7. Rotations in 3-D Space – The Group SO(3)

7.1 Description of the Group SO(3) 7.1.1 The Angle-and-Axis Parameterization 7.1.2 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

7.5.1 Characterization of States 7.5.2 Asymptotic Plane Wave States 7.5.3 Partial Wave Decomposition 7.5.4 Summary

7.6 Transformation Properties of Wave Functions and Operators 7.7 Direct Product Representations and Their Reduction7.8 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 3 which leave the length of coordinate vectors invariant.

( 0 is fixed )

= orthonormal basis vectors along the Cartesian axes. ˆ 1,2,3i i e

3 3: E E

ˆ ˆi ii ix x x e x e x ˆ i

ie x ˆ j ij ie x i i j

jx x

2 i j ii i jx x g x x x

j k i li j k lg x x

ˆ ˆ ˆ ˆ ji i i j i e e e e 0 0 0

2 iix x x

j ii jg x x

3 : i j i jg E

j ii j k l k lg g 3 :

j ii j k l k l E

gij = metric tensor

( is Orthogonal )

det det detT TΘ Θ Θ Θ 2det Θ 1

det 1Θ

1 0 0

0 1 0

0 0 1S

IInversion: det 1S I

~ix column vectorx ~j Ti i jx g x row vector x g

det E

( Orthogonal )

Let be the matrix with ( i , j )th element = i j = i j .

Matrix formulation:

~j i i T Ti j k l i k l k l Θ Θ ΘΘ E

~i j i Ti i jx x g x x x g x

3 : ~i Tix x x xE

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 1Θ can be written as S SR I I R

3 3 SO SO C

Orthogonality condition

can be interpreted as the invariance of the the (2nd) rank (20) tensor

ij :

i k j l i kj l

deti j k lmn i j kl m n Θ

i j k lmn i j kl m nR R R is invariant under rotation

Definition 7.1b: The Special Orthogonal Group SO(3)

SO(3) = Subgroup of O(3) that leaves invariant

Definition 7.1c: The Special Orthogonal Group SO(3)

SO(3) = All 33 orthogonal matrices with unit determinants

Successive rotations:

2 1 2 1ˆ ˆ ji j iR R R Re e 2 1ˆ k j

k j iR Re 2 1ˆ k

k iR Re

Group multiplication ~ Matrix multiplication

Product of orthogonal matrices = orthogonal matrice Closure

Ditto for the existence of identity & inverses.

Each element of SO(3) is specified by 3 (continous) parameters.

7.1.1. The Angle-and-Axis Parameterization

Rotation by angle about the direction

ˆ , ,R R n

ˆ , : n

0 with 0 2

Since ˆ ˆR R n n0 we need only

ˆ , n

Group manifold is a sphere of radius π. SO(3) is a compact group.

ˆ 2R n

Redundancy: ˆ ˆR R n n

Group manifold is doubly connected

i.e., 2 kinds of closed curves

1ˆ ˆR R R R m n

ˆ ˆRm n ˆ ˆR R R R m nx x 1ˆR R R R n x

Theorem 7.1: All R*() belong to the same class

7.1.2. The Euler Angles

1.

31,2,3 , , , ,R

x y z x y z

3R x x x

2.

, , , ,yx y zR

x y z

yR x x x

3. zR x x x

1z y z yR R R R

13 2 3yR R R R

13 3 3zR R R R

3R

3, , z yR R R R

, ,R x x

3 3yR R R

3 2 3, ,R R R R

, 3, ,2,1zx y zR

0 , 2 0

z' = 3

3

cos sin 0

sin cos 0

0 0 1

R

2

cos 0 sin

0 1 0

sin 0 cos

R

1

1 0 0

0 cos sin

0 sin cos

R

Relation between angle-axis parameters & Euler angles:

1

2

tan2tan

sin2

2 2cos 2cos cos 12 2

cos cos cos sin sin cos cos sin sin cos cos sin

, , sin cos cos cos sin sin cos sin cos cos sin sin

sin cos sin sin cos

R

Mathematica: Rotations.nb

7.2. One Parameter Subgroups, Generators, & the Lie Algebra

ˆˆ 0 2i JR e nn is an 1-parameter subgroup isomorphic to SO(2)

Lemma: 1ˆ ˆRR J R J n n 3R SO

Proof: 1ˆ ˆRR R R R n n

ˆ ˆ 1Ri J i Je R e R n n

1ˆi R J Re n QED

1ˆ

0 !

mm

m

iR J R

m

n

1ˆ0 !

mm

m

iR J R

m

n

The 33 matrix Jn transforms like the vector n under rotation.

