Chapter 11. Angular Momentum and the Position Operators

Dennis Morris

A. R. Edmonds

A. R. Edmonds

Peter Joshua Atkins, Ronald Friedman

This chapter discusses how solving the Schrödinger equation leads to the conclusion that orbital angular momentum is quantized and how the same conclusion can also be reached from the angular momentum operators without solving the Schrödinger equation explicitly. It analyses the commutation properties of the angular momentum operators, noting that the same conclusions apply to observables that are described by operators with the same commutation properties. Angular momentum has many more mundane applications, including being central to the discussion of the structures of atoms and the rotation of molecules, as well as to virtually all forms of spectroscopy. The chapter mentions classical mechanics that puts no constraints on the magnitude of the angular momentum, which is consistent with the kinetic energy of rotation being continuously variable. The property of hermiticity can be demonstrated explicitly in the position representation.

Gordon Baym

The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.