7. Rotations in 3-D Space – The Group SO(3)
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
Description of the Group SO(3)
7.1.1 The Angle-and-Axis Parameterization
7.1.2 The Euler Angles
One Parameter Subgroups, Generators, and the Lie Algebra
Irreducible Representations of the SO(3) Lie Algebra
Properties of the Rotational Matrices
Application to Particle in a Central Potential
7.5.1 Characterization of States
7.5.2 Asymptotic Plane Wave States
7.5.3 Partial Wave Decomposition
7.5.4 Summary
Transformation Properties of Wave Functions and Operators
Direct Product Representations and Their Reduction
Irreducible Tensors and the Wigner-Eckart Theorem
7.1. Description of the Group SO(3)
Definition 7.1:
The Orthogonal Group O(3)
O(3) = All continuous linear transformations  in E3 which
leave the length of coordinate vectors invariant.
 eˆ
i  1, 2,3  = orthonormal basis vectors along the Cartesian axes.
i
: E  E
3
3
2
 xi x i  gi j x j x i
 x

eˆ i
eˆi   eˆ i  eˆ j  ji
0
x  eˆ i x i   x   eˆi x i  eˆ j  ji x i
x  eˆ i x i
x
( 0 is fixed )
2
 xi x i  gi j x j x i
gi j  j k i l  gk l
x i  i j x j
E3 :
gij = metric tensor
0  0
gi j  i j
 gi j  j k xk i l x l
E3 : i j  j k i l  k l
( is Orthogonal )
Matrix formulation:
j
T
x

g
x
~
x
 i i j  g  row vector
i
x
  ~ x  column vector
xi xi  gi j x j xi ~ xT g x
E3 : xi xi ~ xT x
Let  be the matrix with ( i , j )th element = i j = i j .
i j  j k i l  ik i l  k l
~
ΘT Θ  Θ ΘT  E
det  ΘT Θ   det ΘT det Θ   det Θ 

Inversion:
2
 det E  1
det Θ  1
 1 0 0 
I S   0 1 0 
 0 0 1 


det IS  1
( Orthogonal )
Definition 7.1a: The Special Orthogonal Group SO(3)
SO(3) = Subgroup of O(3) consisting of elements R whose matrix
representation R satisfies det R = +1
= Rotational group in 3-D
Note: Any element with

det Θ  1 can be written as   R I S  I S R
O 3  SO 3  CS
Orthogonality condition
i j k l 
jl
  ik
can be interpreted as the invariance of the the (2nd) rank (20) tensor ij :
i l  j m  k n  l mn   i jk det Θ
R i l R j m R k n  l mn   i jk
  is invariant under rotation
Definition 7.1b: The Special Orthogonal Group SO(3)
SO(3) = Subgroup of O(3) that leaves  invariant


Successive rotations: R2 R1 eˆ i  R 2 eˆ j R 1
j
i
 eˆ k R2k j R1 j i
 eˆ k  R2 R1 
Group multiplication ~ Matrix multiplication
Product of orthogonal matrices = orthogonal matrice  Closure
Ditto for the existence of identity & inverses.
Definition 7.1c: The Special Orthogonal Group SO(3)
SO(3) = All 33 orthogonal matrices with unit determinants
Each element of SO(3) is specified by 3 (continous) parameters.
k
i
7.1.1. The Angle-and-Axis Parameterization
Rotation by angle  about the direction nˆ    ,   :
Rnˆ    R  , , 
0    2
0  
with
Since Rnˆ    Rnˆ     Rnˆ  2   we need only
nˆ      ,   
0   

Group manifold is a sphere of radius π.
 SO(3) is a compact group.
Redundancy:
Rnˆ    Rnˆ  
Group manifold is doubly connected
i.e.,  2 kinds of closed curves
1
ˆ  R nˆ  R mˆ    R x   R  R nˆ   x   R Rnˆ   R R x
m
Rmˆ    R Rnˆ   R1
Theorem 7.1:
All R*() belong to the same class
7.1.2. The Euler Angles
1.
R3  
1,2,3   x, y, z    x, y, z
x
2.
x  R3   x
R y   



 x , y , z    x, y, z
x
x  R y    x
R y     R3   R2    R31  
3.
R z  




 x , y , z   1, 2, 3
x
x  R z   x
Rz    R y    Rz   Ry1   
z' = 3
Rz    R3   R3   R31    R3  
x  R ,  ,  x

R ,  ,    Rz   R y    R3    R y    R3   R3  
R ,  ,    R3   R2    R3  
0   ,   2
0  
 cos
R 3     sin
 0

 sin
cos
0
0
0 
1 
 cos
R 2     0
  sin

 cos  cos  cos   sin  sin 
R  ,  ,     sin  cos  cos   cos  sin 

 sin  cos 

0 sin 
1
0 
0 cos 
0
1
R1     0 cos
 0 sin

 cos  cos  sin   sin  cos 
 sin  cos  sin   cos  cos 
sin  sin 

 sin 
cos 
0
cos  sin  
sin  sin  
cos  
Mathematica: Rotations.nb
Relation between angle-axis parameters & Euler angles:

1
     
2
tan

2
tan  
 
sin
2
cos  2 cos2

2
cos2
 
2
1
7.2. One Parameter Subgroups, Generators, & the Lie Algebra
R
 i J


e


nˆ
0    2
nˆ

is an 1-parameter subgroup isomorphic to SO(2)
 R  SO 3
R J nˆ R1  J R nˆ
Lemma:
Proof:

RRnˆ    R Rnˆ   R1
i  J R nˆ
e
Re
i  J nˆ
R


1
 i  
m0


m 0
 i  
m!
m
 R J nˆ R
m!

