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FermiGasy
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
Addition of Angular Momenta
1 2
1 2 1 2 1 2 1 2
, ,
: , ,
Angular momenta L L direction undetermined
Projections conserved m m m m m L L L L L
1 2
1 2
11 2
2 2 2 2 2 21 1 2 2
1 2 1 2 1 2
. .
( ), ( ) ( ), ( ), ( )
( )
( ) :
( ) 2
sin sin
sin sin cos( )
i i
const const
t t L t t t
t Larmor frequency
L L t Classical Probability
P L L t L L t
L m L L
L L
( )t
1 2 1 2( ) ( ) :
: , . ( ) ( )
I f L L dipole interaction L couples with L L
Coherent motion m conserved const t L L t
31 2 1 2At large r r r decoupling
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 3
Angular Momentum Coupling
1 2
1 22 21 1 2 2 1 2
1 2
??
ˆ ˆˆ ˆ, : ; , : dim (2 1)(2 1)z z
Quantal angular momentum eigen states j j
j jJ J J J j j dimensionality
m m
1 2 1 2 1 2 1 22
1 2 1 2
?
1 1
?
2 2
ˆ: : ??j j j j j j j j
Max alignment Jj j j j j j j j
22 2 2 2 2
1 2 1 2 1 2 1 2 1 21 2 1 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 2ˆ ˆˆ ˆ
z zUse J J J J J J J JJ J J JJJJ
1 2 1 22 21 1 2 2 1 2
1 2 1 2
1 2 1 22 21 2 1 2
1 2 1 21
0ˆ ( 1) ( 1) 2
( ) ( 1) ( )
0
1J J
j j j jJ j j j j j j
j j j j
j j j jj j j j J J
j j j j
1 2 1 2 1 2 21 2
1 2 1 2 1 2
ˆˆ ˆ( ) ,z z
The firstj j j j j j
J j j m J Jj j j j j j
eigen state
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 4
Constructing J Eigen States
1 2 1 2 1 2 1 21 2 1 2
1 2 1 2 1
1 2 1 21 21 2
1 2 1 2
2 1 2
1 2
ˆ ,
ˆ ˆ
2 21 1
ˆ 2 21
, 11
1
Construct m J spectrum successively by applying J for example
j j j j j j j jJ J J j j
j j j j j j j
j j j jJ j jj j
j j
j
eigen st
j jm
ate
J J j j m J
+
1 1 2 22 , ,
? 2 .
ˆ:
3
This is one specific linear combination of states j m and j m
What about the other There should be orthogonal combinations
And Further application of J yields again only one specific linear
combination ofi ndependent compo
nents
1 2 1 2 1 21 2
1 2 1 2 1 22 1 1 22
? 3 .What about others There shou
j j j j j jJ j j
j j j j j jm J
ld be orthogonal combinations
+ +
Can you show this??
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 5
Constructing J-1 Eigen States
1 2 1 21 21 2
1 2 1 2 1 2
2, 1
21 11
j j j j eigen staJ tj jj j
j j j
e
J j j m Jjm J
+
1 2 1 21 21 2
1 1 21 2
2
12 2 1 ... .
1 1
j j j jJ j jj j
j j j
eigen state
J j jjm J
etc
m J
-
Normalization conditions leave open phase factors choose asymmetrically <|J1z|> ≥ 0 and <a|J2z|b> ≤ 0
Condon-Shortley
1 1 2 2
1 2 1 21 2
1 2 1 2
, , , 2
??2 2
1 11
Two basis states j m j m new orthogonal states
j j j jJj j is orthogonal
j j j jm J
-
1 22 2 21 2 1 2 1 2 1 2
1 2
1 2 1 21 1 1 1 1
1 2 1 2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2
ˆ 1 .
