Orbital Angular Momentum Operator in Spherical Coordinate System

James M. Feagin

Tapan Kumar Das

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

Brian P. Anderson

Tao Xiang

The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.

Charles P. Friedman, Edwin F. Taylor

The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.