The operator **J**, whose Cartesian components satisfy the commutation
relations

[J_{i},J_{j}] = ε_{ijk}iħJ_{k}

is defined as an angular momentum operator.

For such an operator we have [J_{i},J^{2}]
= 0, i.e. the operator J^{2} = J_{x}^{2} + J_{y}^{2}
+ J_{z}^{2} commutes with each Cartesian component of
**J**.
We can therefore find an orthonormal basis of eigenfunctions common to J^{2}
and J_{z}. We denote this basis by {|k,j,m>}.

We have J^{2}|k,j,m>
= j(j + 1)ħ^{2}|k,j,m>,
J_{z}|k,j,m> = mħ|k,j,m>.

The index j can take on only integral and half integral positive values. For a
given j the index m can take on one of 2j + 1 possible values, m = -j, -j + 1,
... , j - 1, j.

We define the ladder operators J_{+} = J_{x} + iJ_{y}
and J_{-} = J_{x} - iJ_{y}.

We then have J_{x}
= ½(J_{+} + J_{-}) and J_{y} = (-i/2)(J_{+}
- J_{-}) .

The operators J_{±}
operating on the basis states {|k,j,m>} yield

J_{±}|k,j,m>
= [j(j+1) - m(m±1)]^{½}ħ|k,j,m±1>.

The operator **L** = **R **×** P
** satisfies
the commutation relations [L_{i},L_{j}] = ε_{ijk}ħL_{k}
and is called the **orbital angular momentum** operator. We denote the
common eigenstates of L^{2} and L_{z} by {|k,l,m>}. In
coordinate representation we have

L_{z} = (ħ/i)∂/∂φ

and

L^{2} = -ħ^{2}[(1/sinθ)∂(sinθ ∂/∂θ)/∂θ + (1/sin^{2}θ)∂^{2}/∂φ^{2}].

The normalized common eigenfunctions of L^{2} and L_{z} are
called the **spherical harmonics**.

**Properties of the spherical harmonics**Y

We have

Y

Y

Y

The Y

∫

and completeness is expressed through

∑

Complex conjugation

Parity

The parity of the spherical harmonics is well defined and depends only on l.

Consider two angular momentum operators **J**_{1} and
**J**_{2}.
**
J**

Let

|j

|j

The C

Properties of the Clebsch-Gordan coefficients

C

C

The state space of a spin ½ particle is two-dimensional.

The common orthonormal eigenbasis of S^{2} and S_{z} is
{|+>, |->}.

Below are matrices of spin operators in this basis.

We write
**S** = (ħ/2)**σ**.
The matrices

are the
**Pauli matrices**.

**Properties of σ _{x} and
σ_{y} and σ_{z}**

det(σ

In general, σ

Therefore

[S

since

The operator S

The matrix of S

The eigenvectors of S

|+>

|->

Therefore |+> = cos(θ/2)|+>

Let E_{s }= E_{s}(1)⊗E_{s}(2)
be the state space of a system of two spin ½ particles.

The tensor product vectors {|++>, |+->, |-+>, |-->} form a basis for
E_{s}.

In the four dimensional state space the operators S_{iz} are product operators.

S_{1z} = S_{1z}(1)
⊗ I(2), S_{2z} = I(1)
⊗ S_{2z}(2),
etc.

The common eigenvectors of S^{2 }= (**S**_{1 }+** S**_{2})^{2}
and S_{z }= S_{1z }+ S_{2z} also form a basis of
E_{s}, which we denote by {|S,S_{z}>},
where s(s + 1)h^{2}
denotes the eigenvalue of S^{2} and m_{s}h denotes the eigenvalue of S_{z}.

We have the **singlet state
**|00> = 2

and the

|11> = |++>,

|10> = 2

|1-1> =|-->.