Commutation relations spin tensor operators

The commutator of two Hermitian operators is an anti-Hermitian operator. From (2.3.2), we can therefore conclude that spin tensor operators are not in general Hermitian. Indeed, the only possible exception to this rule are the operators where M = 0, which may or may not be Hermitian. It is therefore of some interest to examine the Hermitian adjoints of the spin tensor operators. Taking the conjugate of the relations (2.3.1) and (2.3.2), we obtain ... 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j ... 

With Ki a. spherical tensor of arbitrary rank, either spatial or spin, irreducible spherical tensors are defined by their commutation relations with linear Cartesian operators in the following manner... 

Tensors (15.39)—(15.41) meet commutation relations (14.2) for irreducible components of the momentum operator, and, in addition, they commute with the operators of orbital (14.15) and spin (14.16) angular momenta for the lN configuration, since they are scalars in their respective spaces. Accordingly, the states of the lN configuration can be characterized by the eigenvalues of operators L2, Lz, S2, Sz, Q2, Qz. 

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