Spin vectors are usually represented in terms of their Hermitian cartesian component operators
Sometimes, the non-Hermitian ladder operators
are used. The cartesian operators are then given by
Some common commutators are
For a spin S the cartesian and ladder operators are square matrices of dimension 2S+1. They are always represented in the Zeeman basis with states (m=-S,...,S), in short , that satisfy
The state is commonly denoted as , the state as .