FermiGasy. W. Udo Schröder, 2005 Angular Momentum Coupling 2 Addition of Angular Momenta

FermiGasy

W. Udo Schröder, 2005

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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


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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


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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??


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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:


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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 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


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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


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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


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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


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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 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


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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.


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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 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


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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


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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


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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


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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


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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

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


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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


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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


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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.


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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


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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


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2-Particle Isobaric Analog (Isospin Multiplet) States

Corresponding T=1levels in A=14 nuclei

T3=-1

2n holes

T3=+1

2n

T3=0, pn


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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


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


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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


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j-Transfer Through Particle Emission/Absorption

P

T C N

p+T


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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


An

gu

lar

Mom

en

tum

Cou

plin

g 3

1


An

gu

lar

Mom

en

tum

Cou

plin

g 3

2


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)

Documents

FermiGasy. W. Udo Schröder, 2005 Angular Momentum Coupling 2 Addition of Angular Momenta