Spin operators and matrices

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

and

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

For S=1/2

The state is commonly denoted as , the state as .

For S=1

For S=3/2

For S=2

For S=5/2