Rotations and Angular Momentum
Spatial rotations have been encountered repeatedly and in different contexts. Spin spaces, brought to light by the Stern—Gerlach experiment, were treated in Chap. 4. Orbital angular momentum operators were introduced in Chap. 7. In Chap. 13 the rotation group was defined. In the present chapter we will first show that group-theoretical concepts afford a synthesis of all results on rotations that have hitherto been obtained. In so doing we will obtain rotation matrices for arbitrary spin. We will next develop the formalism for the addition of two angular momenta. From a group-theoretical point of view it is the decomposition, into irreducible representations, of a reducible representation of the rotation group. We will introduce the concept of irreducible tensor operator and prove an important theorem about matrix elements of such operators. Finally we will examine two important quantum systems whose symmetry groups are larger than the rotation group.1
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