16. Angular Momentum
1. Angular Momentum Operator
2. Angular Momentum Coupling
3. Spherical Tensors
4. Vector Spherical Harmonics
Principles of Quantum Mechanics
State of a particle is described by a wave function (r,t).
Probability of finding the particle at time t within volume d 3r around r is
P  r, t  d r    r, t 
3
2
d 3r
Dynamics of particle is given by the time-dependent Schrodinger eq.

i
  r, t   H   r, t 
t
H 
SI units:
2
2m
2  V  r, t 
 1.05  1034 J  s
Stationary states satisfy the time-independent Schrodinger eq.
H  r   E  r 

 E r, t   ei Et /  E r 
with
Hamiltonian
V  V r 
Let  be an eigenstate of A with eigenvalue a, i.e.
A  a 
Measurement of A on a particle in state  will give a
and the particle will remain in  afterwards.
 A , B  0

Operators A & B have a set of simultaneous eigenfunctions.
 A stationary state is specified by the eigenvalues
of the maximal set of operators commuting with H.
Measurement of A on a particle in state  will
give one of the eigenvalues a of A with probability
and the particle will be in a afterwards.
 x , px   i

uncertainty principle
a 
2
1.
1
    2 L2
r
2
2
r
Angular Momentum Operator
1  
 
1 2
L 
sin 
 2


sin   
  sin   2
2 2 
  2
r
r r
2
r
p
Quantization rule :
i
2

p2
T

2
Kinetic energy of a particle of mass  :
L  rp 
Angular momentum :
p  rˆ  p  rˆ  rˆ  p
Rotational energy :
i

L2
p p  2
r

L2  
2
L2
TR 

2
2 r
2
2
2
r 
 
2
r

2
2
   
2
2
2
r
2
r2
L2
L2

2
2 r
2
L
2
angular part of T
1  
 
1 2
L 
sin 
 2


sin   
  sin   2
L
2
L 
i
r  

i
eˆ r
r
eˆ 
0
eˆ 
0

r
1 
r 
1 
r sin   
i
r 

1 
 
ˆ
ˆ

e

e


i 
sin   
  

L

L2  L  L  L 2x  L 2y  L z2   2L 2
L 2Yl m  ,   l  l  1 Yl m  , 
with
l  0,1,2,

Ex.3.10.32
L2 Yl m  ,   l  l  1
m  l , l  1,
,l
2
Yl m  , 
L  L
2
Central Force
2
1  
 
1 2
L 
sin 

sin   
  sin 2   2
2
2
2
2
L2
2
2
H  Tr 

V
r




L
V r


r
2
2
2 r
2
2 r
 r , L2   0
Ex.3.10.31 :


 L2 , H   0
 L j , L k   i  j k n L n
 L2 , L j   0
Cartesian commonents
eigenstates of H can be labeled by eigenvalues of L2 & Lz , i.e., by l,m.


 
L x    sin 
 cot  cos 

i 

 


 
cos


cot

sin

i 

  

Lz 
i 
Ex.3.10.29-30
Ly 

L zYl m  m Yl m
Ladder Operators
L  L x  i L y

Ladder operators
 Lz , L     Lz , L x   i  Lz , L y  
i L
y
 Lx 
  Lz , L     L 
Let lm be a normalized eigenfunction of L2 & Lz such that
L2  lm   l

2
 lm
L z  lm  m  lm
 Lz , L    lm   Lz L   L  Lz   lm   Lz L   L  m


i.e.

m
l
  L   lm
Lz L  lm   m  1  L  lm
L   lm is an eigenfunction of Lz with eigenvalue ( m  1)  .
L   lm   cl   lm  1
l
 L are
Raising
operators
Lowering
 L2 , L j   0
L2 L  lm   l

L  lm
2
L   lm   bm  lm
i.e.

m
L  L x  i L y

L 
L  L   L x  i L y  L x

†
i L y   L2x  L2y
L  l
a real 
L  l
m
1
L  L   L L   L2z

2
 L , L   2 Lz
 a  l , m 
L   lm 
i  L x Ly  Ly Lx 
L2 
L L  L2x  L2y  Lz
m
L  lm  a  l , m  lm 1
L
L L  L2  L2z  Lz

l
L   lm is an eigenfunction of L2 with eigenvalue l 2 .