z z
j jApply J J J J J J J J J to J
j j
j j j jUse J j m j m etc
m m m m
We have this state:
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 6
Clebsch-Gordan Coefficients
1 2 1 2
, , 1 2 1 21 2 1 2
, ,1 2
1 2
1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 21 2
: ( )
1 2
1 2m m m m
m m m m
General scheme Unitary transformation between bases
j jj j j j
m m m m
j j j j j j j j
m m m m
j jj
m m
m m m
m m
m
m
j j
m m
j j j
m m m
1 2 1 2 1 2 1 21 1 2 2, 1 2 1 2 1 2 1 2
m m m mj m
j j j j j j j jj j
m m m m m m m mm m
j,m
=1
j j
m m
1 2 1 2
, 1 2 1 21 2
:
j j m mm m
j j j jj j j j
m
Orthogonality relations of CG coefficient
m m mm m m
s
m
1 2 1 2
m ,m 1 2 1 21 2
=1
j j j j
m m m m
1 2
1 2 1 21 2 1 2, ,1 2 1 21 2
ˆ
ˆ ˆ
:j j
j j j jm m j m
j j
Representations ofi
j j j j
m m m m
dentity operato
m
r
m
1
1 1
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 7
Recursion Relations
1 2 1 2 1 2 1 2
, ,1 2 1 2 1 2 1 21 2 1 2
1 2 1 2
, 1 2 1
1 2 1 2
1 2
21 2
,1 2
1
ˆ 1
1 2
,
ˆ ˆ
ˆ1
m m m m
m m
m m
j m
Pj
m
m
j
j j j j j j j jj j j
m m m m m m m mm m m
j j jJ
j j jJ
m - 1
j j j j
m m m mm
j
m m m
m
m
-j j j j
Jm m - 1 m - 1 m
1 2
1
1 11
1 1
2 2
2
1 2
, 1 21 2
11 1 1 1
1 2
, 1 21 2
12
1
2
2
12
2 2 2
2 2
ˆ ˆ ˆ
ˆ
m m
j m j m
m m
j m j m
j j
m -
j jJ
m - 1 m
j jJ
m - 1 m
1 m
j j
j jj
m m - 1
j
m mm m
j j j
m m m
1- 2-J + J
: 1C j m j mjm
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 8
Recursion Relations for CG Coefficients
: 1C j m j mjm 1 2
1 2
1 2 1
1 2
1 2
1 2 1 21 11 1 2 2
1
2
2 1 21 2 1 2
1
1 1
jm
j m j m
j j jC
m m m
j j j jj jC C
m m
j j
m m
j j j j
m m m mm mm m
1 2 1 2 1 21 11 1 2 2
1 2 1 2 1 21 11jm j m j m
j j j j j jj j jC C C
m m m m m mm m m
1 2 1 2 1 21 1 1 2 2
1 2 1 2 1 21 11jm j m j m
j j j j j jj j jC C C
m m m m m mm m m
1 2ˆ ˆ ˆUsing J J J
1 11 1
1 1 1
0 0
:
( 1)1
0 0 2 1
j mj jj j
m mm m
Special values
j
0???
0
Projecting on <j1,j2,m1,m2| yields
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 9
Symmetries of CG Coefficients
1 2 2 11 2
1 2 2 1
( 1) j j jj j j jj j
m m m mm m
3 31 2 2 132 2
3 31 2 2 11
3 31 2 1 231 1
3 31 2 1 22
3 31 2 1 21 2 3
3 31 2 1 2
2 1( 1)
2 1
2 1( 1)
2 1
( 1)
j m
j m
j j j
j jj j j jjm mm m m mj
j jj j j jjm mm m m mj
j jj j j j
m mm m m m
3 1 2( )m m m m
:
,
Calculate CGs m j
Then use recursion re
starting from max alignme
lations to obtain al j
n
l j
t
m
1 2
1 2
ˆ ˆ,
ˆ ˆ: . 0, 0z z
Coupling depends on sequence J J
Phase convention non diag J J
Triangular relation
Condon-Shortley : Matrix elements of J1z and J2z have different signs
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
0
Explicit Expressions
1 2, 1 2
1 2
1 21 2 1 1 2 2
1 2 1 2 1 2 1 1 2 2
1 1 2 11 1
0 1 1 2 1
1 2
1 2
2 1 !( )!( )!( )!( )!
1 ! ! !( )!( )!
( )!( )!1
!( )!( )!( )!
m m m
j m s j m
s
j j j
m m m
j j j j j m j m j m j m
j j j j j j j j j j m j m
j m s j j m s
s j m s j m s j j m s
j j
m m
1 21 2 1 2 1 2 1 2 1 21 2
1 2 1 2 1 1 1 1 2 2 2 2
1 2
1 1
1 21 1 2 1 2
1 2 1 2 1 2 1 2
2 ! 2 ! ! !
2 2 ! ! ! ! !
2 1 ! 2 ! ! !( )!
