In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.
In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.
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 .
Abstract In this chapter we discuss the intrinsic angular momentum called spin. Starting from the magnetic dipole moment related to orbital angular momentum, we focus on the hydrogen atom and consider the effect of an external magnetic field, resulting in Larmor precession and Zeeman splitting. We then turn to the discovery of spin in the Stern-Gerlach experiment. After providing a mathematical description of spin-1/2 systems in terms of two-component spinors and Pauli matrices, we illustrate projective measurements for the example of spin. The relationship between the Lie groups SU(2) and SO(3) is discussed in the context of the transformation of spinors under rotations. We analyse how the presence of spin alters the description of the hydrogen atom. Discussing the addition of angular momenta and the Pauli exclusion principle leads us to the structure of electronic orbitals. Finally, we discuss the occupation of electronic energy levels in heavier nuclei
‘Spin’ considers the movement of electrons orbiting around the nucleus of an atom. Physicists such as Wolfgang Pauli, Pieter Zeeman, Ralph Kronig, George Uhlenbeck, Samuel Goudsmit, Llewellyn Thomas, Otto Stern, and Walther Gerlach spent many years trying to solve the puzzle of electron spin through different experiments. The mathematical theory of spin began to be developed in the 1920s, but it was the work of Paul Dirac in 1928 that solved the problem. By fusing relativity and quantum mechanics, the two great products of the early 20th-century revolution in physics, Dirac had assembled an equation that describes the origin of spin, the fundamental element of the magnetism of the electron.
Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.
Spin and helicityTraditionally, in textbooks on quantum mechanics, spin is introduced via an idealized Stern-Gerlach experiment in which a non-relativistic beam of silver atoms passes through an inhomogeneous magnetic field.Each atom is treated as a single valence electron of charge -e in an s-state.The subsequent splitting of the beam into two indicates the two-valuedness of Sz, which is related to the value 1/2 for s, and the magnitude of the splitting shows that the magnetic moment /l is related to s by e /l = -~s, me the proportionality factor (the gyromagnetic ratio) being twice as big as the factor that classically gives the magnetic moment due to the orbital angular momentum of a point charge.Historically, however, it seems that the early Stern-Gerlach experiments, begun in 1922, had no influence at all upon the discovery of spin, simply because they were too imprecise.Rather, the concept of spin appeared after a long and tedious battle to understand the splitting patterns and separations in line spectra.Several people had for various reasons discussed classical models of rotating charge distributions but Kronig, in 1924, was the first to show that an electron with spin 1/2 would explain the pattern of what we would today call L • S splitting, as well as anomalies in the Zeeman effect.He realized, though, that the gyromagnetic ratio ( -ejmc) needed for the latter would give L • S splittings twice as big as those observed.It is said that Pauli expressed his negative reaction to Kronig's idea with such vehemence that Kronig never published his work (Mehra and Rechenberg, 1982).Soon thereafter, in 1925, the same idea occurred to Uhlenbeck and Goudsmit (1925), who proceeded to a detailed analysis of the splittings, concluding at first that everything worked beautifully, but then becoming aware, as a consequence of a comment by Heisenberg, of the factor-of-2 inconsistency mentioned above.
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.