This chapter provides a brief overview of the quantum theory of angular momentum, which is central to the theory of magnetism. It begins by discussing the inadequacy of motional angular momentum and introduces spin angular momentum. The chapter then develops operator techniques to express angular and spin operators in terms of more primitive fermion or boson operators, treating spin-one-half and spin-one separately for subsequent chapters on magnetism. In the absence of external forces, the classical angular momentum of a point particle is given by \( L = \mathbf{r} \times \mathbf{p} \). In quantum theory, the analogous quantity, denoted as "kinetic angular momentum," is an operator \( L = \mathbf{r} \times \frac{\hbar}{i} \nabla \). For a spherically symmetric wavefunction \( \psi \), the kinetic angular momentum is zero because \( \nabla \psi(r) = \hat{\mathbf{u}}_r \frac{d}{dr} \psi(r) \), leading to \( L \psi(r) = 0 \). Functions with non-trivial angular dependence have non-zero kinetic angular momentum, which requires solving an eigenvalue problem. The components of \( \boldsymbol{L} = (L_x, L_y, L_z) \) do not commute, and their commutation relations are derived. The vector cross product \( \boldsymbol{L} \times \boldsymbol{L} \) is shown to satisfy \( i \hbar \boldsymbol{L} \), emphasizing the importance of this definition for genuine angular momentum operators. The chapter also discusses the pseudovector nature of classical angular momentum and expresses the kinetic angular momentum in spherical polar coordinates.This chapter provides a brief overview of the quantum theory of angular momentum, which is central to the theory of magnetism. It begins by discussing the inadequacy of motional angular momentum and introduces spin angular momentum. The chapter then develops operator techniques to express angular and spin operators in terms of more primitive fermion or boson operators, treating spin-one-half and spin-one separately for subsequent chapters on magnetism. In the absence of external forces, the classical angular momentum of a point particle is given by \( L = \mathbf{r} \times \mathbf{p} \). In quantum theory, the analogous quantity, denoted as "kinetic angular momentum," is an operator \( L = \mathbf{r} \times \frac{\hbar}{i} \nabla \). For a spherically symmetric wavefunction \( \psi \), the kinetic angular momentum is zero because \( \nabla \psi(r) = \hat{\mathbf{u}}_r \frac{d}{dr} \psi(r) \), leading to \( L \psi(r) = 0 \). Functions with non-trivial angular dependence have non-zero kinetic angular momentum, which requires solving an eigenvalue problem. The components of \( \boldsymbol{L} = (L_x, L_y, L_z) \) do not commute, and their commutation relations are derived. The vector cross product \( \boldsymbol{L} \times \boldsymbol{L} \) is shown to satisfy \( i \hbar \boldsymbol{L} \), emphasizing the importance of this definition for genuine angular momentum operators. The chapter also discusses the pseudovector nature of classical angular momentum and expresses the kinetic angular momentum in spherical polar coordinates.