16 - Quantum Theory of Angular Momentum

Summary

In Chapter 15, we showed that invariance under rotation in real three-dimensional position space transforms the state of a system by a unitary transformation generated by the operator which is the sum of the orbital angular momentum operator, and the spin operator which acts on the internal state of the system. The components of obey the same commutation relations as the corresponding components of. The eigenvalue problem of the orbital angular momentum has been solved in Chapter 14 by working in the position representation. We found that the eigenvalues of are

where l = 0, 1, 2,….

However, acts on the internal state of the system and does not have position representative. Hence, for the systems having spin, we need to solve the eigenvalue problem of the angular momentum by invoking only the commutation relations between the components of, the task undertaken in this chapter. We will find that, like the orbital angular momentum, the eigenvalues of are also expressible in the form but, in addition to non-negative integral values, can be a half-odd positive integer. Thus, the spectrum of derived using the commutation relations turns out to be different from that obtained while working in position representation when permissible. This is unlike the case of harmonic oscillator for which the eigenvalues turn out to be same whether the oscillator problem is solved in the position representation (when permissible) or by using only the commutation relations between the harmonic oscillator operators.