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Angular Momentum and the Quantum Gyroscope: The Emergence of Spin

Duncan G. Steel

Published Apr 30, 2021• 0 citations
Publisher

Abstract

Abstract Continuing with understanding the implications of the postulates in Chapter 7 and following the approach in Chapter 8 to use operators find solutions to the time independent Schrödinger equation, we return to the subject of angular momentum, of importance to many problems including the quantum gyroscope. Aside from playing a central role in any spherically symmetric quantum system, it plays a central role in inertial guidance systems from airplanes and rockets to autonomous vehicles. Working with only the operators of the angular momentum vector, L^=L^xx̌+L^yy̌+L^zž and L^2 and the corresponding commutation relations, a procedure similar to that used in Chapter 8 for the nano-vibrator is used to completely identify the eigenvectors and eigenvalues. However, in Chapter 6, we required that the magnetic quantum number, m where L^z|l.m〉=mℏ|l.m〉, be integer, because the eigenfunction Yl,m(l,m)∝eimϕ, and we required that a full rotation around the z-axis give the same result requiring, eimϕ=eim(ϕ+2π). In the operator approach, there is no such requirement, but there is still a constraint on m, namely that m is either integer or half integer. The requirements in Chapter 6 hold, so what is the meaning of half-integer? This was one of the first results to indicate the existence of intrinsic (not associated with real space rotation) angular moment known as spin.

DOI
10.1093/oso/9780192895073.003.0010

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Abstract Continuing with understanding the implications of the postulates in Chapter 7 and following the approach in Chapter 8 to use operators find solutions to the time independent Schrödinger equation, we return to the subject of angular momentum, of importance to many problems including the quantum gyroscope. Aside from playing a central role in any spherically symmetric quantum system, it plays a central role in inertial guidance systems from airplanes and rockets to autonomous vehicles. Working with only the operators of the angular momentum vector, L^=L^xx̌+L^yy̌+L^zž and L^2 and the corresponding commutation relations, a procedure similar to that used in Chapter 8 for the nano-vibrator is used to completely identify the eigenvectors and eigenvalues. However, in Chapter 6, we required that the magnetic quantum number, m where L^z|l.m〉=mℏ|l.m〉, be integer, because the eigenfunction Yl,m(l,m)∝eimϕ, and we required that a full rotation around the z-axis give the same result requiring, eimϕ=eim(ϕ+2π). In the operator approach, there is no such requirement, but there is still a constraint on m, namely that m is either integer or half integer. The requirements in Chapter 6 hold, so what is the meaning of half-integer? This was one of the first results to indicate the existence of intrinsic (not associated with real space rotation) angular moment known as spin.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
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.
Angular momentum plays a similarly important role in atomic systems, such as an electron subject to the central force provided by the Coulomb field of a nucleus. This chapter introduces the properties of angular momentum, both orbital and spin, in the quantum-mechanical scheme of things. The angular momentum of atomic systems is inseparable from the magnetic properties of atoms. The various permitted values of the quantized angular momentum play a central role in determining the number and variety of possible quantum states in atomic systems. The Stern-Gerlach experiment was deliberately designed to test whether or not the angular momentum of individual neutral atoms is quantized in a magnetic field. As in the development of the Schrodinger equation, classical physics gives some hints of the correct forms of these angular momentum operators. The chapter discusses rough estimate using the Neils Bohr theory and attributing the magnetic moment to the orbital motion of the electron.
This chapter presents key results on angular momentum in quantum mechanics, beginning with a theorem on the simultaneous diagonalization of commuting operators. The eigenvalues and eigenstates of both the total angular momentum operator and its z-component are derived. For a quantum rotator in the p-state with zero magnetic quantum number, it is shown that the most probable trajectory lies in a plane containing the z-axis, and the angular momentum vector is perpendicular to it. The procedure for angular momentum addition is explained in detail, including the use of Clebsch–Gordan coefficients to perform the explicit decomposition of tensor products of irreducible representations. The Bloch sphere representation of qubit spin states is introduced. The chapter then provides an experimental background to the discovery of spin, introduces the Pauli matrices, and demonstrates how they act on spin states. It concludes with the formulation of the Pauli equation, which incorporates spin into the framework of nonrelativistic quantum mechanics. Metadata/Keywords: angular momentum, simultaneous diagonalization, magnetic quantum number, Clebsch–Gordan coefficients, spin, Pauli matrices, Bloch sphere, spin states, tensor product, Pauli equation