L   lm   cl   lm  1
 L2 , L   0

2
  lm L L   lm
 l  m  m  1  lm 1

2

lm normalized
l
 m2
m
Ylm thus generated agrees with the
Condon-Shortley phase convention.
L   lm
L   lm 
L   lm
L   lm
2

0
l
 m2
l  m  m  0
For m  0 :
 l  m2  m  0


2

L   lm 



mmax 
1
1 
2
mmin 
1
1
2

mmax  mmin  integer  n  0

1 3
l  0, ,1, ,
2 2
m  l ,
 l, m
1  4 l
1  4 l
1
n n
 l   n 2  2n     1  l  l  1
4
22 
1
1
m


n  l
mmax  n  l
min
2
2
 l  l  l  1
 l  m  m  1  lm 1
 l  m2 m  0

 l, m
For m  0 :
m = 1
m
 0

1  4 l  n  1
l
,l
0
n
2
Multiplicity = 2l+1
Example 16.1.1.
Spherical Harmonics Ladder
 
 
L   e  i  
 i cot 







Y10  ,   
Y
1
1
 ,  
L   lm 
3
cos 
4
3  i
e sin 
8
 L Y
 l  m  m  1  lm 1
0
1

 ,  
Lx 


 

sin


cot

cos



i 

 


 
cos


cot

sin

i 

  

Lz 
L  L x  i L y
i 
Ly 
e
 i
 
 
     i cot     



3
4

2 Y1 1  , 
L  10 

3
cos  
4

e  i  sin  
2  11

 lm  Y l m
for l = 0,1,2,…
Spinors
Intrinsic angular momenta (spin) S of fermions have s = half integers.
E.g., for electrons
s
1
2
S 2  s  s  1 
Eigenspace is 2-D with basis
Or in matrix form :



3
4
1
2
,

1
2
  1  0 
  ,   
  0  1 
ms 
1
1
or 
2
2

 


 ,
spinors
S are proportional to the Pauli matrices.


Example 16.1.2.
s
1
2
Sx 
0 1
2  1 0 
Spinor Ladder
Sy 
 0 i 
2  i 0 
Sz 
1 0 
2  0 1
Fundamental relations that define an angular momentum, i.e.,
Eigenvalue of S2  s  s  1
S j , S k   i  j k n S n
2
can be verified by direct matrix calculation.
S 
Spinors:
0 1
2  0 0 
S 
 0 0
2  1 0 
1
1/2
 1/2
    
 0
s  s  1  m  m  1  0 or
3 1
 1
4 4
Mathematica
 0
1/2
 1/2
    
1

S   0
S   
S   0
S   
Summary, Angular Momentum Formulas
General angular momentum :
Jk , Jl   i
Eigenstates JM :
J  Jx , J y , Jz 
kln Jn
 J 2 , J k   0
 JM  JM    J J   M  M
J 2  JM  J  J  1
2
 JM
J z  JM  M  JM
J  Jx i J y
J   JM 
J  J  J 2  J z2  J z
J  J  1  m  m  1  JM  1
J = 0, 1/2, 1, 3/2, 2, …
M = J, …, J
J , J   2
Jz
2.
Let
Angular Momentum Coupling
j1   j1 x , j1 y , j1z 
j2   j2 x , j2 y , j2 z 
 ,   1, 2

 j k , j l     i  k l n j n
 j , j k   0
k , l , n  x, y , z
2
Implicit summation applies
only to the k,l,n indices
J  j1  j2   J x , J y , J z    j1 x  j2 x , j1 y  j2 y , j1z  j2 z 

 J 2 , J k    J 2 , j1k  j 2 k   0
j1  j 2  j 2  j1

Jk , Jl   i
kln Jn
J 2  j12  j 22  2 j1  j 2  J x2  J y2  J z2
 j k , j l     i  k l n j n
Example 16.2.1.
Commutation Rules for J Components
 j2 , j k   0
Jk , Jl   i
 J 2 , j k    j12  j 22  2 j1  j 2 , j k 
 2  1
 1  2

 2 j l  j l , j k 
 2 i  l k n j l j n
 J 2 , j1 z   2 i
kln Jn
J 2  j12  j 22  2 j1  j 2
  2 j1  j 2 , j k 
e.g.
 J 2 , J k   0
 J 2 , j k   2 i

 j1  j2  z
2i
j
1x
 j  j k
j2 y  j1 y j2 x 
 J 2 , j2    j12  j 22  2 j1  j 2 , j 2    2 j1  j 2 , j 2   2 j k  j k , j2 
 j2 , J k    j2 , j1k  j2 k 

 j2 , J k   0

 J 2 , j 2   0
 j2 , j k   0
 j2 , J k   0
 J 2 , J k   0
 J 2 , j 2   0
 J 2 , j k   2 i
 j  j k
Maximal commuting set of operators :
 j , j , J ,J 
2
1
eigen states :
Adding (coupling)
2
2
or
z
j1 , j2 ; J M


2
1
2
2
1z
, j2 z
j1 m1 , j2 m 2
j1  j2  J
j1 , j2 ; J M
j ,j , j

 j1 m1
j2 m 2
means finding
C m1 m 2
j1 m1 , j2 m 2
m1 , m 2
Solution always exists & unique since

j1 m1 , j2 m 2

is complete.

Vector Model
J z  j1z  j2 z 
M  m1  m 2
j12 , j 22 , J , J z
 j ,j , j
j1 , j2 ; J M

 2 j1  1 2 j2  1    2 J  1
J  J min

2
J min
  j1  j2  1   2 j1  1 2 j2  1   j1  j2 
2
J min  0
i.e.

2
J min  j1  j2
J  j1  j 2 , j1  j 2  1,
, j1  j2
Triangle rule
, j2 z

j1 m1 , j2 m 2
m1 m 2
2
1
1
1
2
0
1
1
1
0
0
0
1
2
1
0
1
0
1
1
0
0
1
 1 2
 1 1
Mathematica 1 0
 2 2
1
1
0
1
0
2 1
3 2
1
1
  J max  J min  1 J max  J min  1
  j1  j2  J min  1 j1  j2  J min  1
1z
M
3
2
2
Total number of states :
J max
2
2
j1  2 ; j 2  1
M max  j1  j 2
J max  j1  j 2

2
1

Clebsch-Gordan Coefficients
j12 , j 22 , J , J z
2
1
2
2
1z
, j2 z
j1 , j2 ; J M
j1 m1 , j2 m 2
J max  j1  j 2
J min  j1  j2
JM
For a given j1 & j2 , we can write the basis as
 j ,j , j
&
m1 , m 2
Both set of basis are complete :
JM

j1
j2
 

m1 , m 2
m1   j1 m 2   j 2
m1 , m 2 
j1  j 2

J  j1  j 2
m1 , m 2 J M
 J M m1 , m 2
Condon-Shortley
phase convention
m1 , m 2 J M
M  m1  m 2
J

JM
M  J
*
m1 , m 2 J M
J M m1 , m 2
Clebsch-Gordan Coefficients (CGC)
 J M m1 , m 2
j1 , J  j1 J J  0

Ladder Operation
Construction
J M

j1

m1   j1
j
m1 , M  m1
m1 , M  m1 J M
j  j  1  m  m  1
jm 
j m 1
J max  j1  j2
J max j1  j2  j1 , j2
J  J max j1  j2   j1  j2  j1 , j2

2  j1  j2 
J max j1  j2  1 
J max j1  j2  1 
2 j1 j1  1 , j2  2 j2 j1 , j2  1
j1
j1  1 , j2 
j1  j2
j2
j1 , j2  1
j1  j2
Repeated applications of J then give the rest of the multiplet

Orthonormality :
J max M ; M   J max ,
J max  1 j1  j2  1  
, J max

j1
j1  1 , j2 
j1  j2
j1 , J  j1 J J  0
j2
j1 , j2  1
j1  j2
Clebsch-Gordan
Coefficients
j1

J M

m1   j1
m1 , M  m1
m1 , M  m1 J M
Full notations :
j1 m1 , j2 m 2 J M
F1 
s
j
1
real
 j 2  J  !  J  j1  j 2  !  J  j 2  j1  !  2 J  1
j
1
 j 2  J  1 !
 J  M !  J  M !  j1  m1 !  j1  m1 !  j 2  m2 !  j 2  m 2 !
F2 
F3  
j
 C  j1 , j2 , J | m1 , m 2 , M   F1 F2 F3
1
 m1  s  !
j
 
2
s
 m2  s !  J  j2  m1  s !  J  j1  m 2  s !  j1  j2  J  s !
Only terms with no negative factorials are included in sum.
Table of Clebsch-Gordan Coefficients
 C  j , j, J | m, m, M 
Ref:
W.K.Tung, “Group Theory in Physics”, World Scientific (1985)
Wigner 3 j - Symbols
 j1 j 2
m m
2
 1
j 3     j1  j 2  m 3

C  j1 , j2 , j 3 | m1 , m 2 ,  m 3 
m 3 
2 j3  1
Advantage :
 j1 j 2
m m
2
 1
more symmetric
jl
jn 

 jk

m m m 
j3  
l
n
 k

m 3  
jl
jn 
j1  j 2  j 3  j k
m m m 
  
l
n
 k

 j1 j 2
m m
2
 1
j3 
j2
j 1  j 2  j 3  j1
  
 m m
m 3 
2
 1
C  j1 , j2 , J | m1 , m 2 , M     
j 2  j1 M
1, 2,3   k , l , n  is even
1, 2,3   k , l , n  is odd
j3 
m 3 
 j1 j 2
2J  1 
 m1 m 2
J 
 M 
Table 16.1 Wigner 3j-Symbols
1 1 
 , ,1  :
2 2 
1 1 
 , ,0 :
2 2 
1,2,3 :
Mathematica
Example 16.2.2.
Two Spinors
j1  j2 
1
2
1 1
11 
,
2 2