1 m
m
R  J nˆ  R 1
m
i  R J nˆ R1
e
QED
The 33 matrix Jn transforms like the vector n under rotation.
Using
R nˆ    0 
 0 0 0
J 1  i  0 0 1 
 0 1 0 


E  i  J nˆ
 0 0 1
J 2  i  0 0 0 
1 0 0 


one gets the basis matrices
 0 1 0
J 3  i  1 0 0 
 0 0 0


J j 
k
m
 i  j k m
Theorem 7.2: Vector Generator J
1.
R J k R1  J m R mk
2.
nˆ  eˆ k n k
m, k  1,2,3
 R  SO 3
J nˆ  J k n k

Proof of 1:
R ,  ,    R3   R2    R3  
Since
R  R2   &
it suffices to prove explicitly the special cases
R  R3  
This is best done using symbolic softwares like Mathematica.
Alternatively,
R i l R j m R k n  l mn   i jk
 pi R p q R i l R j m R k n  l mn   ql R j m R k n  l mn   p i R p q  i jk

R j m R k n  qmn  Ri q  i jk
R
J j 
j
k
m
R n
k
m
 i  j k m
qmn
 Ri 

q
R
j
m
Note : eq(7.2-7) is wrong
i jk
R
A jk  A j k  Aj k  Ajk  Ak j
Numerically,
R Jq R
J   R J 
  J  R
mn
k
n
T
i
q
jk
iq
jk
jk
i
i
q
QED
Proof of 2:
(Tung's version is wrong)
R  ,  , 0  eˆ 3  nˆ   , 
R  ,  , 0


 3  ni
i
 cos  cos  cos   sin  sin 
R  ,  ,     sin  cos  cos   cos  sin 

 sin  cos 

 cos  cos  sin   sin  cos 
 sin  cos  sin   cos  cos 
sin  sin 
cos  sin  
sin  sin  
cos  
From part 1:
J nˆ  R   ,  , 0  J 3 R 1   ,  , 0   J k R   ,  , 0
3
k
 Jk n k
QED
Thus, { Jk | k =1,2,3 } is a basis for the generators of
all 1-parameter subgroups of SO(3), i.e.,
R nˆ    e
 i n k J k
R ,  ,    ei J3 ei  J2 ei J3
Theorem 7.3: Lie Algebra so(3) of SO( 3)
{ Jk | k = 1,2,3 } is also the basis of the Lie algebra
 Jk , Jl   i  k l m J m
Proof:
1
R Jk R  Jm R
m
k
Rl  d  J k R

1
l
 d   J m  Rl  d  
LHS
 E  i d J l   J k  E  i d J l  
RHS
J m  m k  i d  J l 


m
k


m
k
J k  i d  J l J k  J k J l  
J k  d J m l mk 
i  J l J k  J k J l   J m  l mk
J k J l  J l J k   J k , J l   i J m  l mk  i J m  k l m
QED
A Lie algebra is a vector space V endowed with a Lie bracket 
 A, B     B, A 
 A,  B, C    B,  C, A   C,  A, B   0
Jacobi's identity
 A, B, C V
Comments:
• The commutation relations of Jk are equivalent to the group multiplication
rule of R near E.
• Jk determine the local properties of SO(3)
• Global properties are determined by the topology of the group manifold.
E.g.,
Rn(2π) = E, Rn(π) = R–n(π), ….
• It's straightforward to verify that the matrix forms of Jk satisfy the
commutation relations
• The Lie algebra define earlier is indeed an algebra with [ , ] as the
multiplication
• Jk are proportional to components of the angular momentum operator
 H , Rnˆ     0

