Ferm iGasy
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Rotat ional Mat r ices
Effect of: ( , ) ( , )rotation
PJ operates in J space, keeps only components in J space
( , ) , , ( , )L L LM MM M
MY D Y
, , : , ,JMM
M M
PJ
J J J JD D
M M M M
, ,
:
: :
JM
LN J LNL N L N M
JL MN
J JProjection Operator onto J P
M M
L J J Lf c P f c
N M M N
, , , ,JMM
J JD D
M M
J JMM
JP f c
M
( , ) , , ( ,, ) )( J JM M MJ Y YY D
Jz
MM
J
Spherical Tensor
Transform am ong them selvesunder rotations
( , )LMY
Arbitrary f
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Spherical Tensors
' ''
, ,kkm m m km
mT D T' '
', ,k
km m m kmm
T D T
Spherical tensor Tk ( rank k) with 3k components
Irreducible tensor Tk of degree k with 2k+1 componentstransforms under rotations like spherical harmonics
k=0: scalark=1: vector
: 2 1J
tensor of rank JM
Because of central potential, states of nucleus with different structure have different transformation properties under rotations look for different rotational symmetries
Search for all irreducible tensors find all symmetries/exc. modes.
Example tensor Tik of rank 2.
11 12 13
21 22 23
31 32 33
T T T
T T T T
T T T
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I r reducible Representat ions
11 12 13
21 22 23 11 22 33
31 32 33
: 3ik
T T T
T T T T Trace Tr T T T T T
T T T
,
1 12 2
ik ik ik ik ik ik
ik ik ki ik ik ik ki
Decompose into its trace symmetric and antisymmetric parts
T S A with T
S T T and A T T
1
1 1
:
.ii ij ji ij jm mi ij mi jm jji ij ijm ijm j
jm
Unitary U T UTU
T U T U T U U U T T const
1 Trace + 5 indep. symm + 3 indep. antisymm.= 9 components
Each set t ransform s separately: num ber, tensor, axial vector
Have different physical m eaning
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Exam ple: Spherical Harm onics (Dipole)
11
10
11
1 3 1 3 1( , ) sin
2 2 2 2
1 3 1 3( , ) cos
2 2
1 3 1 3 1( , ) sin
2 2 2 2
i
i
rY r e x iy
rY r z
rY r e x iy
Spherical harmonics , irreducible tensor degree k= 1 (Vector)
Structure of generic irr. tensor of degree k= 1 (Vector) in Cartesian coordinates:
11
10
11
1
2
1
2
x y
z
x y
T T iT
T T
T T iT
Construct irr. representation from Cartesian coordinates Tx, Ty, Tz,like spherical harmonics. Then Twill transform like a spherical harmonic
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Exam ple: Quadrupole Operator
2 2 2 2 2 2 2 20 1 1
2 2 2 2 2 22 2
( , ) 3 ( , ) ( ) ( , ) ( )
( , ) ( ) ( , ) ( )
r Y z r r Y z x iy r Y z x iy
r Y x iy r Y x iy
2
. .
1:
3: 0 5 6
2
ik i j j i ij
ii iki
Const ruct irreducible tensor from
s p coordinate vector
T x x x x r
Trace T only out of T
are independent elements rank tensor
1
2
3
x x
r y x
z x
2 216: ( , )
5k kQ r Y
:Quadrupole operator for a nucleon
2
2
k
irreducible rank tensor
transforms rotationally like Y
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Exam ple: 2p WF in p-Orbit
r
V( r) 2l+1= 3 degenerate p states
: ( ) , ( ) , ( )Formal WF f r x f r y f r z
2 : : ( ) ( )ik i kparticle tensor f r f r x x
1 1 2 2 3 3: ( ) ( )
1 1: ( ) ( )
2 31
: ( ) ( )2
ik i k k i ik
ik i k k i
Trace f r f r x x x x x x
Symm S f r f r x x x x
Antisym A f r f r x x x x
2
2
2
0 0
2 1
6 2
ik ik
ik ik
L l relative S state
L A A l relative P state
L S S l relative D state2 2 2 2
( , , )
x
y
z
i j kk i
x y z
L i y xz y
L i z xx z
L i x yy x
cyclic i j k
L i x xx x
L L L L
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Addit ion of Angular Mom enta
1 2
1 2 1 2 1 2 1 2
, ,
: , ,
Angular momenta L L direct ion undeterm ined
Project ions 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 interact ion L couples with L L
Coherent mot ion m conserved const t L L t
31 2 1 2At large r r r decoupling
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Angular Mom entum Coupling
1 2
1 22 21 1 2 2 1 2
1 2
??
, : ; , : 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
: : ??j j j j j j j j
Max alignm ent 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 22 2 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
0( 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
( ) ,z z
The firstj j j j j j
J j j m J Jj j j j j j
eigen state
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Const ruct ing 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
,
2 21 1
2 21
, 11
1
Const ruct m J spect rum 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 .
:
3
This is one specif ic linear combinat ion of states j m and j m
What about the other There should be orthogonal combinations
And Further applicat ion of J yields again only one specific linear
combinat ion of independent components
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 combinat ions
+ +
Can you show this??
