The Angular Momentum as Generators of Rotations: Lie Groups and Lie Algebras

Willi-Hans Steeb

Walter Greiner, Berndt Müller

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials

Kommanaboyana Srinivasa Rao, Vasudevan Lakshminarayanan

BAO JIAN-REN, Zhou Lu-wei, 复旦大学物理系

The association and difference between general angular momentum and orbital angular momentum are discussed. Their association lies in the identity of local property (Lie algebra) between group SU (2) and SO (3) (isomorphism), while their difference is their distinctive integral property. The minimal unit ?/2 of the general angular momentum is determined by the local properties of the groups, namely the commutation relation of Lie algebra (or commutor of angular momentum). However, the minimal unit ? of orbital angular momentum depends upon the integral property of group SO (3).