Problems in 3D

Abstract

In this section I introduce the angular momentum operators and their commutation relations (CR). Later on we will find the eigenfunctions and verify the eigenvalues for some examples. Angular momentum plays an important role in QM. In classical mechanics the angular momentum vector 1 = r × p of a point particle moving in a central potential V(r) is conserved, which implies that the motion is confined to a plane. In QM the angular momentum operator is obtained by application of the usual quantization rule (cf. Sect. 1.1). The Heisenberg uncertainty principle ΔxΔp ≥ ħ makes it impossible to know precisely all three Cartesian components of angular momentum simultaneously. On the basis of the time evolution equation and the commutation relations between the Hamiltonian and the operators for the square of angular momentum and its components it is possible to show that the conservation law carries over into the statement that the magnitude and one projection is conserved and can be determined exactly.