! ! 1 ! ! !
j j j j m m j j m mj j
m m j j j m j m j m j m
j j j
j m j m
j j j j j j j m j m
j j j j j j j j j j j m j m
A. R. Edmonds, Angular Momentum in Quantum Mechanics
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
1
2 Particles in j Shell (jj-Coupling)
1 2 1 2 1 21 2 2 1, 1 21 2
( , ) ( ) ( ) ( ) ( )JM jm jm jm jmm m
j j Jr r N r r r r
m m M
0 2j j J j j j J
JM j j M
1 2 2 11 212
1 2 2 1
( 1) : 1 2j j jj j j jj jUse Pauli Principl and for
m mme
m m m
Which J = j1+j2 (and M) are allowed? antisymmetric WF JM
1 2 1 21 2, 1 2 2 11 2
21 21 2, 1 21 2
( , ) ( ) ( )
1 2
1 ( 1) ( ) ( )
JM jm jmm m
j Jjm jm
m m
j j j jJ Jr r N r r
m m m mM M
j j JN r r
m m M
N
Look for 2-part. wfs of lowest energy in same j-shell, Vpair(r1,r2) < 0
spatially symmetric j1(r) = j2(r). Construct consistent spin wf.
N = normalization factor
1 2
1
5 2
2 5
0,2, 4
For j j
j j j odd
j
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
2
Symmetry of 2-Particle WFs in jj Coupling
1) j1 = j2 = j half-integer spins J =even
wave functions with even 2-p. spin J are antisymmetric
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
1 2 1 21 2, 1 21 2
( , ) ( ) ( )JM jm jmm m
j j Jr r r r J even
m m M
Antisymmetric function of 2 equivalent nucleons (2 neutrons or 2 protons) in j shell in jj coupling.
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
3
Tensor and Scalar Products
1 2
1 21 2 1 21 2, 1 21 2
(1) (2) (1) (2)
k k
kk k k kk
Tensor product of sets of tensorsT and T
k k kT T T T T
000 0
(1) (2)
0 ( 1)(1) (2) (1) (2) (1) (2)
0 2 1
k k
kk k k k k k
Scalar product of sets of tensorsT and T number
k kT T T T T T T
k
0
0
: 1 2 1 3
0
1
3 2 2 2 2
1
3
x y x y x y x yz z
Vectors u and v Rank k k spherical components
Spin scalar product
u iu v iv u iu v ivu v u v
u v
Transforms like a J=0 object = number
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
4
Example: HF Interaction
*1 1 2 2 1 2
*1 1 2 2
:
4cos ( , ) ( , ) ,
2 1
4 41 ( , ) ( , )
2 1 2 1
m mm
mm m
m
Addition Theorem of spherical harmonics
P Y Y r r
Y Y
*
int 1, ,
,
0
1
0
4( , )
2 1
( )
ˆ ( )
( , )1
14
2p p
i p i pp
i p i pi
mm mi ip p
p
p
mi i
i
i
e r Y
Electron nucleus sum over p hyperf
e r Y
ine interactions
e e r rH r P
r rr
scalar product of sepa tT rT a ed t
ensors
protons electrons only only
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
5
Wigner’s 3j Symbols
1 2 3 1 2 3 0Coupling j j j equivalent to symmetric j j j
1j
2j
3j
1j
2j
3 3j j
3 33 3 3 31 2 1 2
3 3 3 31 2 1 23
0 ( 1)0 2 1
j mj j j jj j j j
m m m mm m m mj
1 2 33 31 2 1 2
3 31 2 1 23
3
( 1)
2 1
j j m
Choose additional arbitrary phase factor for j symbol
j jj j j j
m mm m m mj
3 3 31 2 1 2 2 1
3 3 31 2 1 2 2 1
j j jj j j j j jall cyclic
m m mm m m m m m
3 31 2 2 11 2 3
3 31 2 2 1
( 1) 2j j jj jj j j jany columns
m mm m m m
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
6
Explicit Formulas
3 33 3 3 3
3 3 3
1 2 1 23
1 2 1 2
, 1 2
1 1 2 21 2 1 2
1 21 2
3 3
, 3 33 3
12 1
2 1
m m
j m
j j m m
m m m m
j j
m
j j j j
m m m
j
m j
j j j j j
mj
m m m m
m
m
31 2 1 2 31 2 3
31 2
1 2 3 1 2 3 1 2 3
1 2 3
1 1 1 1 2 2 2 2 3 3 3 3
1 2 3 1 1 2 2
13 2 1 3 1 2
1 , 0
! ! !
!
! ! ! ! ! !
1 ! ! ! !
