This chapter begins showing the basic concepts of the magnetic moment and its relation with the angular momentum of an electron with and without external magnetic field applied. The former leading to the Larmor frequency. After that, it is presented the Landé’s core model in which the total angular momentum is a sum of angular momentum of valence and core electrons (plus nuclei for the core). Subsequently, normal and anomalousZeeman effect, their relations with frequency of precession and magnetic moment, and Landé’s g-factor are presented. The Heisenberg’s core model is also presented. Afterwards, Pauli concept of the electron twofoldness and Pauli’s g-factor are given followed by Pauli’s exclusion principle from his contribution to Aufbau principle. Information about the experiments that led to the discovery of the spin are given in the sequence. Finally, Pauli and Dirac’s works on spin yielding the Pauli’s spin matrix and Dirac’s spin matrix are presented along with Dirac’s four spinor equation which gave the theoretical support for the existence of the spin (168 words).
This chapter begins showing the basic concepts of the magnetic moment and its relation with the angular momentum of an electron with and without external magnetic field applied. The former leading to the Larmor frequency. After that, it is presented the Landé’s core model in which the total angular momentum is a sum of angular momentum of valence and core electrons (plus nuclei for the core). Subsequently, normal and anomalousZeeman effect, their relations with frequency of precession and magnetic moment, and Landé’s g-factor are presented. The Heisenberg’s core model is also presented. Afterwards, Pauli concept of the electron twofoldness and Pauli’s g-factor are given followed by Pauli’s exclusion principle from his contribution to Aufbau principle. Information about the experiments that led to the discovery of the spin are given in the sequence. Finally, Pauli and Dirac’s works on spin yielding the Pauli’s spin matrix and Dirac’s spin matrix are presented along with Dirac’s four spinor equation which gave the theoretical support for the existence of the spin (168 words).
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
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 discusses the prime cause of the magnetic moment within an individual atom. In the classical model, the angular momentum of the electrons can be used to determine the magnetic moments of the electrons by invoking the concept of electrical charge in motion. There are two contributions to the electronic magnetic moment: an orbital magnetic moment due to orbital angular momentum and a spin magnetic moment due to electron spin. The component of orbital angular momentum along the axis of a magnetic field is restricted to discrete values by quantum mechanics. Electrons have spin angular momentum, which can be represented by the spin quantum number. As the magnetic field is increased and the field splitting becomes greater than the multiple splitting, the anomalous Zeeman effect changes over to a normal Zeeman effect. The chapter looks at experimental evidence confirming the quantization of magnetic moments, specifically the Zeeman effect and the Stern-Gerlach experiment.
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 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.
Introduction The claim that it is impossible to construct a realistic physical model of quantum matter is often presented as definitively and finally proved in the case of systems with spin ½, or internal angular momentum of magnitude ħ/2, which exhibit a characteristic ‘nonclassical two-valuedness’. Nevertheless, the extension of the causal interpretation to embrace nonrelativistic spin ½ systems was successfully carried through in the 1950s (Bohm, Schiller & Tiomno, 1955; Bohm & Schiller, 1955; Takabayasi, 1955; Takabayasi & Vigier, 1957) and developed more recently (Dewdney, Holland & Kyprianidis, 1986, 1987a, b; Dewdney, Holland, Kyprianidis & Vigier, 1988). The idea is to assume that the spinor wave represents a new type of physical field propagating in spacetime that exerts an influence on a corpuscle moving within it. The additional information conveyed by the spinor beyond that contained in the spinless Schrödinger wave implies that the state of the particle is not exhausted by specifying its position and momentum, but that it is endowed with further properties such as an internal angular momentum. The equation governing the dynamics of the spinor wavefunction is the Pauli equation: where the Hamiltonian operator is given by Here e, m and μ are respectively the charge, mass and magnetic moment of the particle, A 0 and A are the electromagnetic potentials, B = ∇ × A is the magnetic field, V is any other external scalar potential, and σ is a vector whose components are the 2 × 2 Pauli matrices. The Hamiltonian (9.1.2) istherefore a 2 × 2 matrix operator and the wavefunction ψ( x , t ) has two complex components.