# 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.

 Here are the spherical harmonics of rank 1 in terms of the angular coordinates θ and ϕ.
 Here are rules for converting the angular coordinates to Cartesian coordinates.
 The spherical harmonics in terms of Cartesian coordinates, with a new normalization.

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}.

 Find a unitary matrix that transforms the spherical harmonics to {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.