Spin operators and matrices

Spin operators

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

[eqn]

Sometimes, the non-Hermitian ladder operators

[eqn]

are used. The cartesian operators are then given by

[eqn]

Some common commutators are

[eqn]

and

[eqn]
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 [eqn] (m=-S,...,S), in short [eqn], that satisfy

[eqn]
Spin matrices - Explicit matrices

For S=1/2

[eqn]
[eqn]
[eqn]

The [eqn] state is commonly denoted as [eqn], the [eqn] state as [eqn].

For S=1

[eqn]
[eqn]

For S=3/2

[eqn]
[eqn]
[eqn]

For S=2

[eqn]
[eqn]
[eqn]

For S=5/2

[eqn]
[eqn]
[eqn]