Early evidences show that elementary particles like electrons possess an intrinsic degree of freedom which is akin to angular momentum but has no classical description like orbital angular momentum. This is termed spin and is denoted by s . O. Stern and W. Gerlach demonstrated in 1922 that electron has a spin 1/2. We shall consider this landmark experiment in the next section.
Early evidences show that elementary particles like electrons possess an intrinsic degree of freedom which is akin to angular momentum but has no classical description like orbital angular momentum. This is termed spin and is denoted by s . O. Stern and W. Gerlach demonstrated in 1922 that electron has a spin 1/2. We shall consider this landmark experiment in the next section.
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
Unlike orbital angular momentum, the total spin of a system can be integral or half-integral. Fermions such as electrons, positrons, neutrinos, and quarks possess intrinsic angular momentum or spin with a measurable value of ±1/2 (in units of ħ). Composite particles such as protons and neutrons also have measurable spin of ±1/2 and atomic nuclei can have half-integral spin values (1/2, 3/2, 5/2, …). The “spinor” function for half-integral spin is unusual in that rotation by 2π transforms it into the negative of itself. A rotation by 4π is required in order to transform the spin function into itself. While this may at first glance seem unreasonable, there are simple examples that display this property. Take a strip of paper and form a Möbius strip by twisting one end 180° and joining it to the other end. Start at any point on the strip and trace a line through 360°. You do not end up at the starting point, but rather on the other side of the paper strip, as shown in Fig. 5.1. Continue tracing along the surface for another 360° and you will return to the original starting point. Early spectroscopic experiments on hydrogen and hydrogen-like atoms revealed that there were twice as many states as predicted by the solutions of Schrödinger's equation. The idea that an electron could have intrinsic angular momentum (spin) with two possible states was proposed by Kronig, Uhlenbeck, and Goudsmit [5.1] in 1925.
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.
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”.
This chapter analyses the orbital angular momentum to three dimensions and introduces the spin angular momentum of the electron. The description of a localized particle in orbit requires a superposition of eigenfunctions analogous to the packet state that describes motion in one dimension. If the distance between molecules is fixed, all the energy changes are rotational kinetic energy changes, and the quantization of total angular momentum gives rise to discrete rotational energy levels. The analysis of orbital angular momentum is a powerful tool for describing molecular rotation. The Stern-Gerlach experiment and many other lines of evidence show that the measured magnetic moment associated with an electron spin is almost exactly one Bohr magneton. When orbital angular momentum and spin angular momentum both exist in the same atom, the magnetic moments that result from orbital and spin angular momenta interact to cause a splitting in the corresponding energy level.