The Wigner–Eckart Theorem for Reducible Symmetric Cartesian Tensor Operators

We explicitly establish a unitary correspondence between spherical irreducible tensor operators and Cartesian tensor operators of any rank. That unitary relation is implemented by means of a basis of integer-spin wave functions that constitute simultaneously a basis of the spaces of Cartesian and spherical irreducible tensors. As a consequence, we extend the Wigner–Eckart theorem to Cartesian irreducible tensor operators of any rank, and to totally symmetric reducible ones. We also discuss the tensorial structure of several standard spherical irreducible tensors such as ordinary, bipolar and tensor spherical harmonics, spin-polarization operators and multipole operators. As an application, we obtain an explicit expression for the derivatives of any order of spherical harmonics in terms of tensor spherical harmonics.