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Journal of physics. A, Proceedings of the Physical Society. General

Relating the classical and quantal theories of angular momentum and spin

A Norcliffe

Published Feb 1, 1972• 4 citations
Publisher

Abstract

The classical and quantal theories of angular momentum and spin are related using the spherical top as a model. The quantum mechanical spectral operator kernel of the spherical top is evaluated and expressed exactly as a sum over classical paths of terms containing the classical action. A relation between corresponding values of the classical and quantal angular momentum is obtained which holds for all values of the angular momentum quantum number.

DOI
10.1088/0305-4470/5/2/004

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The classical and quantal theories of angular momentum and spin are related using the spherical top as a model. The quantum mechanical spectral operator kernel of the spherical top is evaluated and expressed exactly as a sum over classical paths of terms containing the classical action. A relation between corresponding values of the classical and quantal angular momentum is obtained which holds for all values of the angular momentum quantum number.
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.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
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.