On Angular Momentum

By Julian Schwinger

A concise treatment of angular momentum by an important American physicist, this major work was first published under the auspices of the United States Atomic Energy Commission in 1952 and is finally available to a general audience of students and professionals in the field. Advanced undergraduates and graduate students of physics will particularly benefit from its teachings.
One of the most prominent American physicists of the twentieth century, Julian Schwinger (1918–94) taught at Harvard, MIT, and UCLA, among other institutions. In addition to his many other awards, Schwinger, jointly with Richard Feynman and Shinichiro Tomonaga, received the Nobel Prize in Physics in 1965 for his work in quantum electrodynamics.

• Language: English
• Publisher: Dover Publications
• Release date: Mar 9, 2015
• ISBN: 9780486801889

Book preview

operators.

1. INTRODUCTION

Such a spin assembly, considered as a Bose-Einstein system, can be usefully discussed by the method of second quantization. We shall see that this procedure unites the compact symbolism of the group theoretical approach with the explicit operator techniques of quantum mechanics.

which satisfy

The number of spins and the resultant angular momentum are then given by

With the conventional matrix representation for σ, the components of J appear as

Of course, this realization of the angular momentum commutation properties in terms of those of harmonic oscillators can be introduced without explicit reference to the composition of spins.

To evaluate the square of the total angular momentum

we employ the matrix elements of the spin permutation operator

Thus

and

According to the commutation relations (1.1),

whence

a given number of spins, n=0, 1, 2, ..., possesses a definite angular momentum quantum number,

We further note that, according to (1.3), a state with a fixed number of positive and negative spins also has a definite magnetic quantum number,

Therefore, from the eigenvector of a state with prescribed occupation numbers,

we obtain the angular momentum eigenvector¹

Familiar as a symbolic expression of the transformation properties of angular momentum eigenvectors², this form is here a precise operator construction of the eigenvector.

On multiplying

we obtain, after summation with respect to m, and then with respect to j,

and

in which we have written

To illustrate the utility of (1.16), conceived of as an eigenvector generating function, we shall verify the orthogonality and normalization of the eigenvectors (1.13). Consider, then,

According to the commutation relations we have

whence

We have thus proved that

As a second elementary example, we shall obtain the matrix elements of powers of J± on (1.16). We