The nature of the electron, a point particle according to the Standard Model, presents a paradox that challenges our ability to comprehend the physical reality. This paradox lies in the apparent contradiction between the complexity represented according to mathematical models that fully describe experimental results and the world around us perceived as a representation of an intuitive process. For example, how come all electrons are the same? As we shall see in this chapter, a seemingly simple question stimulated original reflections that later led to significant theoretical developments. Stern and Gerlach’s experiments demonstrated that the electron possesses an intrinsic property, the spin, whose generalization led to the distinction between fermions and bosons. The spin property, however, is apparently not reconcilable with the point properties of the electron. It has, in fact, not any analogue in classical physics. This incongruence also applies to the electron’s magnetic dipole moment. The key to everything is perhaps the spin property, which has Pauli’s Exclusion Principle as its consequence, to which we owe the existence of matter as we perceive it.
The nature of the electron, a point particle according to the Standard Model, presents a paradox that challenges our ability to comprehend the physical reality. This paradox lies in the apparent contradiction between the complexity represented according to mathematical models that fully describe experimental results and the world around us perceived as a representation of an intuitive process. For example, how come all electrons are the same? As we shall see in this chapter, a seemingly simple question stimulated original reflections that later led to significant theoretical developments. Stern and Gerlach’s experiments demonstrated that the electron possesses an intrinsic property, the spin, whose generalization led to the distinction between fermions and bosons. The spin property, however, is apparently not reconcilable with the point properties of the electron. It has, in fact, not any analogue in classical physics. This incongruence also applies to the electron’s magnetic dipole moment. The key to everything is perhaps the spin property, which has Pauli’s Exclusion Principle as its consequence, to which we owe the existence of matter as we perceive it.
The discovery of the electron and its strange properties is the cornerstone of quantum mechanics. Based on their spin properties, photons and electrons, now categorized as fermions and bosons under the Standard Model, exhibit an enigmatic dual nature of wave and particle. This property, the interpretation of which is still debated today, is supported by various experiments, as we saw in Chap. 4 . One of these, in particular, can be considered the basis of any interpretative model on the dual nature of photons and electrons, the double-slit experiment. This experiment will lead to the famous Copenhagen interpretation, which is still a key to explain quantum mechanics today. With all its incredible weirdness, the double-slit experiment is thus the basis of quantum mechanics, as we shall see in this chapter. Experiments and interpretations of the strange quantum properties of the electron have also led to the discovery of other physical phenomena seemingly impossible according to classical physics, such as the tunnel effect and quantum entanglement. These phenomena have no classical equivalent and can be explained only by resorting to quantum properties of electrons.
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 .
Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. It is wildly believed that spin is a purely quantum mechanical concept and has no classical analogue. In fact, elementary particles are conceived as point objects which have no axis to “spin” around. Therefore, there is no explaining how spin arises at the fundamental level, why particles have the values they do, and what underpins the Pauli Exclusion principle and Bose-Einstein behavior. However, spin is like a vector quantity; it has a definite magnitude, and it has a “direction”, in order to spin should be composite. In this paper we propose a physical explanation for spin of the electron at the sub-particle level, relying on the vortex model of the electron. The electron is described as a superfluid frictionless vortex which has a mass, angular momentum and spin to provide a complete explanation of all properties of the electron: it composite, spinning around its own axis, produces a tiny magnetic fields independent of those from its orbital motions. The classical hydrodynamic laws are used to describe the quantum properties of the electron, such as spin, angular momentum, magnetic momentum and a magnetic dipole. The circulation in the vortex is constant, and the angular momentum of the vortex is conserved and has the same value of Planck constant. The direction of the angular momentum of a spinning electron vortex is along the axis of rotation and determined by the direction of spin. The spin quantum number 1/2 has a fixed value which represents the gap between the circulation rate of the core of the vortex and the boundaries of the vortex. The changeable values +1/2 “spin-up” or -1/2 “spin-down” indicate the direction of the magnetic dipole of the vortex. The relation between spin and Planck constant is discussed and the origin h/4pi angular momentum units are revealed.
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
It has been found that a model of extended electrons is more suited to describe theoretical simulations and experimental results obtained via scanning tunnelling microscopes, but while the dynamic properties are easily incorporated, magnetic properties, and in particular electron spin properties pose a problem due to their conceived isotropy in the absence of measurement. The spin of an electron reacts with a magnetic field and thus has the properties of a vector. However, electron spin is also isotropic, suggesting that it does not have the properties of a vector. This central conflict in the description of an electron’s spin, we believe, is the root of many of the paradoxical properties measured and postulated for quantum spin particles. Exploiting a model in which the electron spin is described consistently in real three-dimensional space–an extended electron model–we demonstrate that spin may be described by a vector and still maintain its isotropy. In this framework, we re-evaluate the Stern–Gerlach experiments, the Einstein–Podolsky–Rosen experiments, and the effect of consecutive measurements and find in all cases a fairly intuitive explanation.
Spin, still a concept or property not fully exploited, has already impacted the world in many ways with its many applications. The history of how spin came to light and how it was interpreted and developed as investigations were conducted is of particular interest with the many odd outcomes of experiments concerning this intrinsic property. This article will sort out the historical outline of how spin was first indicated in theories and how it was eventually embraced as a fundamental property of quantum particles in an attempt to holistically review the process of recognition of this remarkable concept. Through this review, it becomes apparent how the recognition and development of a concept involves the intellect of a broad range of physicists who are willing to view things from both classical and quantum perspectives particularly when it comes to new concepts as spin. How elements like luck and chance play a role will also be revealed. As it turns out, our intuitions about the world around us may not be the ultimate truth, especially when it comes to Quantum Mechanics. Space Quantization was once thought of as no more than a theoretical outcome, however, experiments anyway impose this reality on physicists requiring them to abandon stubborn biases. Admittedly, not every speculation was correct, the analogy of “Spin” was an unfortunate one. But this article shows that this is just how understanding the mysteries of the quantum world really unfolds.