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Angular Momentum

Owen L. de Lange, R. E. Raab

Published Nov 28, 1991• 0 citations
Publisher

Abstract

Abstract The main purpose of this chapter is to present shift operators for the quantum numbers of angular momentum. These operators have useful applications, some of which are discussed in this and subsequent chapters. We also give a brief discussion of the non-relativistic treatment of spin angular momentum.
DOI
10.1093/oso/9780198539612.003.0004

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Angular momentum is essential to any advanced discussion of magnetic resonance spectroscopy. A brief history of the modern sub-atomic particle-spin concept is presented. The origin of the spin-1/2 angular momentum quantum number is detailed and angular momentum operators are constructed for spin-1/2 nuclei using the postulates of quantum mechanics which are then used to produce a set of very useful eigenvalue formulas. Angular momentum operators in spherical coordinate form are developed and operator matrices are constructed from Cartesian operators. The time-evolution of these operators is of much interest in later chapters and is explored in some detail here.
Angular momentum is essential to any advanced discussion of magnetic resonance spectroscopy. A brief history of the modern sub-atomic particle-spin concept is presented. The origin of the spin-1/2 angular momentum quantum number is detailed and angular momentum operators are constructed for spin-1/2 nuclei using the postulates of quantum mechanics that are then used to produce a set of very useful eigenvalue formulas. Angular momentum operators in spherical coordinate form are developed and operator matrices are constructed from Cartesian operators. The time-evolution of these operators is of much interest in later chapters and is explored in some detail here.
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.
Introductory Remarks The concepts and techniques of angular momentum theory are by now all but ubiquitous in present-day quantum physics; there is scarcely a single issue of a physics journal that does not in some article directly use angular momentum constructs. It would accordingly be rather fatuous, if not indeed quite impossible, to attempt any definitive (or even very detailed) survey of applications. The purpose of this chapter is therefore rather more modest; we shall focus attention on several typical applications of the theory, in widely different contexts, with an emphasis toward illustrating the basic principles. In discussing any particular field of application, however, we have attempted not to be extensive, but instead have chosen to present a novel, or, at least, unfamiliar, aspect of the subject in some detail. (The discussion in Section 6g – of a result due to Casimir – provides a good example of our intentions.) Basic Principles Underlying the Applications Before turning to specific examples, it will be useful to review briefly the principles underlying the applications. We have seen that a symmetry in quantum physics is a transformation on state vectors (active view – vectors in Hilbert space are transformed and operators are kept fixed) and on observables (passive view – vectors in Hilbert space are kept fixed and operators are transformed) such that all probabilities are preserved (Chapter 3, Section 14). Using the Wigner theorem, we found that this led to either a unitary, or an antiunitary, realization of the symmetry by operators.
Abstract In order to prepare for the discussion of quantum many-particle Hamiltonians, for example the Heisenberg quantum spin chain Hamiltonian, this chapter examines the concept of angular momentum in quantum mechanics, especially the coupling of spin-2 operators for several quantum spins. It begins with the general theory of angular momentum for a single quantum particle, especially for a single spin-1, described by Pauli spin matrices, and then extends to the theory of angular momentum for several particles, again especially for several spins1.