 H , J nˆ   0

Jn is conserved
 Every component of the angular moment is conserved
in a system with spherical symmetry
7.3. IRs of the SO(3) Lie Algebra so(3)
Local properties of Lie group G are given by those of its Lie algebra G
Generators of G = Basis of G
 Rep's of G are also rep's of G.
The converse is also true provided all global restrictions are observed.
Compact Lie group :
1. An invariant measure can be defined so that
all theorems for finite groups can be adopted
2. Its IRs are all "finite" dimensional & equivalent to
unitary reps
3. IR  appears in the regular rep n times
4. Its generators are hermitian operators
SO(3) is compact
Representation space for an IR is a minimal invariant space under G.
Strategy for IR construction (simplest version of Cartan's method):
1. Pick any convenient "standard" vector.
2. Generate the rest of the irreducible basis by repeated
application of selected generators / elements of G.
Natural choice of basis vectors of representation space
= Eigenvectors of a set of mutually commuting operators
Definition 7.2: Casimir Operator
C is a Casimir operator of a Lie group G if [ C, g ] = 0  g  G
Example: SO(3)
Generators J1, J2, J3 do not commute:
J 2  J12  J 22  J 32
 J i , J j   i  i j k J k
is a Casimir operator, i.e.,
Schur's lemma: J 2   E
in any IR
 J 2 , J k   0
Convention: Choose eigenvectors of J2 and J3 as basis.
Raising (J+) & lowering (J–) operators are defined as:
J   J1  i J 2
Useful identities:
 J3 , J    J 
 J  , J    2J 3
J 2  J 32  J 3  J  J   J 32  J 3  J  J 
J †  J
Let | , m  be an normalized eigenvector of J2 & J3 in rep space V:
J 3 , m  m , m
J 2 , m  m , m
If V is a minimal invariant subspace, then
J2 E
on V 
m  
 m
Thus, we can simplify the notation:
J2 m  m
J3 m  m m
J3 J  m   J  J3  J   m

J k m
  m 1  J m
J m
 mk
mk
with
m  k 1
J3 j
j
J2
  J 32  J 3  J  J   j
j
j
mk 0
if
V is finite dimensional   max value j
so that

J

j
0
 j  j  1 j
 m 1
Also,  min value n
n n
J3 n
J2 n
so that

J n
  J 32  J 3  J  J   n
  j  j  1  n  n  1
Hence
J k
Since
we have
j
 j
j k  j

0
 n  n  1 n
nj
for some positive integer k
k
1
3
5
j


0,
,
1,
,
2,
,

2
2
2
2
For a given j, the dimension of V is 2j+1 with basis

m
m  j, j  1,
  j , j 1 ,
,
,  j  1,  j

 j 1 ,  j

Theorem 7.4: IR of Lie Algebra so(3)
The IRs are characterized by j = 0,1/2, 1, 3/2, 2, …. .
Orthonormal basis for the j-rep is

jm
m  j, j  1,
,  j  1,  j

with the following properties:
J2 j m
 j m j  j  1
J
 j m 1
jm
J3 j m
 jm m
j  j  1  m  m  1
Proof:
J
jm
 j m  1 m
j m J J
jm
 j m J 2  J 3  J 3  1 j m
j m JJ
jm
 m m* j m  1
Let
αm is real
j m JJ j m

m 
J j m
 j  j  1  m  m  1
j m 1
j  j  1  m  m  1
 m m* j m  1
 j m  1 m
j m 1
Condon-Shortley convention

m 
j  j  1  m  m  1
Let U(,,) be the unitary operator on V corresponding to R(,,) SO3.
The j-IR is given by
U  ,  ,   j m
 j m  D  j   ,  ,  
m
( Sum over m' only)
m
R  ,  ,    ei J3 ei  J2 ei J3
 U ,  ,   j m  ei J3 ei  J2 ei J3 j m
 ei J3
j m
j m ei  J2
 j m e  i  m d j   

D j  ,  ,  
m
m
m
m
j m ei  m
e i  m
 e  i  md j   
where
m
m
 ei J3 ei  J2
j m ei  m
( m in e– i  m is not a tensor index so it's
excluded from the summation convention)
d j  
m
m

j m e  i  J 2
jm
e i  m
Condon-Shortley convention: Dj(J2) is an imaginary anti-symmetric matrix
 dj() are real & orthogonal
Example 1:
1/ 2
D
j = 1/2
  J3 
 J3   
  J3 
 J3  

 J3  
Basis:
 
1
2

 0


 J  
1  1   1  1 
  1       1
2  2   2  2 
 J  
1  1   1  1 
  1     1
2  2   2  2 
1/ 2
D
1
 J1    D1/ 2  J    D1/ 2  J   
2

0 

1
 
2
1 1
2 2
1
 
 0
1 1 0 
2  0 1
1
1
1  0 1
 
2  1 0 

 
1
3
2
1 1

2 2
Pauli matrix

 0 1
D1/ 2  J    

 0 0

 0 0
D1/ 2  J    

 1 0

1
1
2
D1/ 2  J  
1/ 2
D
1
 J 2    D1/ 2  J    D1/ 2  J  
2i
1

2
 0
 
1
 0 i   1 
2
i 0 
2


1
σ
2
Useful properties of the Pauli matrices:
i  j   i j  i  i jk  k
k
k
2k
2 k 1




