The Wigner-Eckart Theorem for Reducible Symmetric Cartesian Tensor Operators

TL;DR

This paper extends the Wigner-Eckart theorem to include reducible symmetric Cartesian tensor operators by establishing a unitary correspondence with spherical irreducible tensors, enabling new analytical tools for tensor operators.

Contribution

It introduces a unitary correspondence between spherical and Cartesian tensor operators, extending the Wigner-Eckart theorem to higher-rank and reducible tensors.

Findings

Established a basis linking Cartesian and spherical tensors

Extended the Wigner-Eckart theorem to symmetric reducible tensors

Derived explicit expressions for derivatives of spherical harmonics

Abstract

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.