Angular Momentum and Spherical Symmetry

James M. Feagin

Ajoy Kumar Ghatak, S. Lokanathan

Ingvar Lindgren, John Morrison

Ahmad Amer

This chapter addresses quantum mechanical motion in central fields, focusing on the separation of variables in spherical coordinates. The eigenstates are separated into radial and angular components, with the angular part expressed using spherical harmonics. The emergence of Legendre and associated Legendre polynomials is discussed in detail as factors in the angular solutions. The parity properties of spherical harmonics are analyzed. Furthermore, explicit expressions are derived for the components and the square of the angular momentum operator in spherical coordinates. The chapter introduces the orbital angular momentum and magnetic quantum numbers, providing their physical significance and role in quantum systems with central symmetry. Metadata/Keywords: central field, spherical harmonics, Legendre polynomials, orbital angular momentum quantum number, magnetic quantum number, angular momentum operators, parity

‎Louis Marchildon

Efstratios Manousakis

Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.