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.
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One of the methods of treating a general angular momentum in quantum mechanics is to regard it as the superposition of a number of elementary â spins,â or angular momenta with
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.
We introduce spin creation and annihilation operators associated with a given spatial reference system,
and
which satisfy
The number of spins and the resultant angular momentum are then given by
With the conventional matrix representation for Ï, 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 occupat...