2




1
1
1/ 2





d    e
 

E

i


j E
2




 2k  1!  2 
k 0 

  2k  !  2 



cos

sin



2
2
Mathematica:
 E cos  i  2 sin


2
2
Rotations.nb
 sin  cos  
2
2 

i
  2
2

  i 2 
   i 2
 e  cos  e
2

D1/ 2  ,  ,     

 i 
i


 e 2  sin  e 2
2


R nˆ  2   R R yˆ  2  R1


i 



e 2  sin  e 2 
2



 
i 
  i2
2
e  cos  e 
2


i

where
nˆ  R yˆ
D1/ 2  R nˆ  2    D1/ 2  R  e  i   2 D1/ 2  R 1   D1/ 2  R    E  D1/ 2  R 1   E
Since R(2π) = E, D1/2 is a double-valued rep for SO(3)
Example 2:

j=1
 11 J 3 11

D1  J 3    1 0 J 3 11
 1  1 J 11
3

J 1 0  2 1 1
J 1 1  2 1 0
J 1 0  2 1 1
J 1 1  2 1 0
11 J 3 1 0
10 J 3 1 0
1 1 J3 1 0
 0 1 0
 D1  J    2  0 0 1 


 0 0 0


 0 0 0
 D1  J    2  1 0 0 


 0 1 0


 2 sin 
cos 
2 sin 

11 J 3 1  1   1 0 0 

 
1 0 J3 1 1    0 0 0 
1  1 J 3 1  1   0 0 1 
Mathematica: Rotations.nb
 1  cos 
1
1
d      2 sin 
2
 1  cos 

1 1 , 1 0 , 1 1
1  cos  

 2 sin  

1  cos  

 0 1 0 
i 

D1  J 2  
1
0

1

2 

0 1 0 
D1  J 2 
2
 1 0 1 
1
  0 2 0 
2

 1 0 1 
D1  J 2   D 1  J 2 
3
Error in eq(7.3-23)
Theorem 7.5:
IRs of SO( 3)
The IRs of so(3), when applied to SO(3), give rise to
1. Single-valued representations for integer j.
2. Double-valued representations for half-integer j.
k = integer
Proof:
D j  R3  2 
Since

m
m
 D j ei 2 J 3 
m
m
  mm ei 2 m  mm ei 2  jk    mm    2 j
Rnˆ  2   R R3  2  R1
D j  R 3  2       E
2j
where
nˆ  R zˆ
QED
Comments:
• IRs are obtained for region near E w/o considerations of global properties
• SO(3): Group manifold doubly connected  Double-valued IRs
• SO(2): Group manifold infinitely connected  m–valued IRs ( m=1,2,3,… )
7.4 Properties of the Rotational Matrices DJ(,,)
Unitarity:
 D   ,  ,    D  ,  ,  
j
1
j†
 D j   ,  ,  
ei  J 3 ei  J 2 ei  J 3  e i  J 3 e i  J 2 e i  J 3  E
Speciality (Unit Determinant):
det D j  R nˆ     det D j  R R 3   R 1   det D j  R 3   
 det D  e
j
D j  J 3   diag  j, j  1,
j

e
,  j  1,  j

i  J3


i m 
mj
nˆ  R zˆ
j
  e i m  e i m 
1
m 1
wrt basis { | j m  }
Orthogonality of d j() ( Condon-Shortley convention ):
Dj(J2) set to be imaginary & anti-symmetric  Dj(J) are real

i.e.,
d
j
   D
j
 e
i  J 2
 are real & orthogonal
 d      d     d   
j 1
jT
j
1
D 1/ 2  J 2    2
2
Complex Conjugation of Dj ( Condon-Shortley convention ):
D j *  R 3     D j *  e i  J 3   D j  e i  J 3   D j  R 3    
Dj(J3) is real 
 D j  R 2   R 3   R 2    
R2   zˆ  zˆ
Dj(J2) is imaginary 
D j *  R 2     D j *  e i  J 2   D j  e i  J 2   D j  R 2   
Rnˆ  a  Rnˆ b  Rnˆ b Rnˆ  a 
Let
Y  D  R 2   
j
Error in eq(7.4-4)
j
 D j  R 2   R 2   R 2    
Y 

j m
m
  m m   
j m
Ex. 7.7
See: A.R.Edmonds, "Angular momentum in quantum mechanics", p.59
D j  ,  ,    D j  R 3   R 2    R 3     D j  R 3    D j  R 2     D j  R 3   

D j *  ,  ,    Y j D j  ,  ,   Y j 
1
Symmetry Relations of d j() ( Condon-Shortley convention ):
d j  
m
m
 D j e i  J 2 
d
j
 
 m
m
 d j   

m
m
m  m
 d j    
m
d  
j

m
m
m
 Y
j m

Error in eq(7.4-6).
See Edmonds

j m
m
  m m   
j m
Relation to Spherical Harmonics (To be derived in Chapter 8):
Y l m  ,   
1) Integer j = l :
Pl m  cos      
 l  m ! d l  m
 0
 l  m !
m
2l  1 l *
m
D  , ,0 0
4
Pl  cos    Pl 0  cos    d l  
2) Arbitary j :
d j  
m
 Pl  m m , m  m   cos  

m

Jacobi Polynomials
3) Orthonormality & completeness : See § 7.7
0
0
Characters:
All rotations of the same angle  belong to the same class.

   
j

mj

j = 1/2:
j

e
1/ 2
D  R3   
j
i  j 1/ 2  
sin 
sin
j = 1:


m
i  j 1/ 2  
e
ei  / 2  e  i  / 2
  
m
j

e
i m 
mj
e i j  e 

1  ei 
1

sin  j   
2


 2 cos
sin


2
2
2


3
sin cos  cos sin
sin 
2
2
2

 1   


sin
sin
2
2
 2cos  1
i j 1 
 2 cos2

2
 cos
7.5. Application to a Particle in a Central Potential
V = V(r)  Spherical symmetry
 R  SO 3
 H , U  R    0

