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Royal Society of Chemistry eBooks

Lecture 3: The Quantum Description of An Isolated Spin-1/2

Giuseppe Pileio

Published Jun 13, 2022• 0 citations
Publisher

Abstract

In this lecture, I introduce a quantum description of an isolated nuclear spin with spin quantum number 1/2. The discussion introduces the Zeeman Hamiltonian, the Larmor frequency and the nuclear spin angular momentum operators, all of which are fundamental concepts in spin dynamics.
DOI
10.1039/9781837670871-00035

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In this lecture, I lay down the basics for the study of the dynamics of a large collection of identical spin systems by introducing the concept of a density operator and an equation for its evolution. Ensembles of spin systems represent macroscopic samples; in quantum mechanics, these can only be described in statistical terms through a density operator. This latter can be expressed in terms of spin angular momentum operators. The density operator allows all measurable properties of an ensemble of spin systems to be determined.
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.
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.