Angular momentum
The operator J, whose Cartesian components satisfy the commutation
relations 
is defined as an angular momentum
operator. For such an operator we have 
, i.e. the operator 
commutes
with each Cartesian component of J. We can therefore find an orthonormal
basis of eigenfunctions common to J2 and Jz.
We denote
this basis by 
. We have 
.
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, 
. We define the operators 
and 
.
We
then have 
and 
.
The operators 
operating on
the basis states 
yield 
The operator 
satisfies the commutation
relations 
and is called the orbital angular momentum operator.
We denote the
common eigenstates of L2 and Lz by 
.
In coordinate representation
we have 
and 
. The normalized common eigenfunctions of L2 and Lz
are called the spherical harmonics.
.
We have
,
The 
may be written as products of the associated
Legendre functions 
and
. 
,
,
where 
, and Pl(u) is the lth
order Legendre polynomial.
(Note: The choice of phase for the Plm(u) and therefore the
definition of the Ylm in terms of the Plm is not
unique.) The 
form a complete set of functions of
angle on the unit sphere. Orthonormality is expressed through 
, 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 J1 and J2.
J1 operates in E1 and J2
operates in E2. Let J=J1
+J2. J operates in E=E1ÄE2.
Since the operators J12, J1z , J22,
and J2z all commute, a basis of common eigenvectors for E exists.
We denote this basis by {|j1,j2;m1,m2>}.
Since the operators J12, J22, J2,
and Jz all commute, a basis of common eigenvectors for E exists.
We denote this basis by {|j1,j2;j,m>}. We can write the vectors of one basis as linear
combinations of the vectors of the other
basis.
.
The 
are called the Clebsch-Gordon
coefficients.
Properties of the Clebsch-Gordan coefficients:
.
(stretched case).
An observable A is a scalar observable
if 
i.e. A commutes
with all components of J. An example of a scalar observable is J2.
In the standard basis {|k,j,m> } the matrix elements of any scalar operator are
non-zero only between states with the same values of j and m, and they are
independent of m.
A vector observable V is a set
of three observables, Vx, Vy, Vz, which obey the
commutation relations 
An example of a vector
observable is J itself. Inside the subspace E(k,j)
the matrix elements of any vector observable are proportional to the matrix elements of
J.
.
This is the projection theorem, a special
case of the Wigner-Eckart theorem.