ˆ ˆ0R E i J n nUsing one gets the basis matrices

1

0 0 0

0 0 1

0 1 0

J i

2

0 0 1

0 0 0

1 0 0

J i

3

0 1 0

1 0 0

0 0 0

J i

kj j kmmJ i

Theorem 7.2: Vector Generator J

1. 1 mk m kR J R J R , 1,2,3m k

2. ˆk

n kJ J nˆ ˆ kk nn e

Proof of 1:

3R SO

Since 3 2 3, ,R R R R

it suffices to prove explicitly the special cases 2R R & 3R R

This is best done using symbolic softwares like Mathematica.

Alternatively, i j k lmn i j kl m nR R R

p i j k lmn j k lmnpi q l m n ql m nR R R R R R p i j k

p i qR

j k mn i j km n q iqR R R

kj j kmmJ i j kmnj k i

m n q iqR R J R J

j k j kT iq i qR J R J R QED

Note : eq(7.2-7) is wrongj k j k

k j j k k jA A A A A Numerically,j k qmn q i j km n iR R R

Proof of 2: (Tung's version is wrong)

3ˆ ˆ, , 0 ,R e n

3

, , 0i iR n

From part 1: 1

ˆ 3 3, , 0 , , 0 , , 0

k

kJ R J R J R n

kkJ n

QED

Thus, { Jk | k =1,2,3 } is a basis for the generators of all 1-parameter subgroups of SO(3), i.e.,

ˆ

kki n JR e n 3 32, , i J i Ji JR e e e

cos cos cos sin sin cos cos sin sin cos cos sin

, , sin cos cos cos sin sin cos sin cos cos sin sin

sin cos sin sin cos

R

Theorem 7.3: Lie Algebra so(3) of SO( 3)

{ Jk | k = 1,2,3 } is also the basis of the Lie algebra , mk l k lmJ J i J

Proof:

1 mk m kR J R J R 1 m

l k l m l kR d J R d J R d

l k lLHS E i d J J E i d J k l k k lJ i d J J J J

mmm k l k

RHS J i d J mk m l kJ d J

ml k k l m l ki J J J J J

, mk l l k k l lmkJ J J J J J i J m

k lmi J QED

A Lie algebra is a vector space V endowed with a Lie bracket

, ,A B B A

, , , , , , 0A B C B C A C A B

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

ˆ, 0H R n ˆ, 0H J n 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

Generators of G = Basis of

Rep's of G are also rep's of .

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

Example: SO(3)

Generators J1, J2, J3 do not commute:

Definition 7.2: Casimir Operator

C is a Casimir operator of a Lie group G if [ C, g ] = 0 g G

2 2 2 21 2 3J J J J is a Casimir operator, i.e., 2 , 0kJ J

,i j i j k kJ J i J

Schur's lemma: 2J E in any IR

Convention: Choose eigenvectors of J2 and J3 as basis.

Raising (J+) & lowering (J–) operators are defined as: 1 2J J i J

Useful identities: 3 ,J J J 3, 2J J J

†J J 2 2

3 3J J J J J 23 3J J J J

Let | , m be an normalized eigenvector of J2 & J3 in rep space V:

3 , ,J m m m 2 , ,mJ m m

2J E on V m m

If V is a minimal invariant subspace, then

Thus, we can simplify the notation:

