Spin operators and matrices

Spin operators

Spin vectors are usually represented in terms of their Hermitian cartesian component operators

are used. The cartesian operators are then given by

Some common commutators are

and

Spin matrices - General

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

Spin matrices - Explicit matrices

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