Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

FermiGasy
Rotational Matrices
Spherical Tensors
2
z
J
ˆ
Projection Operator onto J : PJ  
M M
J
J
M
M’
Arbitrary f
f :  cLN
L,N
L
J
ˆ
 PJ f :   cLN
N
M
L,N M
J
M
J L
M N
 JL MN
J
ˆ
PJ f   c JM
M
M
PJ operates in J space, keeps
only components in J space
Effect of: (q,f)  (q’,f’) Y J (q , f )  Y J (q , f )  D
ˆ  ,  ,   Y J (q , f )
M
M
M
rotation
DMJ M
J
J
J ˆ
J
J
D

,

,

:

   
  MM 
 ,  ,     D  ,  ,  
M

M
M M
M
M
J ˆ
J
D  ,  ,  
M
M
 PJ
YML (q , f )
W. Udo Schröder, 2005

L
 DMM
M
 ,  ,  
YML(q , f )
“Spherical Tensor”
YML (q , f )
Transform among themselves
under rotations
Spherical Tensors
Because of central potential, states of nucleus with different structure
have different transformation properties under rotations  look for
different rotational symmetries
Spherical tensor Tk (“rank” k) with 3k components
k=0: scalar
k=1: vector
3
Irreducible tensor Tk of “degree” k with 2k+1 components
transforms under rotations like spherical harmonics
Tkm  
Spherical Tensors
m'
k
Dm
'm
 ,  ,  Tkm '
Search for all irreducible tensors
 find all symmetries/exc. modes.
Example tensor Tik of rank 2.
W. Udo Schröder, 2005
J
: tensor of rank 2 J  1
M
 T11 T12 T13 


T   T21 T22 T23 
T

 31 T32 T33 
Irreducible Representations
4
 T11 T12 T13 


T   T21 T22 T23   Trace  Tr Tik   T11  T22  T33 : 3T
T

 31 T32 T33 
Decompose into its trace, symmetric and antisymmetric parts
Tik  Sik  Aik   ik with  ik  T  ik
Sik
1
 Tik  Tki    ik
2
and
Aik
1
 Tik  Tki 
2
Spherical Tensors
1 Trace + 5 indep. symm + 3 indep. antisymm.= 9 components
Each set transforms separately: number, tensor, axial vector
Have different physical meaning
Unitary U : T   UTU 1 
1
1
 Tii   UijT ji   UijT jmUmi   UijUmi T jm   T jj  const.
i
ij
W. Udo Schröder, 2005
ijm
ijm
 jm
j
Example: Spherical Harmonics (Dipole)
Spherical harmonics , irreducible tensor degree k=1 (Vector)
rY11(q , f )  
1 3
1 3 1

r sin q eif 

x

iy



2 2
2   2

rY01(q , f )  
1 3
r cos q
2 
1 3
z
2 
5

rY11(q , f )  
1 3
1 3 1

r sin q e  if 

x

iy



2 2
2  
2

Spherical Tensors
Structure of generic irr. tensor of degree k=1 (Vector) in
Cartesian coordinates:
T11  
1
Tx  iTy 

2
T01  Tz
T11  
W. Udo Schröder, 2005
1
Tx  iTy 

2
Construct irr. representation from
Cartesian coordinates Tx, Ty, Tz,
like spherical harmonics. Then T
will transform like a spherical
harmonic
6
Example: Quadrupole Operator
Construct irreducible tensor from
s.p. coordinate vector
1
Tik : xi x j  x j xi  r 2 ij
3
Trace :  Tii  0  only 5 out of 6 Tik
 x   x1 
   
r   y    x2 
z x 
   3
i
are independent elements  rank  2 tensor
Spherical Tensors
r 2Y02 (q , f )  3z 2  r 2
r 2Y12 (q , f )   z(x  iy ) r 2Y21(q , f )  z(x  iy )
r 2Y22 (q , f )  ( x  iy )2 r 2Y22 (q , f )  ( x  iy )2
Quadrupole operator for a nucleon :
Qk :
W. Udo Schröder, 2005
 irreducible rank  2 tensor
16 2 2
r Yk (q , f )
5
transforms rotationally like Yk2
Example: 2p WF in p-Orbit
V(r)
2l+1= 3 degenerate p states
Formal WF : f (r )  x, f (r )  y, f (r )  z
Spherical Tensors
7
r
ˆ  i  y   x  
L
x
 z
y 

ˆ  i  z   x  
L
y
 x
z 

ˆ  i  x   y  
L
z
 y
x 

cyclic (i, j, k )


