

Mary Dale Alexander Cox

Published Jan 1, 1962• 0 citations
OpenAlex

Abstract

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.

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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.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
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.
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.
Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.
The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.