On Angular Momentum

1. INTRODUCTION

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 spins, 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 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 (1.13) with an analogous monomial constructed from the components of the arbitrary spinor

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 (1.1), and we have

whence

We have thus proved that

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