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Const ruct ing 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 chooseasymmetrically < | 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
2
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:
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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 t ransformat ion 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 relat ions of CG coefficient
m m mm m m
s
m1 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
:j j
j j j jm m j m
j j
Representat ions of i
j j j j
m m m m
dent ity operato
m
r
m
1
1 1
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Recursion Relat ions
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
1
1 2
,
1
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
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
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Recursion Relat ions for CG Coefficients
: 1C j m j mjm1 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 2Using 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
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Sym m etr ies 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 jj
m 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
start ing from max alignme
lat ions to obtain al j
n
l j
t
m
1 2
1 2
,
: . 0, 0z z
Coupling depends on sequence J J
Phase convent ion non diag J J
Triangular relation
Condon-Shortley : Matrix elements of J1z and J2z have different signs
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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
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2- ( j 1= j 2) Part icle j ,m Eigen Funct ion
1 21 2 1 2 1 21 1 1 2 2, 1 21 2
( , ) ( , ) ( ) ( )jm j j m j mm m
j j jr r r r r r
m m m
1 2 1 2j j j j j1 2 1
j
jm j 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 total spins j = j 1+ j 2 (or = (L+ S) ) are allowed?
1 21 2 1 2 2 11 1 2 2, 1 21 2
12 12( , ) ( ) ( ) ( , )jm j m j m jmm m
j j jr r r r r r
m m m
Exchange of part icle coordinates. Spat ially symmetr ic spin antisymmetric j z m
Look for 2-part. wfs of lowest energy in same j -shell, Vpair(r1,r2) < 0spatially symmetric j1( r) = j2( r) . Construct spin wf.
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2- ( j 1= j 2) Part icle j ,m Eigen Funct ion
1 21 2 1 2 1 21 1 1 2 2, 1 21 2
( , ) ( , ) ( ) ( )jm j j m j mm m
j j jr r r r r r
m m m 1 2 1 2j j j j j
1 2 1 2 1. : jm j j jTensor product of sets of spat ially symm WFs for j j
1 2 2 11 2
1 2 2 1
( 1) 1 2j j jPauli Principj j j jj j
Use and form m
em
lmm m
Which total spins j = j 1+ j 2 (or = (L+ S) ) are allowed?
1 21 2 1 21 1 2 2, 1 21 2
2 12 12 2 1 1, 2 11 2
1 21 21 21 1 2 2, 1 21 2
12 12( , ) ( ) ( )
( ) ( )
( 1) ( ) ( )
jm j m j mm m
j m j mm m
j j jj m j m
m m
j j jr r r r
m m m
j j jr r
m m m
j j jr r
m m m
1 212 1 2 1 2( , ) ( 1) ( , )j j j
jm jmr r r r 1 2
1
5 2
2 5
0, 2, 4
For j j
j j j odd
j= antisymmetric !
Exchange of particle j z mcoordinates
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Exchange Sym m etry of 2-Part icle WF
1) j 1 = j2 = half- integer total spinsstates with even 2-p. spin j are antisymmetric
states with odd 2-p. spin j are symmetric
2) Orbital ( integer) angular m om enta l1= l2
states with even 2-p. L are symmetric
states with odd 2-p. L are antisymmetric
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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 tensors T 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 tensors T 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
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Exam ple: HF I nteract ion
*1 1 2 2 1 2
*1 1 2 2
:
4cos ( , ) ( , ) ,
2 1
4 41 ( , ) ( , )
2 1 2 1
m mm
mm m
m
Addit ion Theorem of spherical harmonics
P Y Y r r
Y Y
1
int 1
0
*
, ,
0
( )
14
( ,4
( , )2 1
)2 1
i p i pp
i p i pi
mm i i i i
pp p p p
p
p
m
i
m
Elect ron nucleus hyperfine interact ions
e e r rH r P
r rr
scalar product of
e r
separated te
Y
nT
e r Y
T sors
protons electronsonly only
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Wigner s 3j Sym bols
1 2 3 1 2 3 0Coupling j j j equivalent to symmetric j j j
1j
2j
3j1j
2j
3 3j j3 3
3 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 addit ional arbit rary 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
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Explicit Form ulas
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):
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Spherical Tensors and Reduced Mat r ix Elem ents
' ''
2 1 ( , ...., )
, , .
jm
j j jm m m m
m
Spherical tensor of rank j j operators T m j j
T D T transform ing like angular momentum ops
0 0, : , ,
0 00
jjm
mI jT t ransfers 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
I n 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
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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
Condit ions for non zero
angular momenta j j j
couple to
j
zero m
j
m
T
m
1j
2j
j
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Exam ples 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, , (
2
1)0
1 ( 1)
0 1 2 2
1 2 2
z
j mz j j m m
j m
j j
Look up
Vector operator J J J J
j j j jJ m j J j
m m m m
j j mm m j j j
j J j
Example
j j j
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RMEs of Spherical Harm onics
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 0L
M m
Y
L LLY Y Y
m M
2 122 1 1 2( 1) (2 1) (2 1)
0 0 0L L
Y
I m portant for the calculat ion of gam ma and part icle t ransit ion probabilities
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I sospin Form alism
Charge independence of nuclear forces neutron and proton states of sim ilar WF symm etry have sam e 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
21
: ( 1, 2, 3)2
, ( , , )
i i
i j k
Proton : Neut ron
iI sospin matrices SU
i
Nucleon charge q
I sospin operators t i analog to spin
t t i t cyclic i j k s 11 2 3( ) : ( ) ,pherical tensor vector t t t t t
Transform s in isospin space like angular m om entum in coordinate space use angular momentum formalism for isospin coupling.
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2-Part icle I sospin 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
10: : ( 1, 0, 1)
0 TT
TI so ant isymmetric I so symmetric M
M
1 2, ,1 1 2 2 1 1 2 2, 1 21 2
1(1) (2) ( 1) (2) (1)
2T
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
1 22 1 2, , 1 21 2
(1) (2)jm jmj JM T m m
j j J
m m M