! !
j j m
z
z
jj jm m m
mm m
j j j j j j j j j
j j j
j m j m j m j m j m j m
z j j j z j m z j m z
j j m z j j m z
Explicit (Racah 1942):
All factorials must be ≥ 0
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
7
Spherical Tensors and Reduced Matrix Elements
' '
'
2 1 ( ,...., )
, , .
jm
j j jm m m m
m
Spherical tensor of rank j j operators T m j j
T D T transforming like angular momentum ops
0 0, : , ,
0 00
jjm
mI jT transfers angular
TM mmomentum to I state
= Qu. # characterizing states
3 3 31 1
, , 3 3 3 31 13 3
3 32 1 1 2 1 2
, , 3 32 1 1 2 1 23 3
, , ,
, , , ,
jm
j m
jm
j m
In general LC of basis states
j j degeneracy jj jjT N N
m m not due to m mm mm
j jj j j j j jj jT N N
m mm m m m m mm m
2 1 1 2
2 1 1 2
, ,jm
j j j jjT N dyn geometry
m m m mm
Wigner-Eckart Theorem
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
8
Wigner-Eckart Theorem
2 1 1 2
2 1 1 2
, ,jm
j j j jjT N
m m m mm
2 1 1 22 22 1
2 1 1 2
, , ( 1) j mj jm
j j j jjT j T j
m m m mm
1
2 1
2
2
1
( )
.
:
3 , ,
" "
j
Reduced double bar Matrix Element
contains all physics
Conditions for non zero
angular momenta j j j
couple to
j
zero m
j
m
T
m
1j
2j
j
Take the simplest ME to calculate 2 1jj T j
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 1
9
Examples for Reduced ME
2 1 1 11 12 11 2 1 2
2 1 1 1
1 11 1
1 1 1
2 1 1 1 2
: . 1
0, 1 , ( 1) 1
0
0 ( 1)0 2
1
1
1 2 1
j mj j m m
j m
j j
const operator
j j j jj j
m m m m
j j
m
Ex
Remm
embe
ample
j j j
rj
2 1 1 11 11 2 11 2 1 2
2 1 1 1
1 11 1 1
1 1 1 1 1
2 1 1 1 1 1 2
1, , ( 1)
0
1 ( 1)
0 1 2 2
1 2 2
j mz j j m m
j m
j j
Look up
j j j jJ m j J j
m m m m
j j mm m j j j
j J j j j j
ˆ ˆ ˆ ˆ ˆ:2 usez zsimplest
Vector operator J JExample J J J
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
0
Reduced MEs of Spherical Harmonics
2 1 1 22 22 1
2 1 1 2
*2 112
( 1)
( , ) ( , ) ( , )
mL LM
LM mm
LY Y
m m m mM
d Y Y Y
1 *1111 , 1
( ) ( , )
(2 1)(2 1)(2 1)( , ) ( , ) ( , )
4 0 0 0LM m
Y
L LLY Y Y
m M
2 122 1 1 2( 1) (2 1)(2 1)
0 0 0L L
Y
Important for the calculation of gamma and particle transition probabilities
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
1
Isospin
Charge independence of nuclear forces neutron and proton states of similar WF symmetry have same energy n, p = nucleonsChoose a specific representation in abstract isospin space:
1 2 1 2
1 2 3
3
1 0:
0 1
0 1 0 1 0(2) ; ;
1 0 0 0 1
11
21ˆ : ( 1,2,3)2
ˆ ˆ ˆ, ( , , )
i i
i j k
Proton : Neutron
iIsospin matrices SU
i
Nucleon charge q
Isospin operators t i analog to spin
t t i t cyclic i j k
1
1 2 3ˆ ˆ ˆ ˆ ˆ( ) : ( ),spherical tensor vector t t t it t
Transforms in isospin space like angular momentum in coordinate space use angular momentum formalism for isospin coupling.
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
2
2-Particle Isospin Coupling
1 2 1 21 2 1 2
1 2 1 2
:t t t t
t t t tcan couple to t t T t t
m m m m
Use spin/angular momentum formalism: t (2t+1) iso-projections
1 2 1 2,
, 1 2 1 21 2T MT m m t t t tT Tt t
t t t tT TTotal isospin states
m m m mM M
0, 0 1 2 1 2 1 2 1 2
1, 1 1 2 1 2 1, 1
0: 1( )
0
0 0 1(1) (2) (2) (1)
0 0 2
1: ( 1, 0,1) 3( )
1 1 1 1(1) (2)
1 1 1 1
T
T MT
TT
T M T MT T
TIso antisymmetric singlet state
M
TIso symmetric M triplet states
M
1 2 1 2
1, 0 1 2 1 2 1 2 1 2
(1) (2)
1 1 1(1) (2) (2) (1)
0 0 2T MT
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
3
2-Particle Spin-Isospin Coupling
1 2, ,1 1 2 2 1 1 2 2, 1 21 2
1(1) (2) ( 1) (2) (1)
2J T
JM TM j m j m j m j m T MT Tm m
j j J
m m M
Both nucleons in j shell lowest E states have even J T=1 !