ˆ  i  x   x  
L
i
j
k
xk
xi 

ˆ2  L
ˆ2  L
ˆ2  L
ˆ2
L
x
y
W. Udo Schröder, 2005
z
2  particle tensor :  ik : f (r )f (r )xi xk
Trace :
Symm :
  f (r )f (r )  x1x1  x2 x2  x3 x3 
Antisym : Aik
ˆ2   0  
L
1
1
f (r )f (r )  xi xk  xk xi    ik
2
3
1
 f (r )f (r )  xi xk  xk xi 
2
Sik 
 l  0 relative S state
ˆ2 A  2  A  l  1 relative P state
L
ik
ik
ˆ2 S  6  S  l  2 relative D state
L
ik
ik
Addition of Angular Momenta
Angular momenta L1, L2 , direction undetermined
Projections conserved : m1, m2 , m  m1  m2  L1  L2  L  L1  L2
f(t )
q1  const.
q2  const.
f1(t ), f2(t )  L(t ), q (t ), f(t ) 
8
fi (t )  i Larmor frequency
f2
L  L(t )  Classical Probability :
Spherical Tensors
q2
q
P(L)   L t 
1

2
 2L L t

1
L2  m2  L21 sin2 q1  L22 sin2 q2 
q1
 L1L2 sin q1 sin q2 cos(f1  f2 )
If (L1)  (L2 ) dipole interaction : L1 couples with L2  L
Coherent motion : m  conserved, q  const. f  f(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
9
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
Spherical Tensors
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
10

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
Spherical Tensors
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
11
J  ??
m  J 1
 2 j1
j1
j1  1
j2
j1
- 2 j2
j2
j1
j2
is orthogonal
j2  1
Apply ˆ
J2  ˆ
J12  ˆ
J22  ˆ
J1 ˆ
J2   ˆ
J1 ˆ
J2   2 ˆ
J1z ˆ
J2 z
Spherical Tensors
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
12
j
ˆ
Representations of identity operator 1
j1 j2 :

Spherical Tensors
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
13
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

Spherical Tensors
 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
C j1m1 1
14
j2
j1
j
m2 m  1 m1
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
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
Spherical Tensors
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


15
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
Spherical Tensors
 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

16
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
Spherical Tensors
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-(j1=j2) Particle j,m Eigen Function
Look for 2-part. wfs of lowest energy in same j-shell, Vpair(r1,r2) < 0
 spatially symmetric  j1(r) = j2(r). Construct spin wf.
17
 jm
j
  j1   j2  j1 
m
j1  j2  j  j1  j2
 j1 j2 j 
 jm (r1, r2 )   j1 (r1, r2 )  
 j1m1 (r1 ) j2m2 (r2 )
m1, m2  m1 m2 m 
Spherical Tensors
Which total spins j = j1+j2 (or = (L+S)) are allowed?
Exchange of particle coordinates. Spatially symmetric  spin
antisymmetric jz m
 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 
 j1 j2 j 
r1 ) j2m2 (ˆ
r2 )   jm(ˆ
r2 , ˆ
r1 )

 j1m1 (ˆ
m
m
m1, m2  1
2 m
12 jm (ˆ
r1, ˆ
r2 )  12 
W. Udo Schröder, 2005
2-(j1=j2) Particle j,m Eigen Function
Tensor product of sets of spatially symm. WFs for j1  j2 :  jm   j1   j2  j1
 j1 j2 j 
 j1m1 (r1 ) j2m2 (r2 )
m
m
m
m1, m2  1
2

 jm (r1, r2 )   j1 (r1, r2 )  
j1  j2  j  j1  j2
Spherical Tensors
18
Which total spins j = j1+j2 (or = (L+S)) are allowed?
 j j j
j j j
Use Pauli Principle and  1 2
  (1) 1 2
 m1 m2 m 
 j2 j1 j 

 for 1  2
m
m
m
 2 1 
 j1 j2 j 
12 jm (r1 , r2 )  12  
 j1m1 (r1 ) j2m2 (r2 ) 
m1 , m2  m1 m2 m 
Exchange of
particle jz m
 j2 j1 j 
coordinates
  
 j2m2 (r2 ) j1m1 (r1 ) 
m1 , m2  m2 m1 m 
 j j j
 (1) j1  j2  j   1 2
 j1m1 (r1 ) j2m2 (r2 )
m1 , m2  m1 m2 m 
12 jm (r1, r2 )  (1) j1  j2  j jm(r1, r2 )
= antisymmetric !
W. Udo Schröder, 2005
For j1  j2  5 2 
2 j1  j  5  j  odd
 j  0,2, 4
Exchange Symmetry of 2-Particle WF
1) j1 = j2 = half-integer total spins 
states with even 2-p. spin j are antisymmetric
19
states with odd 2-p. spin j are symmetric
2) Orbital (integer) angular momenta l1= l2 
states with even 2-p. L are symmetric
Spherical Tensors
states with odd 2-p. L are 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


