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Oxford University Press eBooks

Angular Momentum and Rotational Motion in Three Dimensions

Jochen Autschbach

Published Dec 21, 2020• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780190920807.003.0016

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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.
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
This chapter focuses on the concept of rotational motion. It explains that energy and angular momentum for a particle rotating in two- or three-dimensions are quantized and satisfied cyclic boundary conditions, which imply the quantization of angular momentum's magnitude and z-component. The electronic structure of atoms showcases rotational motion because electrons orbit around nuclei and spin on their axis. The chapter details the solutions for the Schrödinger equation and the vector model of angular momentum. It also discusses the concept of space quantization revolving around the quantum mechanical result of a rotating body that may not take up an arbitrary orientation concerning some specified axis.
The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.
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.
This chapter analyses the orbital angular momentum to three dimensions and introduces the spin angular momentum of the electron. The description of a localized particle in orbit requires a superposition of eigenfunctions analogous to the packet state that describes motion in one dimension. If the distance between molecules is fixed, all the energy changes are rotational kinetic energy changes, and the quantization of total angular momentum gives rise to discrete rotational energy levels. The analysis of orbital angular momentum is a powerful tool for describing molecular rotation. The Stern-Gerlach experiment and many other lines of evidence show that the measured magnetic moment associated with an electron spin is almost exactly one Bohr magneton. When orbital angular momentum and spin angular momentum both exist in the same atom, the magnetic moments that result from orbital and spin angular momenta interact to cause a splitting in the corresponding energy level.