Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

FermiGasy
Addition of Angular Momenta
Angular momenta L1, L2 , direction undetermined
Projections conserved : m1, m2 , m  m1  m2  L1  L2  L  L1  L2
(t )
1  const.
2  const.
1(t ), 2(t )  L(t ),  (t ), (t ) 
2
i (t )  i Larmor frequency
2
L  L(t )  Classical Probability :
Angular Momentum Coupling
2

P(L)   L t 
1

2
 2L L t

1
L2  m2  L21 sin2 1  L22 sin2 2 
1
 L1L2 sin 1 sin 2 cos(1  2 )
If (L1)  (L2 ) dipole interaction : L1 couples with L2  L
Coherent motion : m  conserved,   const.   (t ) L  L(t )
W. Udo Schröder, 2005
At large r
1  2  r1  r2
3
 decoupling
Angular Momentum Coupling
Quantal angular momentum eigen states j1  j2  ??
j
j
ˆ
J12 , ˆ
J1z : 1 ; ˆ
J22 , ˆ
J2 z : 2
m1
m2
3
Max alignment :
j1
j1
 dim  (2 j1  1)(2 j2  1) dimensionality
j2
j1
:
j2
j1
j2
j2

ˆ
2 j1
J
j1
j2
j2
j1
 ??
j1
??
j2
j2
2
Angular Momentum Coupling
Use ˆ
J 2   ˆ
J1  ˆ
J2   ˆ
J12  ˆ
J22  2 ˆ
J1  ˆ
J2  ˆ
J12  ˆ
J22  ˆ
J1 ˆ
J2   ˆ
J1 ˆ
J2   2 ˆ
J1z ˆ
J2 z


ˆ
2 j1
J
j1
j2

j2

2
 j1( j1  1) 
2
( j1  j2 )( j1  j2  1)
J
ˆ
Jz
j1
j1
W. Udo Schröder, 2005
j2 ( j2  1)  0  0  2 j1 j2 
J 1
j2
j1
 ( j1  j2 )
j2
j1
j1
j1
j2

j2
j2
j1
 m
j2
j1
2
j2
j2
j1
j1
J( J  1)
j2

j2
j1
j1
j2
j2
 The first

 ˆ
J2, ˆ
Jz
eigen state

Constructing J Eigen States
J , for example
Construct m  J spectrum successively by applying ˆ
ˆ
J
j1
j2
j1
j2
J  j1  j2
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
J2 
J1  ˆ
 ˆ
m  J 1
 2 j1

j1
j2
j1
j2
j1
j2
j1  1
j2

2 j1
+ 2 j2
j1
j2
j1  1
j2

2 j2
j1
j2
j1
j2  1
j1
j2
 eigen state
j1
j2  1
J  j1  j2 , m  J  1
Can you show this??
This is one specific linear combination of 2 states j1 , m1 and
j2 , m2
Angular Momentum Coupling
What about the other ? There should be 2 orthogonal combinations.
And : Further application of ˆ
J yields again only one specific linear
combination of 3 independent components
J  j1  j2
m  J 2
j1

j1  2
j2
j1
+
j2
j1  1
j2
+
j2  1
j1
j1
j2
j2  2
What about others ? There should be 3 orthogonal combinations.
W. Udo Schröder, 2005
Constructing J-1 Eigen States
We have this state:
j1
J  j1  j2
 2 j1
j1  1
m  J 1
j2
j
+ 2 j2 1
j2
j1
j2
j2  1
 eigen state
J  j1  j2 , m  J  1
Two basis states j1 , m1 , j2 , m2  2 new orthogonal states
5
J  ??
m  J 1
 2 j1
j1
j1  1
j2
j1
- 2 j2
j2
j1
j2
is orthogonal
j2  1
Angular Momentum Coupling
Apply ˆ
J2  ˆ
J12  ˆ
J22  ˆ
J1 ˆ
J2   ˆ
J1 ˆ
J2   2 ˆ
J1z ˆ
J2 z
Use
ˆ
J1
j1
m1
j2

m2
 j1
m1   j1  m1  1
j1
J  j1  j2  1
 2 j1
j1  1
m J
j2
j2
- 2 j2
ˆ
J
to
j1
m1
j2
m2
j1
j2
j1
j2  1
j1
j1
j2
j2
etc.
eigen state
J  j1  j2  1 ...etc.
m J
Condon-Shortley
Normalization conditions leave open phase factors  choose
asymmetrically <|J1z|> ≥ 0 and <a|J2z|b> ≤ 0
W. Udo Schröder, 2005
Clebsch-Gordan Coefficients
General scheme : Unitary transformation (between bases)
 j1
 j1
j2 j  j1
j2
j2 j  j1
j2
  
