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CRC Press eBooks

Motion in Central Fields

Ahmad Amer

Published Nov 5, 2025• 0 citations
Publisher

Abstract

This chapter addresses quantum mechanical motion in central fields, focusing on the separation of variables in spherical coordinates. The eigenstates are separated into radial and angular components, with the angular part expressed using spherical harmonics. The emergence of Legendre and associated Legendre polynomials is discussed in detail as factors in the angular solutions. The parity properties of spherical harmonics are analyzed. Furthermore, explicit expressions are derived for the components and the square of the angular momentum operator in spherical coordinates. The chapter introduces the orbital angular momentum and magnetic quantum numbers, providing their physical significance and role in quantum systems with central symmetry. Metadata/Keywords: central field, spherical harmonics, Legendre polynomials, orbital angular momentum quantum number, magnetic quantum number, angular momentum operators, parity

DOI
10.1201/9781003677109-5

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This chapter addresses quantum mechanical motion in central fields, focusing on the separation of variables in spherical coordinates. The eigenstates are separated into radial and angular components, with the angular part expressed using spherical harmonics. The emergence of Legendre and associated Legendre polynomials is discussed in detail as factors in the angular solutions. The parity properties of spherical harmonics are analyzed. Furthermore, explicit expressions are derived for the components and the square of the angular momentum operator in spherical coordinates. The chapter introduces the orbital angular momentum and magnetic quantum numbers, providing their physical significance and role in quantum systems with central symmetry. Metadata/Keywords: central field, spherical harmonics, Legendre polynomials, orbital angular momentum quantum number, magnetic quantum number, angular momentum operators, parity
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 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.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
Angular momentum theory provides a mathematical foundation for the rotation of the eigenvectors of the angular momentum operator, the spherical harmonics functions. A useful property of spherical harmonics is that their rotation in 3D space corresponds to rotation in the virtual space of their azimuthal degrees of freedom. In addition to that, spherical harmonics functions that have different orbital indices do not mix. Given that spherical harmonics functions have known rotation properties, and that any arbitrary function can be expanded in a series of spherical harmonics, this property allows the rotation of an electrostatic potential when it is expanded in the series of spherical harmonics. In this chapter, the exponential form of the rotation operator, the rotation matrix for a 3D rotation of the coordinate system, and the principles of rotation of vectors and functions are described. The chapter provides a detailed self-contained introduction to the theory of angular momentum, derives the commutative properties of the angular momentum operator, and explains shift operators, while minimizing references to quantum mechanics and entirely avoiding group theory.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.