Angularm_utmadd.tn@

• When we put tw systems tegethr ,how does angular momentum add up

?

Example : Orbital + 5pm angular mountain at electron

I = Lots = Eat + Io 5

twos tales :

(1) Il,

s ; Me,

Ms > 7

eigustnks et L2,

52,

Lz ,Sz

(21 lj ,e. s ; m > } HnYatdm?these has

eigenstates et J'

,L2

,52

,Jz .

Note that #, Lz ] to,

[ J2,

Sz ) to

Exanfpk : Two Sph -

' K spms

5=5 , +5,= § at + Iasi

5,z ,

Su basis : IMMD} Agan,

how author related.5

, Sz basis : Is,

M >

Here .si#r:ls=I ,mil ) = |m , it

,m

.= E) I ITT 7 .

Is =L,

m= . I > = Itt )

Isil,

who > = f. [Itt ) + Hp )

Is :O, m :O > = to [ Itt) - It T ) ]

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• How to determine such relationships more generally ?

General problem-

'

.

Consider tensor product V, QVZ

,when :

Hi has angular momentum F,with Ji = jilj , +1 )

TgAssuming V

, ,V. are

irreps .There is no

V ,n " ' ' Jr 1 ' Ji = jz ( jztl ) real loss et

generality in dry this,

Basisnks : ljij , ; M, ,m . > = In

, , mz > ( Don't always write ji , j . . )

M,

is eigenvalued Jiz.

.

Wide : dim ( V,

OXV,) = ( Zj ,

+ 1) ( Zjztl )M ,

n n " Jsz .

( Total # et basis States. )

Note that Ji = j.lj.tl ) ; Ji = jzljztl ) → these operates are

proportional te the identity .

We are interested on egenstates at J'

,Jz

Basis : I

j ,m > ; J

'

ljim ) = jljtllljim > ; Jzlj ,m ) .cm/j,m )

Q : Do jim quantum numbers uniquely label stales ? ( Yes. )

Q : What is lj , m > in term at ljiji ; M, ,m , > States ?

Note : [J2 , Jiz ] to ,[ J2

, Jzz ] to ⇒ Cannot simultaneouslydiagonal ite J

' and Jiz,

Jzz

⇒ two bases are different .

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• Whyte : Rotational symmetry expected de hold for combined

system,net its individual parts .

So total angular memutun J-

should commute with H .But angular momenta

J, ,J , can be approximately conserved

,or might just

be convenient variables to describe States.

So useful to

know how eigenstaks at JYJ £ are related to egmsbaks

of Ji,

Ji,

Jia,

Jzz.

• Reink : V, and V. are rotation maps ,

but in general V, Ok is not

.

⇒ V, OXVZ = R

,Ot R ,

a - . a Rk,

where Ri are meps .

i. ⇐'

Efforts.

A. Emirate.

Tim.

This remark shows the same question ( at adtny quantum numbers )makes sense for any symmetry .

It's related to the math problem et

decomposing tenser products at imps into imps .

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First : Show j , m uniquely Label basis States.

mmm

Ex_avpk : j , =jz =L .

States with m = 2 : I m,

:L,

m,

:L ) ( 1 state )

Mil'

. IM,

:L,

m.io > ; IM, :o) mail ) ( 2 States )

Mio : 3 slates

M :-| : 2 States

m : . 2 : 1 State .

Nile : Jtlm ,=L

,mil > = 0 ⇒ JZIM , =L

,mail ) = jljtl ) 11,1 > with jzz .

j¥.

j= 1 j=o

Mm 1 1,1 >mail.I.tt;Y=a"'it " " '|a¥÷"got:#

" " "( ninth .mn "k . >

mi - I J -311,1 >

M : - 2 J -411,1 )

To tmd the jil States, we mtru that the m=1 state orthgudto

J.

11,17 must have J+ 147=0 ,since there can't be two m=2

stats .

So this state has j =L .

Thin jiz and jil columns exhaust 2 of 9 Mio states, last one

Must have j=o .

⇒ j , m uniquely label States .

( Easy to see that it weeks sane wayfor

any dare of j , ,j{ . )

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-7Same analysis for general ji ,jL ⇒ j = lji - jst, lj ,

- jsltl ,...

, jitjz .÷of : How an the two bases related

-

keyqu_ant.ly : ( jiji ; M, .milj ,

m > = ( m,,m . lj , m >

" Clebsch - Gordan coefficients "

1 j , m > = §m Em,

( m, mzlj ,m ) |m , , mi > .

Pnpeftes : (1) Only defined for j= lji - jsl , ... , jitjz

(2) ( m, ,m, Ij ,m > = 0 it M t m ,

+ mi

Oe ( m , ,mz| ( Jz - Jiz - Jsz ) |j ,m ) = ( M - m

,- mz ) ( m

, ,m . lj , m )

.

= 0 .

(3) ( m, , mlj ,m > is real

.this is a Convention .

The matrix Mm, ,m . ; ; ,m

is unitary and real ⇒ orthogonal matrix ⇒

(4) §y,

( M, , Mzljim ) ( M

,.mil j

'

,m

' > = Sjj ' 8mm '

§÷§. ;

( m, , ml jim ) ( mi , miljim ) = Sm

, ,m ,

' Sm. ,m .

'.

(5) ( j , ,j . j ,

I j , j7 is positive .( Another Convention .)

Determine

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Recall Jtljim > t.FM#Hllj,m+l >

=ljtmlljtmtn lj ,m±l >

Butts : J± ljim ) =Fi±tJz±)ljim )

=lJi±+Jz± )

?fmlmimi) (

miimiljim)

/ /

= Fpm,µTImi±m'Ilmithmikmiimiljim )

+ Fimilft .me#lmi.miHKmi.miljim > ] .

Taking inner.

'

I awee :

FjImTj±m+I'

( m,,mzlj,m±l )

=FjFmTj±mT( m,

-

+ 1. m.lj.my | Rcursren relations,

t -ljsimztllljztmz ) ( M, ,MzF1|j,m )

Greener:

Mnmitlim ( RHS )

www.?iIIY....i:itI.mnnn"M

( LHS )

Atmel�7�

myj - ji , ji

, j

j ,-1

, j - jitl , j

edgy,,m=jtl

,forbidden ( zero )\

.

, .ua ,

L.

" f i

Rus equation gives if( j ,

- I ,j - jitlljj ? in terms of ( j , ,j - j , lj , j )

⇒ Determines all ( mi , mzl j ;D Up to normalization .

jitjs . j

⇒ Fix normal Rutan by [{0¢j ,- k

, j - j ,+ Kljij)/2 =L

Then use µ_ equation to determine coefficients with M=j - I,

etc.