  


m2
m
m1 , m2  m1 m2 m  m1 m2
m1 , m2  m1 m2 m  m1
j1
j2  j1
j2 j 
j1
j2
j1
j2 j
j
 

 
m1 m2 m
m
m1 , m2 m1 m2  m1 m2 m 
m1 , m2 m1 m2
6
j
ˆ
Representations of identity operator 1
j1 j2 :
Angular Momentum Coupling

m1 , m2
j1
m1
j2
m2
j1
j2
ˆ
1
j1 j2
m1 m2

j,m
j
m
j
ˆ
1
j1 j2
m
Orthogonality relations of CG coefficients :
 j j1
j2   j1
j2 j  
 


m1, m2  m m1 m2   m1 m2 m 

m1 ,m2
j2
j  j1
m m1 m2
j1
j2 j
  j j  mm
m1 m2 m
=1
 j1
 
j , m  m1
W. Udo Schröder, 2005
j2 j   j j1

m2 m   m m1
j2 
j1



m2  j,m m1
j2 j
m2 m
j j1
m m1
=1
j2
  m1 m1  m2 m2
m2
Recursion Relations
j2  j1
j2 j 
j1
j2
j1
 




m1
m
m1 , m2 m1 m2  m1 m2 m 
m1 , m2 m1 m2
j1
j2
j1
j2
j
j
j
j
ˆ
ˆ
J 
J
m1 m2 m - 1 m - 1
m
m
m1 , m2 m1 m2
j1
j
j2
m2

j
m
7
ˆ
P
j , m 1

j
m-1
ˆ
J
j
m

m1 , m2
j1
m1
j2
j1
j2
m2 m  1 m1
m2
j

Angular Momentum Coupling
 j  m j  m 1

ˆ
J1- + ˆ
J2-
m
j


m1 , m2
j1
m1
j2 j
m2 m
j1
j1
ˆ
J
m1 - 1 1 m1
j1
j2

m1 - 1 m2
 j1  m1  j1  m1 1

C jm :
 j  m j  m1
W. Udo Schröder, 2005

m1 , m2
j1
m1
j2 j
m2 m
j2
j
ˆ
J2  2
m2 - 1
m2
 j2  m2  j2  m2 1
j1
m1
j2
m2 - 1
Recursion Relations for CG Coefficients
C jm
j1
m1
j1
j2 j
m1  1 m2 m
8
C j1m1 1
j2

m2
j1
m1
j2
j
 C j2m2 1 1
m2
m1
C jm :
 j  m j  m1
j2
j
m2  1 m
j1
m1
j2
m2
Projecting on <j1,j2,m1,m2| yields
C jm
Angular Momentum Coupling
j2
j1
j
m2 m  1 m1
j1
j2
j1
j2 j
j
j2
j
j
 C j1m1 1
 C j2m2 1 1
m1 m2 m  1
m1  1 m2 m
m1 m2  1 m
Using ˆ
J  ˆ
J1  ˆ
J2  
C jm 1
j1
j2
j1
j2 j
j1
j2
j
j
 C j1m1
 C j2m2
m1 m2 m  1
m1  1 m2 m
m1 m2  1 m
Special values :
 j0 j

 1
 m 0 m
W. Udo Schröder, 2005
 j1 j1 0  (1) j1  m1


2 j1  1
 m1 m1 0 
0

 ???
0
Symmetries of CG Coefficients
Coupling depends on sequence ˆ
J1 , Jˆ2


9
Phase convention : non  diag.  ˆ
J1z   0,  ˆ
J2 z   0
 j1 j2 j 
j j j

  (1) 1 2
 m1 m2 m 
 j2 j1 j 


m
m
 2 1 m
Angular Momentum Coupling
 j1 j2 j3 
j3 j1 
j2  m2 2 j3  1  j2



  (1)
m
m
m
m

m
m
2
j

1
3
2
1
1
 1 2 3

 j1 j2 j3 
j1 j2 
j1  m1 2 j3  1  j3



  (1)
m
m
m
m

m
m
2
j

1
1
2
2
 1 2 3
 3
 j1 j2 j3 
j j j

  (1) 1 2 3
 m1 m2 m3 
m(m3 )  m1  m2
Triangular relation
Condon-Shortley :
Matrix elements of
J1z and J2z have
different signs
 j1 j2
j3 



m

m

m
3
1
2

Calculate CGs starting from max alignment : m  j
Then use recursion relations to obtain all
W. Udo Schröder, 2005
j, m  j
Explicit Expressions
A. R. Edmonds, Angular Momentum in Quantum Mechanics
 j1 j2 j 

   m, m1  m2 
m
m
m
 1 2

10
12

2 j  1  j1  j2  j  !( j1  m1 )!( j2  m2 )!( j  m)!( j  m)! 


 
j

j

j

1
!
j

j

j
!

j

j

j
!(
j

m
)!(
j

m
)!






 1
2
1
2
1
2
1
1
2
2 

j m
( j1  m1  s)!( j2  j  m1  s)!
s j m
   1 1 1
s !( j1  m1  s)!( j  m  s)!( j2  j  m1  s)!
s 0
Angular Momentum Coupling
12
 j1 j2 j1  j2   2 j1  ! 2 j2  !  j1  j2  m1  m2  !  j1  j2  m1  m2  ! 



m
m
m

m
2
j

2
j
!
j

m
!
j

m
!
j

m
!
j

m
!












2
1
2
1
1
1
1
2
2
2
2 
 1 2 1
 j1 j2
j


j
m

j
m
1
 1

12


2 j  1 ! 2 j1  !   j1  j2  j  !  j1  j2  m !( j  m)!


j

j

j
!
j

j

j
!
j

j

j

1
!

j

j

m
!
j

m
!










 1

2
1
2
1
2
1
2
W. Udo Schröder, 2005
2 Particles in j Shell (jj-Coupling)
Look for 2-part. wfs of lowest energy in same j-shell, Vpair(r1,r2) < 0 
spatially symmetric  jj1(r) = jj2(r). Construct consistent spin wf.
J
11
 JM   j   j 
M
j  j  0  J  j  j  2j
 j j J
 JM (r1, r2 )  N  
  jm1 (r1 ) jm2 (r2 )   jm2 (r1 ) jm1 (r2 )
m1, m2  m1 m2 M 
N = normalization factor
Angular Momentum Coupling
Which J = j1+j2 (and M) are allowed?  antisymmetric WF JM
 j1 j2 j 
j j j
Use Pauli Principle and 
  (1) 1 2
 m1 m2 m 
 j2 j1 j 

 for 12 : 1  2
m
m
m
 2 1 
 j j J   j j J  
 JM (r1 , r2 )  N  

   jm1 (r1 ) jm2 (r2 )
m
m
m
m
m1 , m2 
 1 2 M   2 1 M  
 j j J
 N 1  (1)2 j  J   
 (r )
(r )

 m ,m m m M  jm1 1 jm2 2 For j1  j2  5 2 
2
1 2 1

2 j1  j  5  j  odd
 N 12
W. Udo Schröder, 2005
 j  0,2, 4
Symmetry of 2-Particle WFs in jj Coupling
Antisymmetric function of 2 equivalent nucleons (2 neutrons or 2
protons) in j shell in jj coupling.
12
 JM (r1, r2 ) 
 j j J
 
  jm1 (r1 ) jm2 (r2 )
m1, m2  m1 m2 M 
J  even
1) j1 = j2 = j half-integer spins  J =even
wave functions with even 2-p. spin J are antisymmetric
Angular Momentum Coupling
wave functions with odd 2-p. spin J are symmetric
jj coupling  LS coupling  equivalent statements
2) l1=l2=l integer orbital angular momenta  L
wave functions with even 2-p. L are spatially symmetric
wave functions with odd 2-p. L are spatially antisymmetric
W. Udo Schröder, 2005
Tensor and Scalar Products
Tensor product of sets of tensors T k1 and T k2
k
T
k
 k1 k2 k  k1
k1
k2
k


 T (1)  T (2)   
T (1)T 2 (2)

2

 1,2 1 2  1


Angular Momentum Coupling
13
Scalar product of sets of tensors T k (1) and T k (2)  number
T00
k 
 k k 0 k
(

1)
k
 T (1)  T (2)   
T (1)T (2)  
Tk (1)Tk (2)


0    0

2k  1


k
k
0
Vectors u and v : Rank k  1  2k  1  3 spherical components
Spin  0 scalar product
1  ux  iuy
u  v 0 
 
3 
2
1

u v
3
0
W. Udo Schröder, 2005

  v x  ivy   ux  iuy   v x  ivy 



  uz v z 
2  
2 
2 


Transforms like a J=0 object = number
Example: HF Interaction
Angular Momentum Coupling
14
Addition Theorem of spherical harmonics :
P  cos   
4
*
 Ym (1 , 1 )Ym (2 , 2 ) 
2 1m
   1
4
4
Y m (2 , 2 )
Ym* (1 , 1 )
2 1
2 1
m
m
 
r1 , r2
Electron  nucleus hyperfine interactions
ei ep
ri  rp
ˆ
Hint  
r P(
)
1 p
ri rp
i , p, ri
   1
m
m,


4
4
*

e
r
Y
(

,

)
e
r




p p m
p
p
i i
 p 2  1
  i 2  1
0
 Tp  Ti   scalar product of separated tensors
0
protons electrons
only
only
W. Udo Schröder, 2005

Y m ( i , i )

1
Wigner’s 3j Symbols
Coupling  j1  j2  j3 equivalent to symmetric  j1  j2  j3  0
j2
15
j1
j3
j2
j1
j3 
 j1 j2 j3   j3 j3 0  (1) j3  m3

 j3  m1 m2 m3   m3 m3 0 
2 j3  1
 j1 j2 j3 



m
m
m
3
1
2


Angular Momentum Coupling
Choose additional arbitrary phase factor for 3 j symbol
 j1 j2 j3  (1) j1  j2  m3


2 j3  1
 m1 m2 m3 
 j1 j2 j3 



m
m
m
3
 1 2
 j1 j2 j3   j3 j1 j2   j2 j3 j1 



 all cyclic
 m1 m2 m3   m3 m1 m2   m2 m3 m1 
 j1 j2 j3 
j1  j2  j3

(

1)


m
m
m
 1 2 3
W. Udo Schröder, 2005
 j2 j1 j3 

 any 2 columns
 m2 m1 m3 
Explicit Formulas
16
 j1 j2 j3   j1 j2 j3 
1

 j3 j3  m3m3
 


  2 j3  1
m1 , m2  m1 m2 m3   m1 m2 m3
 j1 j2 j3   j1 j2 j3 
2
j

1
  3
  m m m   m m m    m1m1  m2m2
j3 , m3
 1 2 3 1 2 3
Explicit (Racah 1942):
Angular Momentum Coupling
 j1 j2 j3 
j1  j2  m3


1
  m1  m2  m3 , 0  

  
 m1 m2 m3 


 j1 
j2  j3  !  j1  j2  j3  !   j1  j2  j3  !

 j1  j2  j3  !
 j1  m1  !  j1  m1  !  j2  m2  !  j2  m2  !  j3  m3  !  j3  m3  ! 
  1  z !  j1  j2  j3  z  !  j1  m1  z  !  j2  m2  z  ! 
z
z
All factorials must be ≥ 0
W. Udo Schröder, 2005
  j3  j2  m1  z  !  j3  j1  m2  z  !
1
Spherical Tensors and Reduced Matrix Elements


Spherical tensor of rank j  2 j  1 operators Tmj (m   j,....,  j )
 j ' transforming like angular  momentum ops.
Tmj   Dmj ' m  ,  ,  T m
17
m'
Tmj transfers angular
momentum to I  0 state
I 0
,
:
M0
Tmj
0
j
,   ,
0
m
, ,  = Qu. #
characterizing
states
In general LC of basis states
Angular Momentum Coupling
Tmj
,
j1
,

m1
j2
m2
Tmj

 , j3 , m3
,
j1
m1

N

j j1 j3
m m1 m3
 , j3 , m3
N
,
j3
N 
m3
j j1 j3
m m1 m3
,
j2
m2
degeneracy
not due to m3
W. Udo Schröder, 2005
m3
j3
j j1 j2
,
 N
m3
m m1 m2
Wigner-Eckart Theorem
,
,
j3
j2
j
j j1 j2
Tmj  , 1  N
 dyn  geometry
m2
m1
m m1 m2
Wigner-Eckart Theorem
j2
j1
j j1 j2
j
,
T ,

N 
m2 m
m1
m m1 m2
18
j2
j1
j
,
Tm  ,
 (1) j2  m2
m2
m1
 j j1 j2 
j

  j2 T  j1
 m m1 m2 
Angular Momentum Coupling
Reduced (double  bar ) Matrix Element
j2
j1
j
 j2 T j  j1 contains all physics.
Conditions for non  zero :
3 angular momenta j1 , j, j2
" couple to zero "  m2  m1  m
Take the simplest ME to calculate
W. Udo Schröder, 2005
 j2 T j  j1
Examples for Reduced ME
Example 1 : const. operator  1
19
,
j2
j
1  , 1    j1 j2  m1m2  (1) j1  m1
m2
m1
 j1 j1 0  (1) j1  m1
Remember 

2 j1  1
 m1 m1 0 

Angular Momentum Coupling
 0 j1 j1 

  j2 1  j1
 0 m1 m1 
 j2 1  j1  2 j1  1   j1 j2

Example 2 : Vector operator  ˆ
J  ˆ
J
,
j2
m2
Jz  ,
j1
m1
 m1  j1 j2  m1m2 
 j 1 j1 
Look up  1

 m1 0 m1 
W. Udo Schröder, 2005

ˆ
J

use
ˆ
ˆ
Jz 

Jz
simplest
(1) j1  m1
 j1 1 j1 

  j2 J  j1

m
m
10 1

(1) j1  m1 m1
j1  j1  1 2 j1  2 
 j2 J  j1 
j1  j1  1 2 j1  2    j1 j2
RMs of Spherical Harmonics
2
m2
YML
1
m1
 L 1 2 
 (1) 2  m2 

m
m

M
1
2

2
YL
1

*
20
  d  Ym2 ( ,  ) YML ( ,  )Ym11( ,  )
2
(2
YML ( ,  )Ym1 ( ,  )  
Angular Momentum Coupling
1
,
2
Y
L
1
1
 1)(2L  1)(2  1)  1 L    1 L    *
Y ( ,  )



4
 m1 M    0 0 0 
 ( ) Y ( , )
 (1)
2
(2
1
 1)(2
2
 2 L 1
 1) 

0
0
0


Important for the calculation of gamma and particle transition probabilities
W. Udo Schröder, 2005
Isospin
Charge independence of nuclear forces  neutron and proton states
of similar WF symmetry have same energy  n, p = nucleons
Choose a specific representation in abstract isospin space:
Angular Momentum Coupling
21
1
Proton : 1 2   
0
0
Neutron : 1 2   
1
0 1
 0 i 
1 0 
Isospin matrices SU(2) 1  
;


;


2
3





1
0
i
0
0

1






1
Nucleon charge q  1   3 
2
1
Isospin operators tˆi :  i (i  1,2,3) analog to spin
2
tˆi , tˆj   i tˆk (cyclic i, j, k )  spherical tensor (vector ) tˆ1 : tˆ  (tˆ1  tˆ2 ), tˆ3


Transforms in isospin space like angular momentum in coordinate space
 use angular momentum formalism for isospin coupling.
W. Udo Schröder, 2005
2-Particle Isospin Coupling
Use spin/angular momentum formalism: t  (2t+1) iso-projections
22
t1
mt1
t2
t1
t2
:
mt 2
mt1 mt 2
Total isospin states T , MT 
0
Iso  antisymmetric :
0
Angular Momentum Coupling
can couple to t1  t2  T  t1  t2
 JM,TMT 
T
MT
 t1 t2 T  t1

 

mt 1 , mt 2  mt 1 mt 2 MT  mt 1
mt 2
T 1
Iso  symmetric :
(MT  1, 0,1)
MT
 j1 j2 J 
T
 
  j1m1 (1) j2m2 (2)  (1)  j1m1 (2) j2m2 (1)T ,MT
2 m1,m2  m1 m2 M 
1
Both nucleons in j shell  lowest E states have even J  T=1 !
For odd J  total isospin T = 0

W. Udo Schröder, 2005
t2
j2 JM,T

j1  j2  j   1
 j1 j2 J 

  jm1 (1) jm2 (2)
m
m
m1, m2  1
2 M

T J
 1
Angular Momentum Coupling
23
Isobaric Analog (Isospin Multiplet) States
W. Udo Schröder, 2005
W. Udo Schröder, 2005
Angular Momentum Coupling
24
25
Electric Quadrupole Moment of Charge Distributions
  r   arbitrary nuclear charge distribution with norm  d3r   r   Z
z
1
Coulomb
V (r )  e2  d 3r   p*  r  
 p  r 
Point
interaction
r  r
Charge e
2

r  r
1 
r 
r  1 
1


2
r
r  r   1    cos    
3 cos   1  ...


|e|Z
r
r
r
2






  r   r 
r  r
1

1   r 
el
2 P2 (cos  )  Q02

r
   P (cos  ) 
quadrupole
r 0  r 
Nuclear Deform
Quadrupole moment Q  T2= Q2 -ME in aligned state m=j
Qz :
j
j
Qz (m) 
Q02
j
m
j
 j j 2
2

 j Q j
j
  j j 0
Q02
Qz (m)  (1) j  m
W. Udo Schröder, 2004
 spectroscopic Qz (m  j )
j 2  j j 2
 j j 2
2
j m   j

j
Q
j

Q
(

1)


 

m

m
m
0

m
m
0

j
j
0


 


j
2
3m  j( j  1)
Q
j(2 j  1)
Look up/calculate
3 j  Symbol  Qz  0 for j  1
Average Transition Probabilities
Fermi ' s Golden Rule : Pi  f 
i
Tk
mf
Tk
ji
mi
f (Ef )
If more than 1 initial state may be populated (e.g. diff. m)
 average over initial states
26
Angular Momentum Coupling
jf
for one i state and f f states
f
1
P 

2 j1  1 mi ,mf
jf
ji
k
T
mf
mi
1
P 
2 j1  1 2k  1
jf T
k
Sum over all components of
2
ji
Tk
2
 ji k jf 
1

 

2 j1  1 mi ,mf  mi  mf 
jf T
k
ji
 const.
2

= total if Tk transition probability
W. Udo Schröder, 2005
2
2
1
P   P 

2 j1  1
jf T
k
ji
2
2
W. Udo Schröder, 2005
Angular Momentum Coupling
27
W. Udo Schröder, 2005
Angular Momentum Coupling
28
Wigner-Eckart Theorem
j2
j1
j j1 j2
j
,
T ,

N 
m2 m
m1
m
m
m 1 2
29
,
j2
j
Tmj  , 1  (1) j2  m2
m2
m1
 j j1 j2 
j

j
T
 j1


2
 m m1 m2 
Reduced Matrix Element  j2 T j  j1 contains all physics.
Angular Momentum Coupling
3 angular momenta j1 , j, j2 " couple to zero ", m2  m1  m
Scalar product of irreducible tensors  jm and  jm :
 j j 0
(1) j  m
 j  m  jm
    0   
  j  m  jm  
m  m m 0 
m 2j 1
Example : Normalization of irreducible set  jm
0
0
 *      *jm  jm   (1) m  j  m  jm   *jm  (1) m j  m

0 m
m
W. Udo Schröder, 2005
Know this for
spherical harmonics
Spherical Tensors and Reduced Matrix Elements


Spherical tensor of rank j  2 j  1 operators Tmj (m   j,....,  j )
 j ' transforming like angular  momentum ops.
Tmj   Dmj ' m  ,  ,  T m
m'
30
Tmj transfers angular momentum to I  0 state  ,
Tmj  ,
I 0
I  j
 ,
M0
Mm
I 0
M 0
 = Qu. # characterizing state
Angular Momentum Coupling
More generally :
T
j
 , j1 : 
m, m1
j1
m1
j2
 N ,
m2
j3
,
Tj
m3
W. Udo Schröder, 2005
 , j1
j j2
j1
j1
j j2
j
T ,
N 
m m2 m
m1
m, m1 m1 m m2
j1 j
,

m1 m
N  normalization factor depending on T j , 
j3
j2
 N ,
,
m3
m2
More General Symmetries: Wigner’s 3j Symbols
Coupling  j1  j2  j3 equivalent to symmetric  j1  j2  j3  0
j2
31
j1
j2
j1
j3
j3 
 j1 j2 j3   j3 j3 0  (1) j3  m3

 j3  m1 m2 m3   m3 m3 0 
2 j3  1
 j1 j2 j3 



m
m
m
3
1
2


Angular Momentum Coupling
Choose additional phase factor for 3 j symbol
 j1 j2 j3  (1) j1  j2  m3


2 j3  1
 m1 m2 m3 
 j1 j2 j3 



m
m
m
3
 1 2
From before:
 j1 j1 0  (1) j1  m1
(1) j  m j
 


m

m
m 2j 1 m
2 j1  1
1 0
 1
(1) j  m
W. Udo Schröder, 2005
j
m

0
0
Invariant
under
rotations
j
j
rot. transforms contragrediently to
m
m
Translations
cmi  m   cni  m VR  n  cmi Ei

Vmn

32

n
m  1,2,3
Angular Momentum Coupling
V(r)
V(x)
W. Udo Schröder, 2005
r
x