The discovery of the spin of the electron by Uhlenbeck and Goudsmit was a major advance in the understanding of physics at the atomic level. Its discovery coincided with the development of both matrix and wave mechanics and its incorporation into the scheme of quantum mechanics and statistics led to deeper understanding of the underlying structure of quantum mechanics. Almost immediately, Heisenberg and Jordan used the new scheme of matrix mechanics to derive the expression for the g -factor which Landé had derived empirically from a very close study of the anomalous Zeeman effect. An important consequence of these developments was that the different approaches of matrix and wave mechanics were brought together. In particular, the discovery of spin as a new quantum number suggested the possibility of understanding systems containing more than one electron. Heisenberg's analysis of the helium atom was to pave the way for the full incorporation of spin into quantum mechanics and quantum statistics. Spin and the Landé g -factor The story of the discovery of the spin of the electron by Uhlenbeck and Goudsmit (1925a) was told in Sect. 8.5. As discussed in that section, their discovery was based upon empirical studies of the regularities observed in the anomalous Zeeman effect, inspired by the intricate analyses of Landé. Although based originally upon the classical concept of a rotating electron, electron spin is a purely quantum mechanical property intrinsic to the electron.
The discovery of the spin of the electron by Uhlenbeck and Goudsmit was a major advance in the understanding of physics at the atomic level. Its discovery coincided with the development of both matrix and wave mechanics and its incorporation into the scheme of quantum mechanics and statistics led to deeper understanding of the underlying structure of quantum mechanics. Almost immediately, Heisenberg and Jordan used the new scheme of matrix mechanics to derive the expression for the g -factor which Landé had derived empirically from a very close study of the anomalous Zeeman effect. An important consequence of these developments was that the different approaches of matrix and wave mechanics were brought together. In particular, the discovery of spin as a new quantum number suggested the possibility of understanding systems containing more than one electron. Heisenberg's analysis of the helium atom was to pave the way for the full incorporation of spin into quantum mechanics and quantum statistics. Spin and the Landé g -factor The story of the discovery of the spin of the electron by Uhlenbeck and Goudsmit (1925a) was told in Sect. 8.5. As discussed in that section, their discovery was based upon empirical studies of the regularities observed in the anomalous Zeeman effect, inspired by the intricate analyses of Landé. Although based originally upon the classical concept of a rotating electron, electron spin is a purely quantum mechanical property intrinsic to the electron.
Abstract This chapter gets into the heart of the discovery of spin in physics as physicists try to make sense of the atomic spectra with the discovery of the anomalous Zeeman effect by Thomas Preston before the dawn of the twentieth century as Nature gave the first signal of the existence of quantum spin. Following Bohr, Sommerfeld, and Lande’s contributions with integers and then half-integers, quantum numbers make appearances, and after Uhlenbeck and Goudsmit’s ground-breaking contributions, physicists finally embrace the idea that electron spins. Pauli’s introduction of quantum spin with Pauli matrices provided a non-relativistic theory of the quantum spin. Hestenes argues that spin is not intrinsically related to Pauli matrices, and contrary to conventional wisdom, he advocates the view that the spin is very much present in the non-relativistic quantum physics.
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.
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 the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
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.
‘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.