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Angular Momentum

Hans-Peter Eckle

Published Jul 25, 2019• 0 citations
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Abstract

Abstract In order to prepare for the discussion of quantum many-particle Hamiltonians, for example the Heisenberg quantum spin chain Hamiltonian, this chapter examines the concept of angular momentum in quantum mechanics, especially the coupling of spin-2 operators for several quantum spins. It begins with the general theory of angular momentum for a single quantum particle, especially for a single spin-1, described by Pauli spin matrices, and then extends to the theory of angular momentum for several particles, again especially for several spins1.

DOI
10.1093/oso/9780199678839.003.0003

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Abstract In order to prepare for the discussion of quantum many-particle Hamiltonians, for example the Heisenberg quantum spin chain Hamiltonian, this chapter examines the concept of angular momentum in quantum mechanics, especially the coupling of spin-2 operators for several quantum spins. It begins with the general theory of angular momentum for a single quantum particle, especially for a single spin-1, described by Pauli spin matrices, and then extends to the theory of angular momentum for several particles, again especially for several spins1.
Angular momentum is essential to any advanced discussion of magnetic resonance spectroscopy. A brief history of the modern sub-atomic particle-spin concept is presented. The origin of the spin-1/2 angular momentum quantum number is detailed and angular momentum operators are constructed for spin-1/2 nuclei using the postulates of quantum mechanics that are then used to produce a set of very useful eigenvalue formulas. Angular momentum operators in spherical coordinate form are developed and operator matrices are constructed from Cartesian operators. The time-evolution of these operators is of much interest in later chapters and is explored in some detail here.
Angular momentum is essential to any advanced discussion of magnetic resonance spectroscopy. A brief history of the modern sub-atomic particle-spin concept is presented. The origin of the spin-1/2 angular momentum quantum number is detailed and angular momentum operators are constructed for spin-1/2 nuclei using the postulates of quantum mechanics which are then used to produce a set of very useful eigenvalue formulas. Angular momentum operators in spherical coordinate form are developed and operator matrices are constructed from Cartesian operators. The time-evolution of these operators is of much interest in later chapters and is explored in some detail here.
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.
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.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.
The chapter begins with the introduction of the two-particle Schrödinger wave equation (SWE) and the solution of this equation for the hydrogen atom. The orbital angular momentum of the electron results from the SWE solution. The Pauli spinors are introduced, and the SWE wavefunctions are modified to account for the spin of the electron. The structure of multielectron atoms is then discussed. The discussion is focused on low-Z atoms for which Russell–Saunders or LS coupling is appropriate. Alternate coupling schemes are briefly discussed. Angular momentum coupling algebra, the Clebsch–Gordan coefficients, and 3j symbols are then introduced. The Wigner–Eckart theorem is discussed, and the use of irreducible spherical tensors for evaluation of quantum mechanical matrix elements is discussed in detail.