Basis states and operators composed of n, in general, different nuclear spins of integer or half-integer values are explicitly constructed. Various coupling schemes are discussed, and transformations between them derived. Following this, a complete set of irreducible tensor operators T(k){V} is constructed which can be used as a basis for expanding operators which depend on n nuclear spins. It is further shown that decomposition of the tensor’s components T(k)q{V} into a sum of products of two irreducible tensor components involves transformation matrices between different coupling schemes. Various properties and commutation relations of the T(k)q{V}’s are given along with a discussion of their reduced matrix elements.
Properties of quantum-mechanical time-dependent correlation functions of pairs of irreducible tensor operators are derived from the transformation properties of the tensor operators under rotations, the assumed rotational invariance of the Hamiltonian of the system, and the Wigner-Eckart theorem. It is shown that many of the possible correlation functions are zero, and that the nonzero correlation functions can be expressed in terms of reduced matrix elements. Some examples of irreducible tensor operators which occur in applications are given, and their reduced matrix elements are calculated.