H,
Ji   0
 i  1, 2,3
7.5.1. Characterization of States
Eigenstates = { | E, l, m  }
CSCO = { H, J2, J3 }
H E, l, m  E, l, m E
J 2 E, l, m  E, l, m l l  1
l  0, 1, 2,
J 3 E, l, m  E, l, m m
m  l ,  l  1,
, l  1, l
 El m  x   x E, l, m
x-rep wave function:
Spherical coordinates:
x  r,  , 
 U ,  , 0 r zˆ
ei  J3 r zˆ  r zˆ
 El m  r,  ,    r,  ,  E, l, m
 r zˆ
 ei  J3 ei  J2 r, 0, 0
J 3 r zˆ  0

 r zˆ U †  ,  , 0 E, l, m
E , l , m  D l †   ,  , 0 
m
m
0 arbitrary
r zˆ
E, l, m  r zˆ ei  J3 E, l, m
E, l, m ei  m
 r zˆ
Since this holds for all , we must have
r zˆ
E, l, m  m 0 r zˆ
E, l , 0
 m 0  El  r 

 E l m  r,  ,    r zˆ
E, l , m  Dl†   ,  , 0 
  El  r   D

l 
 ,  , 0 
  El  r  Yl m  ,  
 El r  El r
4
2l  1
 El  r   r zˆ E, l, 0
m

0
*
m
m
 El r
  E l  r   D
†
l
 ,  , 0 
4
Y l m  ,  
2l  1
0
m
7.5.2. Asymptotic Plane Wave States
V r 
If
Let
1
r
for r  
P p p p
then (x) ~ plane wave as r  
p  p pˆ  ,  
&
Linear momentum
eigenstates
p  p,  ,   U ,  , 0 p zˆ

H plane wave
P2
p 
p
2m
p2
E p 
p
2m
Relation to angular momentum eigenstates (To be derived in Chapter 8):
p, l , m 
2l  1
4
  d
Inverse:

2
0
d  d cos  p,  ,   D

1
1
p,  ,  Y l m  ,  

p,  ,   
l

l  0 m  l
p, l , m Y l*m  ,  
l
 ,  , 0 
m
0


*
d   d d cos
7.5.3. Partial Wave Decomposition
Scattering of a particle by V(r):
pi  p zˆ   p, i  0, i  arbitrary
Initial state:
p f T pi
Scattering amplitude:
final state: p f   p,  , 


p,  ,  T
p zˆ
T is the T-matrix. In the Born approximation, T = V.
 T , J 2    T , J 3   0
V = V(r)  T is invariant under rotation, i.e.,
p, l, m T p, l, m   ll  mm Tl  p


p,  ,   
l

p, l , m Y
l  0 m  l

p f T pi 

l m l
*
lm
 ,  
Yl m  ,   p, l , m T
  Yl 0  ,   p, l , 0 T p, l , 0 Yl *0  0,0
l
where
Yl 0  ,   
2l  1
Pl  cos  
4

p zˆ   p, l , 0 Y l*0  0, 0
l 0
p, l , 0 Yl *0  0,0 

l
2l  1
Tl  E  Pl  cos  
4
Tl  E   p, l, 0 T p, l, 0

7.5.4. Summary
Group theoretical technique:
• Separates kinematic ( symmetry related ) & dynamic effects.
• For problems with spherical symmetry,
angular part ~ symmetry
radial part ~ dynamics
Computational tips:

l

lm
 d
lm I
l  0 m  l
  l m  Y l m  ,   
r,  , 

*
2l  1  l
m

D  ,  , 0 0


4
E l m  r zˆ
E, l, 0
,  l m
 I
zˆ l m  Y l 0  ,    m 0
7.6. Transformation Properties of Wave Functions & Operators
x  R x

   d 3x x
 '  U  R 
x i  R i j x j
x 
U  R x  R x  x
R  SO 3
  d 3x x  x
  d 3x x  ' x
Theorem 7.6: Transformation Formula for Wave Functions
 '  x     R 1 x 
 '  x    x
Proof:
 '  U  R    d 3 x U  R x   x 
  d 3 x x   x 
  d 3 x x   R 1x   d 3 x x   R 1x 
  d 3x x  ' x
QED
3
3
d
x

d

 x
since detR = 1
Example 1:
 p x  x p
 ei px
 p  x   x  p
 x U  R  p
  p  R 1 x   e i p  R
Example 2:
p  p
Plane Waves
1
x
e
i p RT x
 x U  R p
 e i  R p  x
Elm
 '  U  R E l m
 r zˆ
,  l m
E, l , 0
 E l m D  l   R 
  p  x 
m
m
m
   R 1 x    E l  r  Y l m  R 1 xˆ 
Y l m  R 1 xˆ   Y l m  xˆ  D l   R 
m
m
p  R p
  E l  r  Yl m  xˆ 
m
l 
 '  x  r,  ,   '   E l  r  Y l m  xˆ  D  R 