2J m m 3J m m m

3 3J J m J J J m 1m J m 1J m m

kJ m m k

V is finite dimensional max value j

3J j j j 0J j

so that 2 23 3J j J J J J j 1j j j

1m k m k if 0m k with

Also, min value n

3J n n n 0J n

so that 2 23 3J n J J J J n 1n n n

Hence 1 1j j n n n j

Since kJ j j for some positive integer k

we have j k j 2

kj

1 3 50, , 1, , 2, ,2 2 2

For a given j, the dimension of V is 2j+1 with basis

, 1, , 1,m m j j j j

, 1 , , 1 ,j j j 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 , 1, , 1,j m m j j j j

with the following properties:

2 1J j m j m j j 3J j m j m m

1 1 1J j m j m j j m m

Proof:

Let 1 mJ j m j m 1 mJ j m j m

23 3 1j m J J j m j m J J J j m 1 1j j m m

* 1 1m mj m J J j m j m j m

1 1m j j m m

* 1 1m mj m J J j m j m j m 1 1m j j m m

αm is real Condon-Shortley convention

Let U(,,) be the unitary operator on V corresponding to R(,,) SO3.

The j-IR is given by

, , , ,mj

mU j m j m D

3 32, , i J i Ji JR e e e

3 32, , i J i Ji JU j m e e e j m 3 2i J i J i me e j m e

3 2i J i J i me j m j m e j m e

mi m j i m

mj m e d e 2

m i Jj

md j m e j m where

, ,m mj i m j i m

m mD e d e

( m in e– i m is not a tensor index so it's excluded from the summation convention)

Condon-Shortley convention: Dj(J2) is an imaginary anti-symmetric matrix

dj() are real & orthogonal

( Sum over m' only)

Example 1: j = 1/2 Basis:

11 1

02 2

3 31/ 23

3 3

J JD J

J J

10

21

02

1 01

0 12

1 1 1 11 1

2 2 2 2J

1/ 2 0 1

0 0D J

1

1 1 1 11 1

2 2 2 2J

1/ 2 0 0

1 0D J

1

3

1

2

1/ 2 1/ 2 1/ 21

1

2D J D J D J

0 11

1 02

1

1

2

01

02

i

i

2

1

2

Pauli matrix

1/ 2 1/ 2 1/ 22

1

2D J D J D J

i

01 1

12 2

1/ 2 1

2D J σ

Useful properties of the Pauli matrices: i j i j i j k ki

21/ 2 2

i

d e

2 2 1

20

1 1

2 ! 2 2 1 ! 2

k kk k

k

E ik k

2

j E

2cos sin2 2

E i

cos sin2 2

sin cos2 2

2 2 2 2

1/ 2

2 2 2 2

cos sin2 2

, ,

sin cos2 2

i i i i

i i i i

e e e e

D

e e e e

1ˆ ˆ2 2R R R R n y where ˆ ˆRn y

21/ 2 1/ 2 1/ 2 1ˆ 2

iD R D R e D R n 1/ 2 1/ 2 1D R E D R E

Since R(2π) = E, D1/2 is a double-valued rep for SO(3)

Mathematica: Rotations.nb

Example 2: j = 1 1 1 , 1 0 , 1 1

3 3 3

13 3 3 3

3 3 3

11 11 11 1 0 11 1 1

10 11 10 10 10 1 1

1 1 11 1 1 1 0 1 1 1 1

J J J

D J J J J

J J J

1 0 0

0 0 0

0 0 1

1

0 1 0

2 0 0 1

0 0 0

D J

1 0 2 1 1J

1 1 2 1 0J

1

0 0 0

2 1 0 0

0 1 0

D J

1 0 2 1 1J

1 1 2 1 0J

12

0 1 0

1 0 120 1 0

iD J

212

1 0 110 2 0

21 0 1

D J

31 12 2D J D J

Mathematica: Rotations.nb

1

1 cos 2 sin 1 cos1

2 sin cos 2 sin21 cos 2 sin 1 cos

d

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.