For odd J total isospin T = 0 1 2 1 1T J
j j j
2 1,, 1 ,1 2
0 1
0(1) (2) (2) (1) 1, 0,1
0 jm j m jm j m T M TTj J M T MT m m
J T
j jM
m m
3 states (MT=-1,0,+1) are degenerate, if what should be true (nn, np forces are same)
ˆ ˆ, 0H T
Different MT states belong to different nuclei T3 = (N-Z)/2
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
4
2-Particle Isobaric Analog (Isospin Multiplet) States
Corresponding T=1levels in A=14 nuclei
T3=-1
2n holes
T3=+1
2n
T3=0, pn
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
5
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
6
Separation of Variables: HF Interaction
*1 1 2 2 1 2
*1 1 2 2
:
4cos ( , ) ( , )
2 1
4 41 ( , ) ( , )
2
,
2 1 1
m mm
mm m
m
Addition Theorem of spherical harmonics
P Y Y
Y Y
r r
*
int 1, ,
,
0
1
0
4( , )
2 1
( )
ˆ ( )
( , )1
14
2p p
i p i pp
i p i pi
mm mi ip p
p
p
mi i
i
i
e r Y
Electron nucleus sum over p hyperf
e r Y
ine interactions
e e r rH r P
r rr
scalar product of sepa tT rT a ed t
ensors
protons electrons only only
W. Udo Schröder, 2004
Nu
clear
Defo
rm 2
7
Electric Quadrupole Moment of Charge Distributions
|e|Z
e
z
r
r
r r r
r
arbitrary nuclear charge distribution with norm 3d r r Z
Coulomb interaction
2 3 * 1( ) p pV r e d r r r
r r
2 22 0
21 2
1
0
1 11 cos 3cos 1
(cos
...2
1(c s )o ) el
quadrupole
r rr r
r r r
rr PP Qr
r rr
Point Charge
Quadrupole moment Q T2= Q2 -ME in aligned state m=j
2 20
2 20
2
2: ( )
0
2 2 2( ) (
3 ( 1)( ) ( 1)
(2 1)
1)0 0 0
j mz
z z
j mz
j j j jQ Q j Q j spectroscopic Q m j
j j j j
j j j j j j j jQ m Q j Q j Q
m m m
m j j
m m
Q m Qj j
m j j
Look up/calculate
3 0 1zQj Sy ol rmb fo j
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
8
Angular-Momentum Decomposition: Plane Waves
Plane wave can be decomposed into spherical elementary wavesk
r
r k
z
cos
0
: ( , )ikz ikrm m
m
m m
in z direction e e c Y
symmetric about z axis c
cos0 0
0( ) 2 sin ( ) 4 (2 1) ( )ikrc r d Y e i j kr
Spherical Bessel function ( )j kr
cos0 0 0( , ) 4 (2 1) ( ) ( , )ikz ikre e c Y i j kr Y
*
:
4 (2 1) ( ) ( , ) ( , )
m
mik rm r r m k k
m
For arbitrary direction k use Addition Theorem for Y
e i j kr Y Y
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 2
9
j-Transfer Through Particle Emission/Absorption
P
T C N
p+T
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 3
0
Average Transition Probabilities
22
' : ( )fi ki ff f
fi
f
j jFermi s Golden Rule P T E
m m
for one i state and f states
f
i
kT If more than 1 initial state may be populated (e.g. diff. m) average over initial states
22
,1 1.
,
2
1
21 1
2 1 2 1
12 1 2 1
fi k kfi
m m fii fcons
i f
m m i ft
i f
kfi
j jj jP T j T j
m mj j
P j T
m
j
k
m
jk
21
2 1k
fii
P P j T jj
Sum over all components of Tk
= total if Tk transition probability
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 3
1
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 3
2
W. Udo Schröder, 2005
An
gu
lar
Mom
en
tum
Cou
plin
g 3
3
Translations
1,2,3mi m ni m R n mi in
Vmn
c c V c E m
x
V(x)
r
V(r)