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Springer eBooks

Angular Momentum

Will Raven

Published Oct 17, 2024• 0 citations
Publisher

Abstract

Abstract In this chapter, we explore angular momentum, a key concept in quantum mechanics and atomic physics. We cover the quantization of angular momentum, the role of quantum numbers, and their impact on atomic states. We will learn about the different types of angular momentum, the states representing these types of angular momentum, and how angular momenta are added together in quantum mechanics. The chapter also examines the compatibility of different observables, such as the magnitude and components of angular momentum, and the impact of measurements on quantum states.

DOI
10.1007/978-3-031-69507-0_7

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Abstract In this chapter, we explore angular momentum, a key concept in quantum mechanics and atomic physics. We cover the quantization of angular momentum, the role of quantum numbers, and their impact on atomic states. We will learn about the different types of angular momentum, the states representing these types of angular momentum, and how angular momenta are added together in quantum mechanics. The chapter also examines the compatibility of different observables, such as the magnitude and components of angular momentum, and the impact of measurements on quantum states.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.
Angular momentum plays a similarly important role in atomic systems, such as an electron subject to the central force provided by the Coulomb field of a nucleus. This chapter introduces the properties of angular momentum, both orbital and spin, in the quantum-mechanical scheme of things. The angular momentum of atomic systems is inseparable from the magnetic properties of atoms. The various permitted values of the quantized angular momentum play a central role in determining the number and variety of possible quantum states in atomic systems. The Stern-Gerlach experiment was deliberately designed to test whether or not the angular momentum of individual neutral atoms is quantized in a magnetic field. As in the development of the Schrodinger equation, classical physics gives some hints of the correct forms of these angular momentum operators. The chapter discusses rough estimate using the Neils Bohr theory and attributing the magnetic moment to the orbital motion of the electron.
This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.
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.
One of the most important applications of the quantum mechanical formalism developed in the previous chapter is to angular momentum. This is because many microsystems are spherically symmetric, or approximately so, and angular momentum is therefore conserved. This chapter discusses that a careful study of quantized angular momentum leads to a completely new physical phenomenon – intrinsic spin. The fact that the angular momentum vector of a quantum particle must point in certain restricted directions is a bizarre quantum property. In some respects the abstract idea of an internal angular momentum is reminiscent of the classical concept of a spinning body. In classical mechanics angular momentum is added vectorially but in the quantum case the issue is more subtle. The procedure will be illustrated by the example of the addition of two spins. As in classical mechanics a charged quantum system with angular momentum possesses a magnetic moment μ.