Rp
  Elm
Angular Momentum States
  x  r,  , 
 x
( See § 8.6 )
Extension:
Pauli Spinors
U  R  x, 
 R x,  D1/ 2  R 
   d 3 x x, 
 '  U  R 
x, 
x, 
Basis vectors:


  d 3 x x,  
  d 3 x R x,  D1/ 2  R   





x
x
1
R
 x
  d 3 x x,   '   x 

'
 x   D  R      R 1 x 
1/ 2
1
2

  d 3 x x,  D1/ 2  R   

 

This forms a representation for SO(3).
See Problem 7.10
sum over 
implied
Definition 7.3: Irreducible Wave Functions & Fields

 m  x  m   j,
, j

is an irreducible wave function or field of spin j
if it transforms under rotations as
 ' m  x   D j  R  n  n  R 1 x 
m
Examples:
Spin 1 ( vector ) fields: E, B, v.
Spin ½ fields: Pauli spinors.
Direct sum of two spin ½ fields: Dirac spinors
Spin 2 ( tensor ) fields: Stress tensor
X x  x x
Coordinate operators
Theorem 7.7:
X j x  x xj
Transformation Formula for Vector Operators
U  R X i U  R  X j R ji
1
i, j = 1, 2, 3
Proof:
U  R X i x  U  R X i U  R U  R
1
 U  R x x  x x
i

i
U  R X U  R   R
i
1
R 
1 i
 x

1 i
j
X
j
 U  R X i U  R
x
j
x
 X Rj
j
 x
j
i
R 
1 i
j
X
QED
1
x
j
c.f.
x i  R i j x j
This also forms a representation of SO(3) on the operator space
Any operator that transforms like X is a vector operator.
E.g.,
U  R  Pi U  R   Pj R j i
1
Other tensor operators can be similarly defined
Field operators
Pauli-spinor field operator   x  annihilates a particle of spin  at x
0   x       x 
| 0  = vacuum
 0 U  R  U  R    x  U  R  U  R  
1
1
 0 U  R    x  U  R 
D
1/ 2
1 
 R 
 D1/ 2  R 1 


'   R x


[ (x) is a spin ½ field ]
0   R x   '

1 
1
D
1 
 R 
1/ 2 

U  R 0  0
'
U  R     x U  R   D1/ 2  R 
1/ 2

1
U  R    '
D
 R




 D
  R x 
1/ 2
 R 

 *
U  R      x  U  R   D1/ 2  R      R x 
1
 '   x   D1/ 2  R      R 1 x 

U 1  U 

U  R    x  U  R      R x  D1/ 2  R 
1
c.f.


c.f.
1
U  R X i U  R  X j R ji
Generalization
Let
A
m
U  R A
m
x
m  1, 2, ..., N
 x U  R 
1

 D  R 1 
transforms under SO(3) as
m
n
An  R x 
D(R) is N-D
If D is an IR equivalent to j = s, then A is a spin–s field.
Examples:
• E(x), B(x), A(x) are spin-1 fields
• Dirac spinors: D = D½  D½
7.7. Direct Product Representations and their Reduction
Let Dj & Dj be IRs of SO(3) on V & V, with basis | j m  & | j m  , resp.
The direct product rep Dj j on VV, wrt basis
j m  j m  m, m
is given by
U  R  m, m  U  R  j m


j n D
j
  U  R 
 R n m
 n, n D j  R 
j j'
 R
i.e., D
Dj j is
n , n
m , m
 D j  R
n
n
m
m
 
j  n D 

D j '  R
D j '  R

j  m
n
m
 R n m 

 n, n D j  j '  R 
n
m
single-valued if j + j = integer,
double-valued if j + j = half-integer
Dj j is reducible if neither j nor j = 0.
j '
n , n
m , m
D j  j '  R  D j  R   D j '  R 
| m m'  = | + + , | + – , | – + , | – – 
Example: j = j = ½
a    
Let


 ,   D  R D  R  D  R D  R 
 ,   D  R D  R  D  R D  R 
 ,   D  R D  R  D  R D  R 
 ,   D  R D  R  D  R D  R 
,   ,    D  R D  R  D  R D  R 
U  R a  n, n
D 1/ 2  R
n

D 1/ 2  R

1/ 2
n

1/ 2





 a det D 1/ 2  R 
1/ 2
1/ 2



1/ 2



1/ 2


1/ 2


1/ 2


1/ 2

1/ 2


1/ 2



1/ 2
n

1/ 2


1/ 2

D 1/ 2  R


1/ 2
n

1/ 2


1/ 2

 D 1/ 2  R

1/ 2


1/ 2



1/ 2



 a
 | a  spans a 1-D subspace invariant under SO(3) .
 D½  ½ is reducible.
To be proved:
D1/ 2  1/ 2  D 0  D 1
Theorem 7.8:
J n j  j '  J n j  E j '  E j  J nj '  J n j  J n j '
Proof:
D j  R n  d    D j '  R n  d    D j  j '  R n  d  
LHS   E j  i d J n j 
   E j '  i d J nj ' 
 E j  E j '  i d  J nj  E j '  E j  J nj '  
RHS  E j  j '  i d J nj  j ' 

