. 
, 
, 
. We write
. The matrices
, 
, 
are the Pauli matrices.The state space of a spin ½ particle is two-dimensional. The common orthonormal
eigenbasis of S2 and Sz is {|+>,|->}.
In this basis
. , , .
We write .
The matrices , , are the Pauli matrices.
 | Properties of  :
The operator Su is defined through The matrix of Su is |
Let Es=Es(1)
Es(2) be the state space of
a system of two spin ½ particles. The tensor product vectors {|++>, |+->,
|-+>, |-->} form a basis for Es.
In the
four dimensional state space Siz is product operators. 
etc.
The matrix of any product operator A(1)B(2) in a basis of tensor product vectors {|i(1)>|j(2)>=|i,j>} is
,
where the Aij are the matrix elements of A in the {|i(1)>}
basis of E1 and 
is the matrix of B in the {|j(2)>} basis of E2.
The common eigenvectors of S2=(S1+S2)2
and Sz=S1z+S2z also form a basis of Es, which we denote by {|S,Sz>},
where 
denotes the eigenvalue of S2
and 
denotes the eigenvalue of Sz.
We have the singlet state 
and the triplet states
.
, 
.
.
.
The eigenvectors of Su are 
.