1. 7. Rotations in 3-D Space – The Group SO(3)
7.1 Description of the Group SO(3)
The Angle-and-Axis Parameterization
The Euler Angles
7.2 One Parameter Subgroups, Generators, and the Lie Algebra
7.3 Irreducible Representations of the SO(3) Lie Algebra
7.4 Properties of the Rotational Matrices
7.5 Application to Particle in a Central Potential
Characterization of States
Asymptotic Plane Wave States
Partial Wave Decomposition
Summary
7.6 Transformation Properties of Wave Functions and Operators
7.7 Direct Product Representations and Their Reduction
7.8 Irreducible Tensors and the Wigner-Eckart Theorem

2. 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.
( 0 is fixed )
= orthonormal basis vectors along the Cartesian axes.
gij = metric tensor 
( is Orthogonal )

3. Matrix formulation:
Let  be the matrix with ( i , j )th element = i j = i j .
( Orthogonal ) 
Inversion:

4. 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
can be written as 
Orthogonality condition
can be interpreted as the invariance of the the (2nd) rank (20) tensor ij :
  is invariant under rotation
Definition 7.1b: The Special Orthogonal Group SO(3)
SO(3) = Subgroup of O(3) that leaves  invariant

5. Successive rotations:
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.

6. 7.1.1. The Angle-and-Axis Parameterization
Rotation by angle  about the direction
with
Since
we need only
Group manifold is a sphere of radius π.
 SO(3) is a compact group.
Redundancy:
Group manifold is doubly connected
i.e.,  2 kinds of closed curves 
Theorem 7.1: All R*() belong to the same class

7. The Euler Angles 1. 2. 3.
z' = 3 

8. Mathematica: Rotations.nb
Relation between angle-axis parameters & Euler angles:

9. 7.2. One Parameter Subgroups, Generators, & the Lie Algebra
is an 1-parameter subgroup isomorphic to SO(2)
Lemma:
Proof: 
QED
The 33 matrix Jn transforms like the vector n under rotation.
Using
one gets the basis matrices

10. Theorem 7.2: Vector Generator J1. 2. 
Proof of 1:
Since
it suffices to prove explicitly the special cases
&
This is best done using symbolic softwares like Mathematica.
Alternatively, 
Note : eq(7.2-7) is wrong
Numerically, 
QED

11. Proof of 2: (Tung's version is wrong)
From part 1:
QED
Thus, { Jk | k =1,2,3 } is a basis for the generators of all 1-parameter subgroups of SO(3), i.e.,

12. Theorem 7.3: Lie Algebra so(3) of SO( 3)
{ Jk | k = 1,2,3 } is also the basis of the Lie algebra
Proof:  
QED
A Lie algebra is a vector space V endowed with a Lie bracket 
Jacobi's identity

13. 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 
Jn is conserved 
Every component of the angular moment is conserved in a system with spherical symmetry

14. 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 :
An invariant measure can be defined so that all theorems for finite groups can be adopted
Its IRs are all "finite" dimensional & equivalent to unitary reps
IR  appears in the regular rep n times
Its generators are hermitian operators
SO(3) is compact

15. Representation space for an IR is a minimal invariant space under G.
Strategy for IR construction (simplest version of Cartan's method):
Pick any convenient "standard" vector.
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:
is a Casimir operator, i.e.,
Schur's lemma:
in any IR

16. Convention: Choose eigenvectors of J2 and J3 as basis.
Raising (J+) & lowering (J–) operators are defined as:
Useful identities:
Let | , m  be an normalized eigenvector of J2 & J3 in rep space V:
If V is a minimal invariant subspace, then
on V 
 m
Thus, we can simplify the notation:

17.  
with if
V is finite dimensional   max value j 
so that

18. Also,  min value n 
so that
Hence 
Since
for some positive integer k
we have 
For a given j, the dimension of V is 2j+1 with basis

19. 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
with the following properties:
Proof:
Let 
Condon-Shortley convention
αm is real 

20.  dj() are real & orthogonal
Let U(,,) be the unitary operator on V corresponding to R(,,) SO3.
The j-IR is given by
( Sum over m' only) 
( m in e– i  m is not a tensor index so it's excluded from the summation convention)
where 
Condon-Shortley convention: Dj(J2) is an imaginary anti-symmetric matrix
 dj() are real & orthogonal

21. Example 1: j = 1/2
Basis:
Pauli matrix  

22. Useful properties of the Pauli matrices:
Mathematica: Rotations.nb
where 
Since R(2π) = E, D1/2 is a double-valued rep for SO(3)

23. Example 2: j = 1   
Mathematica: Rotations.nb
Error in eq(7.3-23)

24. Theorem 7.5: IRs of SO( 3)
The IRs of so(3), when applied to SO(3), give rise to
Single-valued representations for integer j.
Double-valued representations for half-integer j.
k = integer
Proof:
Since
where 
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,… )

25. 7.4 Properties of the Rotational Matrices DJ(,,)
Unitarity:
Speciality (Unit Determinant):
wrt basis { | j m  }
Orthogonality of d j() ( Condon-Shortley convention ):
Dj(J2) set to be imaginary & anti-symmetric  Dj(J) are real 
are real & orthogonal
i.e.,