J n j  j '  J n j  E j '  E j  J nj '

Reduction of Dj  j ' :
J 3 m, m    J 3 j  E j '  E j  J 3 j '  m  m 
 J 3 j m  E j ' m  E j m  J 3 j ' m
 m m  m
 m  m m
  m  m m, m
 j  m  j,  j '  m  j '
  m  m m  m
 M m, m

max M  j  j '
with 1 state
J 3 j, j '   j  j '  j, j '
M  j  j ' 1
with 2 states
J 3 j  1, j '   j  j ' 1 j  1, j '
J 3 j, j ' 1   j  j ' 1 j, j ' 1
M   j  j ' 1


with 2 states
J 3  j  1,  j '    j  j ' 1  j  1,  j '
J 3  j,  j ' 1    j  j ' 1  j,  j ' 1
min M   j  j '
with 1 state
J 3  j,  j '    j  j '   j ,  j '
Let || J M  be eigenstates of { J2, J3 } 
J 2 J, M
 J  J  1 J , M
J3 J , M
 M J, M
Linked states have same M.
Only 1 state for M = j + j '  it belongs to J = j + j ' &
j  j ', j  j '
 j, j '
Justification:
J3
j  j ', j  j '
  J 3j  J 3j '  j, j '
  j  j '
J
2
j  j ', j  j '
 J  J
j
  j  j '  j, j '
j  j ', j  j '

j' 2
j, j '
  j  j '  j  j ' 1
(Problem 7.8)
j  j ', j  j '
Other members in the multiplet

j  j' , M
M  j  j ', j  j ' 1,
,  j  j ' 1,  j  j '

can be generated by repeated use of J– . E.g.,
j  j ', j  j '
J

  J j  J j '  j, j '

2  j  j '

j  j ', j  j ' 1
2j

J  J  1  J  J  1  2J
j  j ', j  j ' 1
j  1, j ' 
j
j  j'
2 j ' j, j ' 1
j  1, j ' 
j'
j  j'
j , j ' 1
{ || j+j', M  } thus generated spans an [ 2(j+j')+1 ]–D invariant
subspace corresponding to J = j + j'.
(Problem 7.8)
Using a linear combination of
that is orthogonal to
j  1, j '
j  j ', j  j ' 1
&
j, j ' 1
as
j  j ' 1, j  j ' 1
we can generate the multiplet corresponding to J = j + j' – 1.
Arbitrary phase factor to be fixed by, say, the Condon-Shortley convention.
Dimension of D j  j = ( 2 j+1 ) ( 2 j+1 )
j  j'
  2J  1   j  j '
J  j  j'
j  j '   j  j ' 1  j  j '    j  j ' 1  j  j ' 
  j  j ' 1  j  j '   j  j ' 1  j  j ' 
  j  j ' 1   j  j '    2 j  1 2 j ' 1
2

D
j  j'

2
j  j'

DJ
J  j j '
Transformation between | J M  & | m, m' :
J M  m, m
m, m J M  m, m
m, m  J M
JM
m, m
 J M
m m  j j '  J M
J M  j j '  m m
Clebsch–Gordan Coefficients:
m m  j j '  J M
 J M  j j '  m m
*
M  m  m
Condon-Shortley convention:
Both { | m, m'  } and { | J M  } are orthonormal.
m m  j j '  J M
 J M  j j '  m m  real
j, J  j  j j '  J J
0
 j, j ', J
( Largest M & m )
Other notations for the CGCs:
J M  j j '  m m  J M
j j ', m m
 J M j m, j ' m
 C  J M ; j m, j ' m   C  J j j ' ; M m m  
D½  ½ re-visited:
| m, m'  = | + + , | + – , | – + , | – – 
11   
10 
1
2
J  11  2 1 0

  

J = 1, 0
  J 1/ 2  J 1/ 2   
J  1 0  2 1,  1

1
2
   

  
1 1   
00 
1
2

  

( orthogonal to | 1 0  )
CGCs:
1 1 1 1
11 

2 2 2 2
1 1 1 1
10 
 
2
2

2 2
1
1 1 1 1
1 1 
 
2 2 2 2
1
1 1 1 1
00
 
2 2 2 2
1 1 1 1
 10 

2
2

 22
1

2
1 1 1 1
 00

2 2 2 2

1
2

Appendix V
A square root  is to be understood over every coefficient.
1 1 1 1
00
 
2 2 2 2
1 1 1 1
 00

2 2 2 2

1
2
Other methods to calculate the CGCs are discussed in books by
Edmond, Hamermesh, Rose, ….
Some special values we've calculated:
j j '  j j '  j  j ', j  j '  1
j  1, j '  j j '  j  j ', j  j ' 1 
j, j ' 1  j j '  j  j ', j  j ' 1 
j
j  j'
j'
j  j'
General Properties of the CGCs
Angular Momentum Selection Rule:
m m  j j '  J M
unless
0
m  m  M
and
j  j'  J  j  j'
Orthogonality and Completeness:
J M  j j '  m m
m m  j j '  J M
m m  j j '  J  M    J J   M M 
J M  j j '  n n    m n  m n 
Symmetry Relations:
m m  j j '  J M
  
