By Professor Dr. Walter Greiner, Professor Dr. Joachim A. Maruhn (auth.)
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160) qm where a and f3 refer to any other quantum numbers affected by the operator. The resulting angular momenta are determined by the selection rules implicit in the Clebsch-Gordan coefficients. 154) above. Of course for operators one has to keep track of the ordering because of their possible noncommutation. In this way the scalar and vector products of operators can be defined as above as the coupling to zero or unit angular momentum. The distinction between operators and fields is crucial for some considerations and the "same" mathematical expression may have to be treated differently depending on how it is used.
146) 29 30 2. 147) which describes a field rotating in the mathematically positive sense about the z axis and thus nicely conforming to the idea of positive intrinsic angular momentum about that axis. A second important case of internal angular momentum is that of spin fields. In this case the wave function has two components at each point in space that transform under rotations according to the representation of the rotation group with angular momentum j = ~. There are no classical analogies and in this case we do not have to worry about the choice of the basis: since there is no predefined physical set of basis states, we can immediately use the eigenstates of angular momentum sz.
181) The operator j2 has the eigenvalue! 182) A Hamiltonian is isospin invariant if it commutes with the operators t. , any mixture of proton and neutron state, whereas in nature only the eigenstates of iz , the proton and neutron, are realized. However, requiring this more general invariance is quite successful and there is no need to consider special transformations exchanging proton and neutron only. Because isospin is not an angular momentum, the dimensional factor Ii is not necessary and is dropped from all the formulas.