Proof:

323 2

mm i Jj j

m mD R D e

2i m

m m e

2i j km m e

2 jm m

Since 1ˆ 32 2R R R R n where ˆ ˆRn z

23 2jjD R E 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,… )

k = integer

7.4 Properties of the Rotational Matrices DJ(,,)

Unitarity: 1 †, , , ,j jD D

, ,jD

3 3 3 32 2i J i J i J i Ji J i Je e e e e e E

Speciality (Unit Determinant):

1ˆ 3det detj jD R D R R R n ˆ ˆRn z 3det jD R

1j

i m

m j

e

1

ji m i m

m

e e

3det i JjD e

3 , 1, , 1,jD J diag j j j j wrt basis { | j m }

Orthogonality of d j() ( Condon-Shortley convention ):

Dj(J2) set to be imaginary & anti-symmetric Dj(J) are real

2i Jj jd D e are real & orthogonal

i.e., 1j j T jd d d

1/ 22 2

1

2D J

Complex Conjugation of Dj ( Condon-Shortley convention ):

3* *3

i Jj jD R D e Dj(J3) is real 3i JjD e 3jD R

2 3 2jD R R R 2 ˆ ˆR z z

2* *2

i Jj jD R D e Dj(J2) is imaginary 2i JjD e 2jD R

2 2 2jD R R R ˆ ˆ ˆ ˆR a R b R b R an n n n

Let 2j jY D R m j mj m

mmY

1* , , , ,j j j jD Y D Y

3 2 3, ,j jD D R R R 3 2 3j j jD R D R D R

Ex. 7.7

Error in eq(7.4-4) See: A.R.Edmonds, "Angular momentum in quantum mechanics", p.59

Symmetry Relations of d j() ( Condon-Shortley convention ):

2m i Jj j

md D e mj

md m j mj

md

m m mj

md

mm j mj j m

mm md Y

Relation to Spherical Harmonics (To be derived in Chapter 8):

1) Integer j = l :

*

0

2 1, , ,0

4ml

lm

lY D

0

!cos

!m ml

lm

l mP d

l m

00 0cos cos l

l lP P d

2) Arbitary j :

, cosm m m m mj

lmd P Jacobi Polynomials

3) Orthonormality & completeness : See § 7.7

Error in eq(7.4-6). See Edmonds

Characters:

All rotations of the same angle belong to the same class.

3

jmj j

mm j

D R

j

i m

m j

e

1

1

i ji j

i

e e

e

1/ 2 1/ 2

/ 2 / 2

i j i j

i i

e e

e e

1sin

2

sin2

j

j = 1/2:

1/ 2 sin

sin2

2cos2

j = 1: 1

3sin2

sin2

sin cos cos sin2 2

sin2

22cos cos

2

2cos 1

7.5. Application to a Particle in a Central Potential

V = V(r) Spherical symmetry

, 0H U R 3R SO

, 0iH J 1,2,3i

7.5.1. Characterization of States

CSCO = { H, J2, J3 }

, , , ,H E l m E l m E

Eigenstates = { | E, l, m }

2 , , , , 1J E l m E l m l l

3 , , , ,J E l m E l m m

0,1, 2,l

, 1, , 1,m l l l l

x-rep wave function: , ,Elm E l m x x

Spherical coordinates:

, ,r x ˆ, , 0U r z 3 20, 0,i J i Je e r

3 ˆ ˆi Je r r z z 3 ˆ 0J r z

, , , , , ,E lm r r E l m †ˆ , , 0 , ,r U E l m z

†ˆ , , , , 0ml

mr E l m D

z

0 arbitrary

3ˆ ˆ, , , ,i Jr E l m r e E l m z z ˆ , , i mr E l m e z

Since this holds for all , we must have

0ˆ ˆ, , , , 0mr E l m r E l z z

0m El r ˆ , , 0E l r r E l z

†ˆ, , , , , , 0m

Elm l mr r E l m D

z 0† , , 0El l m

r D

*

0, , 0

mlE l r D 4

,2 1El lmr Yl

,El lmr Y

4

2 1El E lr rl

7.5.2. Asymptotic Plane Wave States

1V r

r then (x) ~ plane wave as r

If for r

Let P p p p

, ,p p

& ˆ ,p p p

ˆ, , 0U p z

*2 1

00 1

2 1, , cos , , , , 0

4mll

p l m d d p D

2

2plane waveHm

P

p p2

2E

m

pp p

Relation to angular momentum eigenstates (To be derived in Chapter 8):

Linear momentum eigenstates

, , ,lmd p Y cosd d d

Inverse: *

0

, , , , ,l

lml m l

p p l m Y

7.5.3. Partial Wave Decomposition

Scattering of a particle by V(r):

Initial state: ˆ , 0,i i ip p arbitrary p z final state:

, ,f p p

Scattering amplitude: ˆ, ,f iT p T p p p z

T is the T-matrix. In the Born approximation, T = V.

