Angular Momentum

Abstract

Apart from the Hamiltonian, the angular momentum operator is of one of the most important Hermitian operators in quantum mechanics. In this chapter we consider its eigenvalues and eigenfunctions in more detail. We begin this chapter with the consideration of the orbital angular momentum. This gives rise to a general definition of angular momenta. We derive the eigenvalue spectrum of the orbital angular momentum with an algebraic method. After a brief presentation of the eigenfunctions of the orbital angular momentum in the position representation, we outline some concepts for the addition of angular momenta. 16.1 Orbital Angular Momentum Operator The orbital angular momentum is given by l = r × p. (16.1) As we have seen in Chap. 3, Vol. 1, it is not necessary to symmetrize for the translation into quantum mechanics (spatial representation). It follows directly that l = i r × ∇, (16.2) or, in components, l x = i y ∂ ∂z − z ∂ ∂ y (16.3) plus cyclic permutations (x → y → z → x → · · ·). All the components of l are observables.