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American Journal of Physics

Pedagogy of spin in nonrelativistic quantum mechanics

Brian Williams, Terence Robinson

Published Jul 1, 1981• 3 citations
Publisher

Abstract

The quantum-mechanical properties of electron spin are developed systematically from a nonrelativistic Hamiltonian. The correct values of the magnetic moment of the electron, the spin-orbit coupling energy of atomic electrons, the Darwin correction, and the angular momentum of the electron are obtained directly without explicitly including any relativity beyond classical electromagnetism.

DOI
10.1119/1.12452

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The quantum-mechanical properties of electron spin are developed systematically from a nonrelativistic Hamiltonian. The correct values of the magnetic moment of the electron, the spin-orbit coupling energy of atomic electrons, the Darwin correction, and the angular momentum of the electron are obtained directly without explicitly including any relativity beyond classical electromagnetism.
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 .
In nonrelativistic quantum mechanics, the total (i.e. orbital plus spin) angular momentum of a charged particle with spin that moves in a Coulomb plus spin–orbit-coupling potential is conserved. In a classical nonrelativistic treatment of this problem, in which the Lagrange equations determine the orbital motion and the Thomas equation yields the rate of change of the spin, the particle's total angular momentum in which the orbital angular momentum is defined in terms of the kinetic momentum is generally not conserved. However, a generalized total angular momentum, in which the orbital part is defined in terms of the canonical momentum, is conserved. This illustrates the fact that the quantum-mechanical operator of momentum corresponds to the canonical momentum of classical mechanics.
According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave.
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.
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.
Spin is well known to be an intrinsically relativistic property of particles. Nonetheless its effects are seen in many physical situations which are not obviously relativistic, perhaps the most obvious examples being magnets and the quantum mechanically permitted electronic configurations of the elements in the periodic table. The view of spin adopted in these and other problems is that the electron has a quantized spin ( s = 1/2) and that is a fundamental tenet of the theory, rather than something that has to be explained. In this chapter we are going to discuss the behaviour of spin-1/2 particles (electrons, protons and neutrons for example) without much direct reference to relativistic quantum theory and the origins of spin. This chapter should be instructive in its own right and as a guide to understanding spin when we come to discuss it in a fully relativistic context at various stages throughout this book. Unless otherwise specified, we will refer to electrons, but it should be borne in mind that the theory is equally applicable to any spin-1/2 particle. Students of quantum theory cannot delve very deeply into the subject without coming across the quantization of angular momentum. The orbital angular momentum usually surfaces in the theory leading up to the quantum description of the hydrogen atom.