26. Complex Conjugation of Dj ( Condon-Shortley convention ):
Dj(J3) is real 
Dj(J2) is imaginary 
Let
Ex. 7.7
Error in eq(7.4-4)
See: A.R.Edmonds, "Angular momentum in quantum mechanics", p.59 

27. Symmetry Relations of d j() ( Condon-Shortley convention ):
Error in eq(7.4-6). See Edmonds
Relation to Spherical Harmonics (To be derived in Chapter 8):
1) Integer j = l :
2) Arbitary j :
Jacobi Polynomials
3) Orthonormality & completeness : See § 7.7

28. Characters:
All rotations of the same angle  belong to the same class. 
j = 1/2:
j = 1:

29. 7.5. Application to a Particle in a Central Potential
V = V(r)  Spherical symmetry 

30. 7.5.1. Characterization of States
CSCO = { H, J2, J3 }
Eigenstates = { | E, l, m  }
x-rep wave function:
Spherical coordinates:
0 arbitrary 

31. Since this holds for all , we must have

32. 7.5.2. Asymptotic Plane Wave StatesIf
for r  
then (x) ~ plane wave as r  
Linear momentum eigenstates
Let
& 
Relation to angular momentum eigenstates (To be derived in Chapter 8):
Inverse:

33. 7.5.3. Partial Wave Decomposition
Scattering of a particle by V(r):
final state:
Initial state:
Scattering amplitude:
T is the T-matrix. In the Born approximation, T = V.
V = V(r)  T is invariant under rotation, i.e.,  
where

34. 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:

35. 7.6. Transformation Properties of Wave Functions & Operators
Theorem 7.6: Transformation Formula for Wave Functions
Proof:
QED
since detR = 1

36. Example 1: Plane Waves
Example 2: Angular Momentum States 
( See § 8.6 )

37. Extension: Pauli Spinors Basis vectors:
sum over  implied 
This forms a representation for SO(3). See Problem 7.10

38. Definition 7.3: Irreducible Wave Functions & Fields
is an irreducible wave function or field of spin j
if it transforms under rotations as
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

39. Coordinate operators
Theorem 7.7: Transformation Formula for Vector Operators
i, j = 1, 2, 3
Proof:
c.f. 
QED
This also forms a representation of SO(3) on the operator space
Any operator that transforms like X is a vector operator.
E.g.,
Other tensor operators can be similarly defined

40. Field operators
Pauli-spinor field operator
annihilates a particle of spin  at x
| 0  = vacuum
[ (x) is a spin ½ field ]
c.f.  
c.f.

41. Generalization Let transforms under SO(3) as 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

42. 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
is given by
i.e.,
Dj j is single-valued if j + j = integer,
double-valued if j + j = half-integer
Dj j is reducible if neither j nor j = 0.

43. Example: j = j = ½
| m m'  = | + + , | + – , | – + , | – – 
Let
 | a  spans a 1-D subspace invariant under SO(3) .
 D½  ½ is reducible.
To be proved:

44. Theorem 7.8:
Proof: 

45. Reduction of Dj  j ' : 
with 1 state
with 2 states
 
with 2 states
with 1 state

46. Let || J M  be eigenstates of { J2, J3 } 
Linked states have same M.
Only 1 state for M = j + j '  it belongs to J = j + j ' &
Justification:
(Problem 7.8)

47. Other members in the multiplet
can be generated by repeated use of J– . E.g., 
{ || 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 as
we can generate the multiplet corresponding to J = j + j' – 1.
Arbitrary phase factor to be fixed by, say, the Condon-Shortley convention.

48. Dimension of D j  j = ( 2 j+1 ) ( 2 j+1 )
Transformation between | J M  & | m, m' :
Clebsch–Gordan Coefficients:

49. Condon-Shortley convention:
Both { | m, m'  } and { | J M  } are orthonormal.
( Largest M & m )
Other notations for the CGCs:

50. D½  ½ re-visited:
| m, m'  = | + + , | + – , | – + , | – – 
J = 1, 0
( orthogonal to | 1 0  )
CGCs:

51. Appendix V
A square root  is to be understood over every coefficient.

52. Other methods to calculate the CGCs are discussed in books by Edmond, Hamermesh, Rose, ….
Some special values we've calculated:

53. General Properties of the CGCs
Angular Momentum Selection Rule:
unless
and
Orthogonality and Completeness:

54. Symmetry Relations:
Wigner 3-j Symbols:
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.

55. Reduction of a direct product representation ( c.f. Theorem 3.13 )

56. 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
Os is the th spherical component of the tensor.

57. Theorem 7.9: Differential Characterization of Irreducible Spherical tensor
Proof:
For an infinitesimal rotation about the kth axis, 
Using
completes the proof.

58. Examples: 1. 
is an irreducible spherical vector with s = 1 &  = { 1, 0, –1} 2.
This is easily proved using

59. Definition 7.5: Vector Operator – Cartesian Components
1. Operators
are the Cartesian components of a vector if 2.
are the Cartesian components of a nth rank tensor if
Actually, the above can be derived from the more familiar definition of Cartesian tensors in terms of rotations in E3
c.f. Theorems 7.2, 3
using
and

60. 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. or
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 )

61. 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 )
Selection Rules:
unless
&
Branching Ratios: