Angular momentum

The operator J, whose Cartesian components satisfy the commutation relations
[Ji,Jj] = εijkiħJk  
is defined as an angular momentum operator.  
For such an operator we have [Ji,J2] = 0, i.e. the operator J2 = Jx2 + Jy2 + Jz2 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 {|k,j,m>}.  
We have J2|k,j,m> = j(j + 1)ħ2|k,j,m>,  Jz|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+ = Jx + iJy and J- = Jx - iJy.  
We then have Jx = ½(J+ + J-) and Jy = (-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)]½ħ|k,j,m±1>.


Orbital angular momentum

The operator L = R × P  satisfies the commutation relations [Li,Lj] = εijkħLk and is called the orbital angular momentum operator.  We denote the  common eigenstates of L2 and Lz by {|k,l,m>}.  In coordinate representation we have
Lz = (ħ/i)∂/∂φ
and
L2 = -ħ2[(1/sinθ)∂(sinθ ∂/∂θ)/∂θ + (1/sin2θ))∂2/∂φ2].
The normalized common eigenfunctions of L2 and Lz are called the spherical harmonics.

Properties of the spherical harmonics
Ylm(θ,φ) = [(-1)l/(2l l!)][(2l+1) (l+m)!/(4π (l-m)!)]½eimφ(sinθ)-mdl-m(sinθ)2l/d(cosθ)l-m.
We have
Y00 = (4π),  Y1±1 = ∓(3/8π)½sinθ exp(±iφ),  Y10 = (3/4π)½cosθ,
Y2±2 = (15/32π)½sin2θ exp(±i2φ),  Y2±1 = ∓(15/8π)½sinθ cosθ exp(±iφ),
Y20 = (5/16π)½(3cos2θ - 1).

The Ylm(θ,φ) form a complete set of functions of angle on the unit sphere.  Orthonormality is expressed through
0πsinθ dθ∫0dφ Y*l'm'(θ,φ)Ylm(θ,φ) = δl'lδm'm.
and completeness is expressed through
l=0m=-ll Y*lm(θ,φ)Ylm(θ',φ') = δ(cosθ - cosθ')δ(φ - φ') = δ(θ - θ')δ(φ - φ')/sinθ.

Complex conjugation
Y*lm(θ,φ) = (-1)mYl(-m)(θ,φ).

Parity
PYlm(θ,φ) = Ylm(π - θ,π + φ) = (-1)lYlm(θ,φ).
The parity of the spherical harmonics is well defined and depends only on l.


Addition of angular momentum

Consider two angular momentum operators J1 and J2.  
J
1 operates in E1 and J2 operates in E2
Let J = J1 +J2J operates in E = E1⊗ E2.  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.
|j1,j2;j,m> = ∑m1m2 Cjm1m2|j1,j2;m1,m2>,
|j1,j2;m1,m2> = ∑jm Cjm1m2|j1,j2;j,m>.
The Cjm1m2 = <j1,j2;m1,m2|j1,j2;j,m> are called the Clebsch-Gordon coefficients.

Properties of the Clebsch-Gordan coefficients

Cjm1m2 = real, Cjm1m2 = 0 unless m1 + m2 = m and |j1 - j2| ≤ j ≤|j1 + j2|.
Cjm1=j1 m2=j2 = 1  (stretched case).


Spin ½

The state space of a spin ½ particle is two-dimensional.
The common orthonormal eigenbasis of S2 and Sz is {|+>, |->}.
Below are matrices of spin operators in this basis.

spin matrix z   spin matrix x   spin matrix y   spin matrix s squared

We write S = (ħ/2)σ.  The matrices

Pauli matrices

are the Pauli matrices.

Properties of σx and σy and σz
det(σi) = -1,  Tr{σi} = 0,  σi2 = I,  σxσy = -σyσx = iσz.
In general, σiσj =  iεijkσk + δijI,   [σij] = i2εijkσk.
Therefore 
[Sx,Sy] = iħSz,   [Sy,Sz] = iħSx,    [Sz,Sx] = iħSy,
since S = (ħ/2)σ.

The operator Su is defined through Su = Sxsinθcosφ + Sysinθsinφ + Szcosθ.
The matrix of Su is (Su) = (Sx)sinθcosφ + (Sy)sinθsinφ + (Sz)cosθ.

The eigenvectors of Su are
|+>u = cos(θ/2)exp(-iφ/2)|+> + sin(θ/2)exp(iφ/2)|->,
|->u = -sin(θ/2)exp(-iφ/2)|+> + cos(θ/2)exp(iφ/2)|->.
Therefore |+> = cos(θ/2)|+>u - sin(θ/2)|->u,  |-> = sin(θ/2)|+>u + cos(θ/2)|->u


A system of two spin ½ particles

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 the operators Siz are product operators.
S1z = S1z(1) ⊗ I(2), S2z = I(1) ⊗ S2z(2), etc. 
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 s(s + 1)h2 denotes the eigenvalue of S2 and msh denotes the eigenvalue of Sz.
We have the singlet state
|00> = 2(|+-> - |-+>)
and the triplet states
|11> = |++>,
|10> = 2(|+-> + |-+>),
|1-1> =|-->.