# The Wigner-Eckart Theorem

The Wigner-Eckart theorem gives the matrix elements of irreducible tensor operators in terms of their reduced matrix elements:

Here is the 3- symbol and (∘||∘||∘) is the reduced matrix element, which is independent of spatial indices (M, , and q).

WignerEckart[sys,{op,k,q}] | the matrix representation of the operator op with rank k and index q for atomic system sys |

WignerEckart[sys,{op,k}] | the matrix components of the operator op with rank k for atomic system sys |

WignerEckart[s_{1},{op,k,q},s_{2}] | the matrix element of the operator op between atomic states s_{1} and s_{2} |

Matrix representation of operators.

If the reduced matrix element for an operator is predefined, as for example for the electronic-angular-momentum operator J, this definition is used by WignerEckart to produce explicit matrix elements for the operator.

The J operator is a rank 1 (κ=1) spherical tensor operator. Thus it has 2κ+1=3 components. This gives the matrices for the J _{1}, J_{0}, and J_{-1} operators: |

operator | rank | |

Zero | 0 | the zero operator |

Identity | 0 | the identity operator |

Energy | 0 | the internal (unperturbed) energy operator |

Dipole | 1 | the E1 electric dipole operator |

J | 1 | the angular momentum operator |

MagneticMoment | 1 | the magnetic moment operator |

Polarizability | 0 | the effective Hamiltonian for scalar polarizability |

Polarizability | 1 | the effective Hamiltonian for vector polarizability |

Polarizability | 2 | the effective Hamiltonian for tensor polarizability |

Polarization | κ | the polarization operator of rank κ |

WignerEckart can be used with an undefined operator: the reduced matrix elements are then given symbolically as ReducedME objects.

ReducedME[s_{1},{op,k},s_{2}] | a symbolic reduced matrix element of the rank-k operator op between atomic states s_{1} and s_{2} |

Symbolic reduced matrix element.

We can complete the definition of a tensor operator by giving a definition for its reduced matrix element. For example, consider a scalar, parity non-conserving interaction mixing two opposite parity states, written in terms of a real, parity-violating mixing parameter δ.

Define the ReducedME for a scalar PNC operator. The parity-violating matrix elements are imaginary. WignerEckart assumes that symbolic parameters are real, so a factor of must be included explicitly. |