j j 'J
  
j j 'J
m m  j ' j  J M
 m,  m  j j '  J ,  M
  
jJ  m '
M ,  m  J j '  j m
Wigner 3-j Symbols:
J  
 j j'
 m m  M  
2 j 1


j j 'M
m m  j j '  J M
is invariant under:
• Cyclic permutation of the columns.
• Change sign of 2nd row & multiply by (–) j+j'+J
• Transpose 2 columns & multiply by (–) j+j'+J
See Edmond / Hamermesh / Messiah for proof.
2J  1
2 j 1
Reduction of a direct product representation ( c.f. Theorem 3.13 )
U  R n, n  U  R J N
m, m  D j  R 
D j  R
D j  R
m
m
j'
D
 R
n
j'
D
 R
n
 J J  D J  R
m
M
M
m
n
j'
D
 R
n
m
n
JN
m
n
n, n
 J M D J R
 m, m J M
 m m  j j '  J M
M
N
D J  R
M
D J  R
 J M  j j '  m m D j  R 
m
n
N
J N
n, n
J N
n, n
M
N
D j ' R
J N  j ' j  n n
m
n
n n  j ' j  J  M 
7.8. Irreducible Tensors & the Wigner-Eckart Theorem
Definition 7.4:
Irreducible Spherical tensor
Operators { Os |  = –s, …, s } form an irreducible spherical tensor of
angular momentum s wrt SO(3) if
U  R  Os U  R   Os D s  R 
1


Os is the th spherical component of the tensor.
 R  SO 3
Theorem 7.9:
Differential Characterization of Irreducible Spherical tensor
 J 2 , Os   s  s  1 Os
 J  , Os  
 J 3 , Os    Os
s  s  1      1 Os  1
U  R  Os U  R   Os D s  R 
1
Proof:


For an infinitesimal rotation about the kth axis,
LHS   E  i d J k 
 Os  E  i d J k  

RHS  Os    i d D s  J k 
 J k , Os   Os D s  J k 

D s J3
Using
D J 
s



     1





 Os  i d Os D s  J k 





    
s  s  1      1
completes the proof.
 Os  i d  J k , Os  
J 2  J  J  J 3  J 3 1
Examples:
 R  SO  3
1.
 O s , U  R    0
2.
1
1

 is an irreducible spherical vector with

J
,
J
,
J


3

2
2

 s = 1 &  = { 1, 0, –1}
This is easily proved using
 J 3 , O    O
s
 J  , Os  
s

 J3 , J    J 

 J  
 J 
   

 J3 , 
2
2






s0
 J  , J    2J 3
 J3 , J3   0
s  s  1      1 Os  1
J 

J
,

0
 

2 

J 

 J ,  2   2 J3



 J 
J , J3   2    
2


 J 
J , J3   2   
 2
J 

 J , 2   2 J3


J 

J
,
0
 

2 

Definition 7.5:
Vector Operator – Cartesian Components
1. Operators
A
2.
T
l  1, 2, 3
l
 J k , Al   i 
l1
ln
l j  1, 2, 3

klm
 are the Cartesian components of a vector if
Am
are the Cartesian components of a nth rank tensor if

 J k , Tl
1

k l1 m


i

Tm l 2
ln 
ln


k ln m
Tl 1
l n 1 m

Actually, the above can be derived from the more familiar definition of
Cartesian tensors in terms of rotations in E3
R k   Al R k1    Am R k  
R k   Tl1
using
ln
R k1    Tl1
Rk    e i  Jk
ln
m
c.f. Theorems 7.2, 3
l
R k  
and
m1
l1
R k  
 Ji 
j
k
mn
ln
 i  i j k
Examples:
• { Jk } are Cartesian components of a vector operator (Theorems 7.2)
• Ditto { Pk } .
• A 2nd rank ( Cartesian) tensor Tj k transforms under rotation according
to the D11 rep.

It is reducible.
D11  D 0  D1  D 2
or
D 11  D 0  D 1  D 2
Properties of a 2nd Rank Cartesian Tensor:
•
Its trace is invariant under SO(3); it transforms as D0.
•
The 3 independent components of its anti-symmetric part transforms
like a spherical vector ( as D1 ) under SO(3).
•
The 5 independent components of its traceless symmetric part
transforms like a spherical tensor of s = 2 ( as D2 ) under SO(3).
Higher rank Cartesian tensors can be similarly reduced ( Chap 8 )
A physical system admits a symmetry group

Operators belonging to the same IR are related

Observables must be irreducible tensors

Matrix elements of { Os } satisfy the Wigner-Eckart theorem ( § 4.3 )
j ' m Os
jm 
j ' m  s, j   m
j'
Os
j
Selection Rules:
j ' m Os
jm 0
unless
j  s  j'  j  s
&
Branching Ratios:
j ' m Os
j ' n Os
jm

jn
j ' m  s , j   m
j ' n  s, j   m
m    m