Spherical Tensors
20
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
21
Addition Theorem of spherical harmonics :
P  cos q  
4
*
 Ym (q1 , f1 )Ym (q2 , f2 ) 
2 1m
   1
4
4
Y m (q2 , f2 )
Ym* (q1 , f1 )
2 1
2 1
m
m
q 
r1 , r2
Spherical Tensors
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
(
q
,
f
)
e
r




p p m
p
p
i i
 p 2  1
  p 2  1
0
 Tp  Ti   scalar product of separated tensors
0
protons electrons
only
only
W. Udo Schröder, 2005

Y m (q i , fi )

1
Wigner’s 3j Symbols
Coupling  j1  j2  j3 equivalent to symmetric  j1  j2  j3  0
j2
22
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


Spherical Tensors
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
23
 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):
Spherical Tensors
 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
  j3  j2  m1  z  !  j3  j1  m2  z  !
W. Udo Schröder, 2005
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
24
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
Spherical Tensors
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
Spherical Tensors
25
j2
j1
j
,
Tm  ,
 (1) j2  m2
m2
m1
j2
j1
j
 j j1 j2 
j

  j2 T  j1
 m m1 m2 
Reduced (double  bar ) Matrix Element
 j2 T j  j1 contains all physics.
Conditions for non  zero :
3 angular momenta j1 , j, j2
" couple to zero "  m2  m1  m
W. Udo Schröder, 2005
Examples for Reduced ME
Example 1 : const. operator  1
26
,
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 

 j2 1  j1  2 j1  1   j1 j2
Example 2 : Vector operator  J   J
Spherical Tensors
 0 j1 j1 

  j2 1  j1
 0 m1 m1 
,
j2
m2
Jz  ,
j1
m1
 m1  j1 j2  m1m2 
 j1 1 j1 
Look up 


m
m
10 1


W. Udo Schröder, 2005
J
Jz 
(1) j1  m1
 j1 1 j1 

  j2 J  j1
 m1 0 m1 
(1) j1  m1 m1
j1  j1  1 2 j1  2 
 j2 J  j1 
j1  j1  1 2 j1  2    j1 j2
RMEs of Spherical Harmonics
2
m2
YML
1
m1
 L 1 2 
 (1) 2  m2 

m
m

M
1
2

2
YL
1

*
27
  d  Ym2 (q , f ) YML (q , f )Ym11(q , f )
2
(2
YML (q , f )Ym1 (q , f )  
Spherical Tensors
1
,
2
Y
L
1
1
 1)(2L  1)(2  1)  1 L    1 L    *
Y (q , f )



4
 m1 M    0 0 0 
 ( ) Y (q ,f )
 (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 Formalism
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:
Spherical Tensors
28
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
29
t1
mt1
t2
t1
t2
:
mt 2
mt1 mt 2
Total isospin states T , MT 
0
Iso  antisymmetric :
0
Spherical Tensors
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
30
Spherical Tensors
W. Udo Schröder, 2005
 1 2
 1
Addition Theorem N 1 2Ym (r )   
Ym1 (r )Ym22 (r )
m1m2  m1 m2 m 
Spherical Tensors
31
Normalization N 1 2  (1)
W. Udo Schröder, 2005
2
1
 1 2
4
2
 1  1 2 


0
0
0


Wigner-Eckart Theorem
j2
j1
j j1 j2
j
,
T ,

N 
m2 m
m1
m
m
m 1 2
32
,
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.
3 angular momenta j1 , j, j2 " couple to zero ", m2  m1  m
Spherical Tensors
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'
33
Tmj transfers angular momentum to I  0 state  ,
Tmj  ,
I 0
I  j
 ,
M0
Mm
I 0
M 0
 = Qu. # characterizing state
More generally :
Spherical Tensors
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
34
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


Spherical Tensors
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

35

n
m  1,2,3
V(r)
Spherical Tensors
V(x)
W. Udo Schröder, 2005
r
x