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American Journal of Physics

Orbital Angular Momentum in Quantum Mechanics

M. L. Whippman

Published Aug 1, 1966• 8 citations
Publisher

Abstract

Various methods of proving that half-integral values of the orbital angular momentum do not occur are discussed and a complete proof is given.
DOI
10.1119/1.1973196

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That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.
This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.