V = V(r) T is invariant under rotation, i.e.,

, , , , l l mm lp l m T p l m T p

23, , 0T J T J

*0, , , , , 0 0,0f i lm l

l m l

T Y p l m T p l Y

p p

*

0

, , , , ,l

lml m l

p p l m Y

*0 0, , , 0 , , 0 0,0l l

l

Y p l T p l Y

*0

0

ˆ , , 0 0, 0ll

p p l Y

z

0

2 1, cos

4l l

lY P

2 1cos

4 l ll

lT E P

where , , 0 , , 0lT E p l T p l

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 ml m Y

0

l

l m l

l m l m I

d I

0 0ˆ ,l ml m Y z *

0

2 1, , 0

4mll

D

ˆ, , , , 0 ,r E l m r E l l m z

7.6. Transformation Properties of Wave Functions & Operators

U R R x x xRx x i i jjx R x 3R SO

3d x x x 3d x x x

' U R 3 'd x x x

Theorem 7.6: Transformation Formula for Wave Functions

1' R x x

Proof:

3' U R d x U R x x 3d x x x

3 1d x R x x 3 1d x R x x

3 'd x x x QED

3 3d x d x

' x x

since detR = 1

Example 1: Plane Waves p p

p x x p ie p x

p px x

1R p x1i Re p x Ti Re p x i Re p x

U R px U R x p R x p

p x Rp p

Example 2: Angular Momentum States E l m

, ,r E l m x ˆE l l mr Y x

' U R E l m ml

mE l m D R

' , , 'r x ˆ mlE l l m mr Y D R

x

1R x 1 ˆE l l mr Y R x

1 ˆ ˆml

l m l m mY R Y D R

x x ( See § 8.6 )

ˆ , , 0 ,r E l l m z

Extension: Pauli Spinors Basis vectors:

,x 1

2

1/ 2, ,U R R D R

x x

3 , ,d x x x 3 ,d x x x

' U R 3 1/ 2,d x R D R x x

sum over implied

3 1/ 2 1,d x D R R x x

3 , 'd x x x

1/ 2 1' D R R x x

This forms a representation for SO(3). See Problem 7.10

Definition 7.3: Irreducible Wave Functions & Fields

1'mm j n

nD R R x x

, ,m m j j x 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

Coordinate operators X x x x

Theorem 7.7: Transformation Formula for Vector Operators

j jX xx x

1 ji j iU R X U R X R

i, j = 1, 2, 3

Proof:

1i iU R X U R X U R U Rx x 1iU R X U R

x

i iU R x x x x 1 i j

jR x x

1 1 ii j

jU R X U R R X

1 i j

jR X x

j ijX R QED

This also forms a representation of SO(3) on the operator space

Any operator that transforms like X is a vector operator.

1 ji j iU R P U R P R

E.g.,

Other tensor operators can be similarly defined

c.f.i i j

jx R x

Field operators

xPauli-spinor field operator

0 x x

annihilates a particle of spin at x

| 0 = vacuum

1 10 U R U R U R U R x

10 'U R U R x

'U R

0 0U R

1/ 2 1 'D R R

x [ (x) is a spin ½

field ] 1/ 2 1 0 'D R R

x

1 1/ 2 1U R U R D R R

x x 1/ 2 1' D R R x x

c.f.

1/ 2 1 1/ 2D R D R

*1/ 2D R

1 1/ 2U R U R R D R

x x

1 1/ 2U R U R D R R

x x

1 ji j iU R X U R X R

c.f.

1U U

Generalization

1, 2, ...,mA m Nx

1 1 mm n

nU R A U R D R A R

x x

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

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 VV, wrt basis

,U R m m U R j m U R j m

',n nj j

m mn n D R D R

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.

