fermigasy - university of rochester€¦ · jmjm j jjjjjjmjm jjjjjjjjj jjmjm a. r. edmonds, angular...

Ferm iGasy

W. Udo Schröder, 2005

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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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0

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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2

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

fermigasy - university of rochester€¦ · jmjm j jjjjjjmjm jjjjjjjjj jjmjm a. r. edmonds, angular...

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