# Conversion Between the Cartesian and Spherical Bases

The spherical harmonics of a particular rank are covariant components of an irreducible tensor. This can be used to find the prescription for converting between the spherical and Cartesian bases.

This loads the package with coordinate systems. (Quiet suppresses some shadowing warnings that will occur if the ADM package is already loaded.) |

The spherical harmonics are still in the spherical basis, but they are written in terms of the coordinates x, y, and z. To put them in the Cartesian basis, we want to find a linear (unitary) transformation whose result transforms like a Cartesian vector, i.e., like {x,y,z}.

The ADM package has functions that use this matrix to convert between the spherical and Cartesian bases.

ToCovariant[expr] | convert expr to a covariant spherical tensor |

ToContravariant[expr] | convert expr to a contravariant spherical tensor |

ToCartesian[expr] | convert expr to Cartesian basis |

Converting between standard and Cartesian bases.

ToCartesian converts either set back to Cartesian components. |

The conversion functions also work on vectors whose components are operators.