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Cambridge University Press eBooks

Introduction to Spin, Magnetic Resonance and Polarization

Tapio O. Niinikoski

Published Jan 16, 2020• 1 citation
Publisher

Abstract

The concepts of angular momentum, spin and magnetic moment are worked out using standard quantum mechanical formalism. The concepts of intrinsic spin of a pointlike particle is contrasted with the intrinsic angular momentum of composite particles. The Larmor frequency and the magnetic resonance of non-interacting spins are introduced. The quantum statistics of a system of spins is overviewed, before introducing the thermodynamics of a spin system in a static frame of reference. Nuclear magnetic phase transitions are briefly reviewed.

DOI
10.1017/9781108567435.002

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The concepts of angular momentum, spin and magnetic moment are worked out using standard quantum mechanical formalism. The concepts of intrinsic spin of a pointlike particle is contrasted with the intrinsic angular momentum of composite particles. The Larmor frequency and the magnetic resonance of non-interacting spins are introduced. The quantum statistics of a system of spins is overviewed, before introducing the thermodynamics of a spin system in a static frame of reference. Nuclear magnetic phase transitions are briefly reviewed.
First, the experimental foundations for the existence of a new dynamical variable, the spin of particles, are presented. The Pauli equation is then derived in detail, two applications are given and the energy levels of a particle with spin in a constant magnetic field are studied. This is the analysis of Landau levels, which can be performed by using the known results on the spectrum of harmonic oscillators. In this chapter we also study in detail the addition of orbital angular momentum and spin, as well as the spin–orbit interaction with the resulting Thomas precession. Stern-Gerlach experiment and electron spin The hypothesis that the electron has an intrinsic magnetic moment and an angular momentum, in short a spin , was first suggested in Uhlenbeck and Goudsmit (1926). They noticed, even before the discovery of quantum mechanics, that a complete description of spectra was not possible unless a magnetic moment and a mechanical moment were ascribed to the electron, and hence the concept of an electron as a point charge was insufficient. First, let us therefore try to understand how a magnetic moment can be associated to an atomic system. For this purpose, consider for simplicity the Bohr model of an hydrogen atom, where the electron moves along a circular orbit and rotates with orbital angular momentum L .
This chapter outlines the quantum ideas that underpin magnetic resonance and explains their link to classical physics. It discusses some of the essential concepts governing the quantum behaviour of ensembles of nuclear spins, including the nature of quantum states, the eigenvalue equation and observables, the measurement process, dynamics, and the Schroedinger equation. A detailed description of angular momentum is given along with discussion of the role of symmetry, rotation operators, and both product and total angular momentum representations. Statistical ensembles of spins are introduced and the density matrix and Liouville formalism outlined, along with the various tensor bases for the description of spin systems, both independent and coupled. The spin Hamiltonian is described in detail and the evolution of spin systems under various Hamiltonian terms is discussed using the density matrix description, along with suitable recession diagrams.
Abstract The concept of angular momentum in quantum mechanics has many counterintuitive aspects, such as assuming only certain discrete values and only certain orientations in space in the presence of external fields. Spin is more esoteric yet; it is hard to imagine how point‐particles with no spatial extent can have an intrinsic angular momentum. Nevertheless, spin is the most fundamental concept in magnetic resonance. Appreciation of the spin concept requires some understanding of the relativistic quantum mechanics developed by P. A. M. Dirac. The history of the evolution of the spin concept is traced—from the turn‐of‐the‐century observation of the Zeeman effect, through the old and new quantum mechanics, and on to Dirac's theory of the electron, which was published in the late 1920s. The special relativity theory is discussed, and there is a brief explanation of how spin emerges naturally in a relativistically correct formulation of quantum mechanics.
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 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.