1 1
2 10   ,
2 2
2 1 1 
1
1 1
00 
 ,
2 2
2

j1

m1 , M  m1
m1   j1
C  j1 , j2 , J | m1 , m 2 , M     

 1


1

 32
 1
2
1
2
1
2

1

1

j 2  j1 M
 j1 j 2
2J  1 
 m1 m 2
 1

 3 2
  1
 2
1 1
1 1
C  , ,1  , ,0
2 2
2 2
10 

j m 1
m1 , M  m1 J M
j1 , J  j1 J J  0
1 1
,
2 2

1
1 1
 ,
2 2
2
1 1
1 1
C  , ,0 ,  ,0
2 2
2 2
J M
J  1, 0
1 1
1 1
C  , ,1 , ,1
2 2
2 2
j  j  1  m  m  1
jm 
j
1
1 1
 ,
2 2
2
11  
1 1

 1 1

1 1   2 2 0  2 2 0
 1
1 1
 C  , ,0  , ,0   



2 2   1  1 0    1 1 0 
2

2 2
2
2

 2
2

1
2
1

2

1

0 

 1

2

1 1 1
,
2
2
2

1
1
,
2
2
1 1
1 1
C  , ,1  ,  , 1
2 2
2 2
1 1 1

,
2 2 2

1
1
2
 3
 1
0 

2
1 1

1 1
  C  2 , 2 ,1 2 ,  2 ,0


1
1 1
 ,
2 2
2

1
2
1
2
J 
 M 

 1

 1

 3 2
  1
 2
1

1
2

1 

1
2 
Simpler Notations
1

2



11 

1
1 1
 ,
2 2
2

11  
1
1
,
2
2
 
00 
1
1 1
 ,
2 2
2
10 
1 1
,
2 2
1 1 1
,
2 2 2

1 1 1
,
2 2 2
 

1
1
 

2
2
1
1

 

2
2
Example 16.2.3.
Coupling of p & d Electrons
C  j1 , j2 , J | m1 , m 2 , M     
JM
j1


m1   j1
j1  1 , j2  2

Simpler notations :
 j1 j 2
2J  1 
 m1 m 2
j 2  j1 M
m1 , M  m1
m1 , M  m1 J M
J  3, 2,1
lm  m
where
l
0
1
2
3
 s
p
d
f
3 3  1, 2  p1 d 2
3 2  C 1, 2,3 0, 2, 2  0 , 2  C 1, 2,3 1, 1, 2  1 , 1
1 2 3 
1 2 3 
 7
0
,
2

7

 1 1 2  1 , 1 
0
2

2




1
p0 d 2 
3
2
p1 d1
3
31  C 1, 2,3  1, 2, 1  1 , 2  C 1, 2,3 0, 1, 1 0 , 1  C 1, 2,3 1, 0, 1 1 , 0
1 2 3
1 2 3 
1 2 3 
 7

1
,
2

7
0
,
1

7

 0 1 1
1 0 1 1 , 0

1
2

1







1
8
p1 d 2 
p0 d1 
15
15
2
p1 d 0
5
Mathematica
J 
 M 
C  j1 , j2 , J | m1 , m 2 , M     
JM

j1

m1   j1
j 2  j1 M
m1 , M  m1
 j1 j 2
2J  1 
 m1 m 2
J 
 M 
m1 , M  m1 J M
2 2  C 1, 2, 2 0, 2, 2  0 , 2  C 1, 2, 2 1, 1, 2  1 , 1
2
1 2 2 
1 2 2 
 5
0
,
2

5
1
,
1


p0 d 2 

1 1 2 
0
2

2
3




1
p1 d1
3
2 1  C 1, 2, 2  1, 2, 1 1 , 2  C 1, 2, 2 0, 1, 1 0 , 1  C 1, 2, 2 1, 0, 1 1 , 0
1 2 2
1 2 2 
1 2 2 
 5

1
,
2

5
0
,
1

5

 0 1 1
 1 0 1 1 , 0

1
2

1







1
1
p1 d 2 
p0 d1 
3
6
1
p1 d 0
2
11  C 1, 2,1  1, 2, 1 1 , 2  C 1, 2,1 0, 1, 1 0 , 1  C 1, 2,1 1, 0, 1 1 , 0
1 2 1
1 2 1 
1 2 1 
 3

1
,
2

3
0
,
1

3

 0 1 1
 1 0 1 1 , 0

1
2

1







3
3
1
p1 d 2 
p0 d1 
p1 d0
5
10
10
Mathematica