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EPJ Web of Conferences

Spin as an intrinsic form of top angular momentum

I. B. Pestov

Published Jan 1, 2019• 1 citation
Publisher

Abstract

In quantum mechanics and particle physics, Spin is considered as an intrinsic form of the quantum orbital angular momentum. The goal of this paper is to demonstrate that in accordance with the creative original idea of Kronig, Uhlenbeck and Goudsmit, Spin can be represented as an intrinsic form of quantum Spherical Top angular momentum. It will be shown that this internal symmetry can be realized on a set of the simplest geometrical quantities, which themselves do not exhibit this emergent property. That is why this phenomenon will be called Emergent Spin. The concept of Spin as an emergent property is more general than the habitual concept of Spin and, hence, it can be interesting in terms of discussion of possible ways to look for a physics beyond the Standard Model. Now, there is no doubt that new physics really exists and we need clear guidance on the best place to look.

DOI
10.1051/epjconf/201920402004

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Cambridge University Press eBooks•Dec 18, 2013•0 citations
In quantum mechanics and particle physics, Spin is considered as an intrinsic form of the quantum orbital angular momentum. The goal of this paper is to demonstrate that in accordance with the creative original idea of Kronig, Uhlenbeck and Goudsmit, Spin can be represented as an intrinsic form of quantum Spherical Top angular momentum. It will be shown that this internal symmetry can be realized on a set of the simplest geometrical quantities, which themselves do not exhibit this emergent property. That is why this phenomenon will be called Emergent Spin. The concept of Spin as an emergent property is more general than the habitual concept of Spin and, hence, it can be interesting in terms of discussion of possible ways to look for a physics beyond the Standard Model. Now, there is no doubt that new physics really exists and we need clear guidance on the best place to look.
In quantum mechanics and particle physics, spin is considered as an intrinsic form of orbital angular momentum. Our goal here is to demonstrate that the well known fundamental idea of Kronig, Uhlenbek and Goudsmit can be presented as an intrinsic form of symmetric top rotation motion. It will be shown that internal symmetry of intrinsic top can be realized on a set of the simplest geometrical quantities, which themselves do not exhibit this property. That is why an intrinsic form of top angular momentum will be called emergent spin. The concept of emergent spin is more general than the concept of spin and, hence, it can be interesting in terms of discussion of the possible ways to look for a physics beyond the Standard Model.
One of the most important applications of the quantum mechanical formalism developed in the previous chapter is to angular momentum. This is because many microsystems are spherically symmetric, or approximately so, and angular momentum is therefore conserved. This chapter discusses that a careful study of quantized angular momentum leads to a completely new physical phenomenon – intrinsic spin. The fact that the angular momentum vector of a quantum particle must point in certain restricted directions is a bizarre quantum property. In some respects the abstract idea of an internal angular momentum is reminiscent of the classical concept of a spinning body. In classical mechanics angular momentum is added vectorially but in the quantum case the issue is more subtle. The procedure will be illustrated by the example of the addition of two spins. As in classical mechanics a charged quantum system with angular momentum possesses a magnetic moment μ.
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 this chapter we shall treat two of the most important observables of quantum mechanical systems: angular momentum and spin. We shall argue that angular momentum is the generator of rotations. In quantummechanics, angular momentum is quantized in units of the reduced Planck constant. The quantization of angular momentum is dealt with by finding the angular momentum eigenvalues and eigenstates. We shall then apply the theory of angular momentum to study the hydrogen atom. Spin is an intrinsic angular momentum of subatomic particles and therefore corresponds to an important intrinsic degree of freedom. We shall discuss briefly how spin was discovered and many of its important properties. Finally, we shall also explore here some further subtleties of quantum theory.a When we consider quantum systems in two or three spatial dimensions, we need to take into account also the effect of rotations. In this case, we need to introduce a physical quantity called the angular momentum, that is, themomentum induced by or connected with rotations. In classical physics the angular momentum of a particle about a given origin is defined by [see Fig. 8.1]
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
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.