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Royal Society of Chemistry eBooks

The Quantum Mechanics of Angular Momentum

Published Sep 12, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1039/9781837678785-00380

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The Quantum Mechanics of Angular Momentum

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.

Royal Society of Chemistry eBooks•Dec 13, 2016•1 citation

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Abstract The concept of angular momentum in quantum mechanics has many counterintuitive aspects, such as assuming only certain discrete values and only certain orientations in space in the presence of external fields. Spin is more esoteric yet; it is hard to imagine how point‐particles with no spatial extent can have an intrinsic angular momentum. Nevertheless, spin is the most fundamental concept in magnetic resonance. Appreciation of the spin concept requires some understanding of the relativistic quantum mechanics developed by P. A. M. Dirac. The history of the evolution of the spin concept is traced—from the turn‐of‐the‐century observation of the Zeeman effect, through the old and new quantum mechanics, and on to Dirac's theory of the electron, which was published in the late 1920s. The special relativity theory is discussed, and there is a brief explanation of how spin emerges naturally in a relativistically correct formulation of quantum mechanics.

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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.

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- Angular Momentum I

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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.

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Angular Momentum

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Abstract The main purpose of this chapter is to present shift operators for the quantum numbers of angular momentum. These operators have useful applications, some of which are discussed in this and subsequent chapters. We also give a brief discussion of the non-relativistic treatment of spin angular momentum.

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Angular Momentum and Spin

James K. Freericks

In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.

Springer eBooks•Jan 1, 2026•0 citations
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 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.
Abstract The concept of angular momentum in quantum mechanics has many counterintuitive aspects, such as assuming only certain discrete values and only certain orientations in space in the presence of external fields. Spin is more esoteric yet; it is hard to imagine how point‐particles with no spatial extent can have an intrinsic angular momentum. Nevertheless, spin is the most fundamental concept in magnetic resonance. Appreciation of the spin concept requires some understanding of the relativistic quantum mechanics developed by P. A. M. Dirac. The history of the evolution of the spin concept is traced—from the turn‐of‐the‐century observation of the Zeeman effect, through the old and new quantum mechanics, and on to Dirac's theory of the electron, which was published in the late 1920s. The special relativity theory is discussed, and there is a brief explanation of how spin emerges naturally in a relativistically correct formulation of quantum mechanics.
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 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.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.