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Cambridge University Press eBooks

Spin ½: the Pauli theory

Peter Holland

Published May 13, 1993• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511622687.010

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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.
The Pauli spin matrices have many useful properties. The spin of a particle is a degree of freedom additional to the spatial degrees of freedom. It is independent of the spatial degrees of freedom in the sense that one can simultaneously specify the spin state and the spatial state of a particle. Non-relativistically one must use more indirect methods of observing the spin. Classically, a spinning distribution of charge will have a magnetic moment; quantum mechanical spinning particles also have magnetic moments associated with their spins, and one tries to observe these magnetic moments in experiments. Spins in magnetic fields have very interesting dynamical properties. The right side is just the torque exerted by a magnetic field on a magnetic moment. This equation says in operator language that the rate of change of the spin angular momentum is the applied torque.
We re-examine the square of the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , in Equation 15.7 and also the Dirac approach in finding the relativistic Hamiltonian. The classical Hamiltonian for the free particle should be the positive square root of Equation 15.7 which is not a linear function of the momenta. Applying this to the problem, say for example in Equation 15.10, we have the resultant wave equation that is asymmetrical with respect to space and time derivatives and so the necessity to linearize the equation in the momentum operators. So for Equation 15.10, it is impossible to include the Pauli spin matrices without destroying the invariance of the theory since the spin matrices transform the components of a three-dimensional rather than a four-dimensional vector. In addition, the function, https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , has one component rather than two components like spinors and so the Schrödinger relativistic equation represents a particle with no spin. So for a potential to be added in Equation 15.10, the Lorentz-transformation properties of such an addition should be guaranteed by matrix algebra which is the requirement for the Hamiltonian https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to be linear in the momentum operators and guarantees relativistic invariance. This matrix algebra which permits the linearization of the Hamiltonian in the momentum operator was introduced by Dirac with the help of the Pauli matrices.
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 .
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
The quantum theory of particles of indefinite mass investigated earlier by Hostler [J. Math. Phys. 21, 2461 (1980); 22, 2307 (1981)] is adapted to apply to spin- 1/2 particles. The proposed wave equation is { 1/2 Π⋅(1+iσ)⋅Π+(1/i)(∂/∂λ)} φ=0, Πμ=pμ−eAμ, in which φ is a 2×1 Pauli spinor and σ is a spin tensor whose Lorentz components are 2×2 Pauli spin matrices. Particle scattering by an external c-number static field is computed within the framework of the new formalism, and agreement with the known physics of a charged spin- 1/2 particle is obtained for this special case.
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).