Chapter 9 The Two-Body Problem

1.5. Angular Momentum Eigenvalues Determined from the Eigenvalues of Two Uncoupled Oscillators
An elegant way of finding the eigenvalues and eigenvectors of angular momentum in terms of the creation and annihilation operators for the uncoupled oscillators has been derived by Schwinger [9].
Let us consider two simple harmonic oscillators for which the creation and annihilation operators are denoted by and . The number operators for and oscillators are

and these operators satisfy the commutation relations

where in each case we have to take all plus signs or all minus signs. Since all these oscillators are uncoupled we have

and so forth. The two number operators and commute, a result that simply follows from (9.129). Hence they can be diagonalized simultaneously. Let represent a state where the oscillator has an eigenvalue and the oscillator has an eiganvalue , then we have

The action of the creation and annihilation operators on these states will change the numbers and , i.e.

and

Next we define and by

and

From the commutation relations of the operators we can verify that and satisfy the angular momentum commutation relations;

and

Moreover if denotes the sum of and

then we have

Now using Eqs. (9.131)-(9.134) we determine the action of the operators and on

and

We note that is an eigenstate of the operator . To write these in the notation of Sec. 9.2 we replace and by

then

and

A physical picture of the connection between and oscillators and the eigenvalues of angular momentum has been discussed by Sakurai [10].
An object of high can be visualized as a collection of spin particles, of them with spin up and of them with spin down. Thus if we have spin particles we can add them up in different ways to get states with angular momenta . If we have just two particles each with a spin then we can have either a system of or . The problem with this interpretation is that
(a) - we have to start with an even number of particles, , and
(b) - the and particles are in fact bosons and not fermions.