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In the theory at angular momentum,

a special role is played bytenser operates . Kmwmy about tenser operates simplifies the calculatorof some matrix elements , and they are related to selection rules .

A set at operators OI ( with I some index ) is a tenser operateit :

exp ( II. F) O , expf : EE ) = 4)l8D*±s Os = Osftoys.

when DCI ) is a ap at the rotation gap

. D 18 ) may or may net be an inep . If D (8) is an imp ,we

say OI is an irreducible tenor.

• If the OI are Hwmitnn,

D (8) must be real ⇒

exp lies IOI exp ties ) =[DTODI , Os .

Writing 1) ( Of = exp tilt ' J ')

,when ( JP

,J't ]= ieijkj 't

. , we have :

[ Ji,

O :] = Os [ II.]s± . ( Get this by expanding to firstorder in 8.)

What 's special wheat tense operates ?

Let I Him) he has :S States for angular meantime imp with J2

ij Cjtl )

with ( Him IF 14 in ' ) = Jfnnm,

1 angular nomutunj matrices ).

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Casey : OI 14in >

Ji OI 14in > = [ Ji,

OI ] Him ) + OIJ ; Him >

= ( Os # let Him > + O± 14in ' > Hi'

]m 'm

= (Os Mimi > ) ( IT ? ] # Sam + S# l JHI ]m 'm )

This is the same action et angular momentum as in a state

II ) 0×14 in ) ,when F = J '

a I + I 0×5 's'

'.

⇒ Tensor operators add angular momentum just like taking the

tenser product at two states.

mm

Exacts :

Exit : Vector operator vi.ci#ViEi&J=RijCOJvj[ Ji

, Vj ] = i Eijkvk . Irreducible tenser ( j = 1 imp . )

As we did before,

can also cheek j :L by computing [ Ji,

1 Ji , vj ] ]

= Zvj .

= llltllv ;

with l= 1.

EXI : Scalar operates .

ei&TSei8± = S ⇐ [ J; ,

s ] = O .

Irreducible tenser with j=o .

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EI : Cartesian tensor operate , e.g. Ti ;k ( can haveany # at indus )

ei&T Tijk Eit's

= 12181in. ' 1218 ljjl RC8) uk' Tijk .

C⇒ [ Ji, Tjke ] = if Eijjl Tyne + E iv. n' Tjh 'd + Eieeitjke ' ] .

Not irreducible without extra conditions atjh .

Can show a Cartesian tenser is irreducible if :

(1) It's completely symmetric , e.g. Tijh=Tjik=Tikj =Tkj ; etc

(2) It is tnuess over any parrot intern,

e.g. Sij Tijk = °.

~

fiktijk = 0 .

mmmm

SphericaltensrsieCartesian tensors often inconvenient to work with

,even when they an

irreducible

•"

Spherical tensors" transform under rotations like Yen 's

.

A spherical tense with angular momentum K is a set at operators

Tlklq ,with q= - k

,- KH

, ... ,K

, ( k ~j , qrm ).

when :[ Ji ,TWg]= Thai Ht "

]qq .

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This implies : ei&T Tag e-it's

= Tlkgt AMI . 8D qq .

In # A : [ Jz,

TWG ]= 9th .

[ J± , Tklg ] = Fktfktf Tfh'±i .

Spheroid tenses from spherical harmonics.

We can use spherical hormones togive Us many examples et

spherical hormones.

• S.nu Yen = Yin 10,01 ,we can think of Yin as a function at

a unit vector it, e.g. Yinln )

.Then we can replace A by any

vector we Want .

• Exude : lit Yilnl if,

na Yi'

In = Ifstalnxtiny )

Nn let 8 be aey vector operator ,then Tkgilm

"

= Yd ( i ) is a

kit spherical denser :

Yilv ) = Fyavz ; Y,

"= IFE

,lvxtivyl .

If we put 8 = F ( position operate ),

then these operators measure

the dipole moment of the probability density ( or charge density ,smu

this is proportional to probability density for a charged pwtrk ).

Similarly ,Yzm ( F ) are operators measuring th quadruple meant .

Winger50

Let I d, j , m > label a basis et states with total angular

momentum quantum members j ,m .

X labels " all other"

quantum members

( e.g. trth hydrogen atom it would be n,

th principal qeautum # ).

Then for Tlklq a spherical tensor operator :

( a 'j' HTMH aj )( a'

, j'

,m' IT Ng It ,j ,

m ) = ( jik ; mqlj '

,m' 7 -

Ft

• First term is ( G coefficient for Abby j and k angular momenta to

get j'

total angular momentum

• Themeaning do ( dj 'll TM Hxj ) is just that this is some

quantity independent of mm'

and q. In practice he fmk

this quantity by competing th matrix element for one chore of

9in'

, m,

and dividing by the CG Coeft :c .at .Thin all other chains

atq ,

on'

,m are determined completely by symmetry .

( The Fj factor is a convent an . )

• This is a powerful result .As we vary in ,m

', q ,

there an

( 2jtN2j '+ 1) (26+1) different Matrix elements here

.

The Wigner - Eckart

them tells he need only compete one , and look up CG coefficients for ntfe,

Prat :

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Consider the state |k , g) Qlj ,m ) = |k , j ; q ,

m ) .

Cheery ( j'

,m' I k

,j'

, q ,m > is just th CG coefficient in th

RHS et the Wigner - Eckart theorem.

Now,

J ; acts an this state oh exactly the same way as on

Tlkf I L, jim ) ,

i.e.

J ilk, jiqm ) =

14, jig

'

,m' 7 [ HY'

}qq Sm 'm + Say Hfn ]m 'm ] .

Ji Tlhdlaijim ? = Thlqlhj ,n' 7[ HY'

)qq Sm 'm + Say Htm ]wm ] .

For I kiji , g. m ),

tho } equation is th basis at th recursion

relations, which determine all the CG coefficients for fixed j , j

'

,k on tens

at one at them .The recursion relations Completely determine ratios of

the different ( Cr coefficients ,

The means that ratios of different Matrix elements

( d '

, jl ,m

' I Tlhlq I &, j ,

m ) ( for fixed d, &

'

,k

, j , jl ) are determined

by sauce recursion relations,

these rates are just rates at C 6

Coefficients.

the result follows immediately foam this statement.

( See Sakura : for a were detailed but equivalent argument . )