,j m j m m m is given by

'n nj j

m mj n D R j n D R

,'

,,

n nj j

m mn n D R

i.e.,

,' '

,

n n n nj j j j

m m m mD R D R D R

' 'j j j jD R D R D R

Example: j = j = ½ | m m' = | + + , | + – , | – + , | – –

Let a

1/ 2 1/ 2 1/ 2 1/ 2,n n n n

U R a n n D R D R D R D R

1/ 2 1/ 2 1/ 2 1/ 2, D R D R D R D R

1/ 2 1/ 2 1/ 2 1/ 2, D R D R D R D R

1/ 2 1/ 2 1/ 2 1/ 2, D R D R D R D R

1/ 2 1/ 2 1/ 2 1/ 2, D R D R D R D R

1/ 2 1/ 2 1/ 2 1/ 2, , D R D R D R D R

1/ 2deta D R a

| a spans a 1-D subspace invariant under SO(3) .

1/ 2 1/ 2 0 1D D D D½ ½ is reducible. To be proved:

Proof:

' 'j j j jD R d D R d D R d n n n

' 'j j j jLHS E i d J E i d J n n

' ' 'j j j j j jE E i d J E E J n n

' ' 'j j j j j jJ J E E J n n n

' 'j j j jRHS E i d J n

'j jJ J n n

Theorem 7.8:

' ' 'j j j j j jJ J E E J n n n

Reduction of Dj j ' :

' '3 3 3, j j j jJ m m J E E J m m

' '3 3j j j jJ m E m E m J m

m m m m m m m m m m

,m m m m

, ' 'j m j j m j

,M m m

max 'M j j with 1 state 3 , ' ' , 'J j j j j j j

' 1M j j with 2 states 3 1, ' ' 1 1, 'J j j j j j j

3 , ' 1 ' 1 , ' 1J j j j j j j

min 'M j j with 1 state 3 , ' ' , 'J j j j j j j

' 1M j j with 2 states 3 1, ' ' 1 1, 'J j j j j j j

3 , ' 1 ' 1 , ' 1J j j j j j j

Let || J M be eigenstates of { J2, J3 }

2 , 1 ,J J M J J J M

3 , ,J 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

22 '', ' , 'j jJ j j j j J J j j

Justification:

'3 3 3', ' , 'j jJ j j j j J J j j ' , 'j j j j

' ' 1 ', 'j j j j j j j j

(Problem 7.8)

' ', 'j j j j j j

Other members in the multiplet

' , ', ' 1, , ' 1, 'j j M M j j j j j j j j

can be generated by repeated use of J– . E.g., ', ' 2 ' ', ' 1J j j j j j j j j j j 1 1 2J J J J J

' , 'j jJ J j j 2 1, ' 2 ' , ' 1j j j j j j

'

', ' 1 1, ' , ' 1' '

j jj j j j j j j j

j j j j

{ || j+j', M } thus generated spans an [ 2(j+j')+1 ]–D invariant subspace corresponding to J = j + j'.

Using a linear combination of

1, ' & , ' 1j j j j

that is orthogonal to ', ' 1j j j j

we can generate the multiplet corresponding to J = j + j' – 1.

as ' 1, ' 1j j j j

Arbitrary phase factor to be fixed by, say, the Condon-Shortley convention.

(Problem 7.8)

Dimension of D j j = ( 2 j+1 ) ( 2 j+1 )

'

'

2 1 ' ' ' 1 ' ' 1 'j j

J j j

J j j j j j j j j j j j j

' 1 ' ' 1 'j j j j j j j j

2 2' 1 'j j j j 2 1 2 ' 1j j

'

'

'

j jj j J

J j j

D D

Clebsch–Gordan Coefficients:

, ,J M m m m m J M , 'm m m m j j J M

, ,m m J M J M m m 'J M J M j j m m

Transformation between | J M & | m, m' :

*' '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

, ' 0j J j j j J J , ',j j J

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

( Largest M & m )

D½ ½ re-visited: | m, m' = | + + , | + – , | – + , | – –

11

J = 1, 0

11 2 10J 1/ 2 1/ 2J J

110

2 10 2 1, 1J 1

2

1 1

10 0

2 ( orthogonal to | 1 0 )

1 1 1 111 1

2 2 2 2

1 1 1 1 1 1 1 1 110 10

2 2 2 2 2 2 2 2 2

1 1 1 11 1 1

2 2 2 2

1 1 1 1 1 1 1 1 10 0 0 0

2 2 2 2 2 2 2 2 2

CGCs:

Appendix V

A square root is to be understood over every coefficient.

1 1 1 1 1 1 1 1 10 0 0 0

2 2 2 2 2 2 2 2 2

Other methods to calculate the CGCs are discussed in books by Edmond, Hamermesh, Rose, ….

Some special values we've calculated:

' ' ', ' 1j j j j j j j j

1, ' ' ', ' 1'

jj j j j j j j j

j j

', ' 1 ' ', ' 1

'

jj j j j j j j j

j j

General Properties of the CGCs

Angular Momentum Selection Rule:

' 0m m j j J M

unless m m M and ' 'j j J j j

Orthogonality and Completeness:

' ' J MJ MJ M j j m m m m j j J M

' ' m mn nm m j j J M J M j j n n

Symmetry Relations:

'' '

j j Jm m j j J M m m j j J M

', ' ,

j j Jm m j j J M

' 2 1, '

2 1j J m J

M m J j j mj

Wigner 3-j Symbols:

'

''

2 1

j j Mj j J

m m j j J Mm m M j

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.

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

' ' 'm m Mj j J

n n ND R D R m m j j J M D R J N j j n n

'' 'M m mJ J j j

J M n nD R J M j j m m D R D R n n j j J M

, ,U R n n U R J N J N n n

', ,m m Mj j J

n n Nm m D R D R J M D R J N n n

' , ,m m Mj j J

n n ND R D R m m J M D R J N n n

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

1s s sU R O U R O D R

3R SO

Os is the th spherical component of the tensor.

Theorem 7.9: Differential Characterization of Irreducible Spherical tensor

2 , 1s sJ O s s O 3 ,s sJ O O

1, 1 1s sJ O s s O

Proof:

For an infinitesimal rotation about the kth axis,

1s s sU R O U R O D R

sk kLHS E i d J O E i d J ,s s

kO i d J O

s skRHS O i d D J

s s s

kO i d O D J

, s s sk kJ O O D J

3sD J

1 1 1sD J s s

Using

23 3 1J J J J J

completes the proof.

Examples:

, 0sO U R 3R SO 0s 1.

2. 3

1 1, ,

2 2J J J

is an irreducible spherical vector with s = 1 & = { 1, 0, –1}

3 ,J J J

, 02

JJ

3, 2

2

JJ J

3, 2J J J

3 3, 0J J

This is easily proved using

3 ,s sJ O O 3 ,

2 2

J JJ

1, 1 1s sJ O s s O

3, 22

JJ J

3, 22

JJ J

3, 22

JJ J

, 02

JJ

Definition 7.5: Vector Operator – Cartesian Components

1. Operators 1, 2, 3lA l are the Cartesian components of a vector if

, k l mk l mJ A i A

2.

11, 2, 3

nl l jT l are the Cartesian components of a nth rank tensor if

1

1 2 1 1, n

n n n

k l m k l m

k l l m l l l l mJ T i T T

Actually, the above can be derived from the more familiar definition of Cartesian tensors in terms of rotations in E3

1 m

k l k m k lR A R A R

using ki JkR e and ji i j kk

J i

1

1 1 1

1 n

n n n

m m

k l l k l l k kl lR T R T R R

c.f. Theorems 7.2, 3

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 D11 rep.

It is reducible.1 1 0 1 2D D D D 1 1 0 1 2D D D D 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 )

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 )

' ' , 's sj m O j m j m s j m j O j

Selection Rules:

' 0sj m O j m

unless 'j s j j s & m m

Branching Ratios:

' ,'

' ' ,

s

s

j m s j mj m O j m

j n O j n j n s j m