Having introduced the Clebsch-Gordon(C.G) coefficients, the triangular relations between and the m-sum rule in the preceding module, we proceed to study here the ortho-normal properties and certain symmetry relations between the C-G coefficients. We shall also indicate briefly a method based on deriving a recurrence relation in the projection quantum numbers for fixed values of and apply it to show how this relation can be used to determine the expressions for the C-G coefficients in a few cases. We present a couple of tables for some typical cases.

3. Clebsch-Gordon Coefficients

3.1 Orthogonality Relations

Let us rewrite Eq. (28.9) in the form











1 2 1

1 2 1 1 1

1 2 1

1 2 1 1 2

1 2 1

m m m

m m j m j jm m m m j j

m m j m j jm m m j j jm

(29.1)

where from now onwards, we denote the Clebsch-Gordon coefficients, j j j m m Cm¹ ²

2 1

, as jm

m m j

j1 2 1 2 ). Further, in writing the latter part of the right hand side, we observe that since are not independent but are given by the relation m= , we use it to take care of the summation over and writing for wherever it appears.

As stated in the preceding module, the C.G coefficients are elements of unitary transformation and therefore must satisfy the orthonormal properties. These can be shown in a straight forward way, using the fact that the eigenstates are ortho-normal, i.e.,

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5

jm jm jjmm (29.2)

With the standard phase convention chosen such that the C.G coefficients are real, we substitute Eq.(29.1) in Eq. (29.2) to have

1 2

1 2 1 1 1 1 ,

, 1 , 1 1 2 1 1

1 2 1

1 1

m m j m m j m j m j m m

m j m m m m j j jm m m m j m j

m j m j

j jm

 

 



 



 

 

 



 ^ ^



(29.3)

Using the fact that the eigen states are ortho normal, viz.,

j m j m j mm j mm mmmm

1

1 1

2 1 2

1 1 1

1 (29.4)

Eq.(29.3) reduces to

j j m

m j m m m j j jm m m m j

j     



^

1

, 1 1 2 1 1

1 2 1

(29.5)

With this orthogonality relation, one can obtain the inverse expansion of (29.1):



^





j

jm jm m m m j j m

m j m

j₁ ₁ ₂ ₁ ₁ ₂ ₁ ₁ (29.6) Thus multiplying both sides of this equation by j₁j₂m₁mm₁ jm and summing over , we get:

 

^ ^ ^ ^



^ ^ ^

j m m

m m j m j m j m m m j j jm

m m m j j m j m m m j j jm

1 1

1 2 1 1 1

1 2 1 1

1 2

1 )

( (29.7)

The quantity in the parenthesis on the left hand side is just _j_j_, according to Eq.(29.5). Thus we have:



 

 



 

 



 

1 1

1 2 1 1 1

1 2 1

1 2 1 1 1

1 2 1

m

j m

j j

m m j m j m j m m m j j m

j

m m j m j m j m m m j j jm 

(29.8)

which is the same as Eq.(29.1).

Using Eq.(29.6), another orthogonality condition, which can be seen as an expression on the orthonormality condition on the rows and columns of a unitary matrix, can also be deduced in the same way. We here write this equation without deriving it, i.e.,

m m m m j

j        



1 2 1 1 1 2 1, 1 ^ 1 1^

(29.9)

3.2 Symmetry relations of Clebsch-Gordon coefficients

Among the three number pairs, j1,m1;j2m2 and jmm1m2, we observe certain symmetry as a result of the triangular relation between j1,j2 and j and the m- sum-rule. We mention some simple relations which exist between C-coefficients when the roles of the angular

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6

momenta are interchanged. These relations may prove useful in evaluating the C-coefficients from those already determined.

, 1 1 , 2 2, 2 / 1 1 1 2

1 ) 2

(

, 2 , 2

1 1 2 / 1 2 1 2

1 ) 2

(

1 , 2, 1, 2

2 , 1, 2, ) 1

(

1 2 1 ) 2

2 ( 1 2 1

2 2

1 1

2 1

m j m m j jj

m j j

m j m jm j j

m j j

m j m m j j

m j m m j j j j j

jm m m j j j j jm j

m m j j



 



 







 







 



 







 











 

 





 



(29.10)

Note that the 3j-symbol, introduced by Wigner and used often in many text books is related to the notation used here as:

₁ ₂ ₁ ₂ ₃ ₃

3 3 2 1

3 2

1 ₁ ₂ ₃

) 1 1( 2

1 j j mm j m

m j m m

j j

j _j _j _m



 





 



 _ _

(29.11)

3.3 Determination of the Clebsch-Gordon Coefficients

Let us outline briefly the procedure for evaluating the C-G coefficients. The simplified derivation is based on a recurrence formula for the coefficients. The first step is to apply Jˆ2to both sides of the Eq29.1), using on the right hand side

J z J z J

J J J J J

ˆ2 ˆ1 2 2

1 ˆ 2 ˆ 1 ˆ ˆ 2 ˆ2 2 ˆ1

ˆ2 1. 2ˆ 2 ˆ2 2 ˆ1 ˆ2

 

 



 







  

(29.12)

We recall from the study of previous modules that while state vectors jm are the eigen states of Jz

and

Jˆ2 ˆ , the states, j1m1 and j2m2(ormm1) are respectively the eigen states of J z

J z and J

J 2, ˆ2

ˆ2 1 ,

,ˆ 2

ˆ1 . We use the relations given in Eqs. (28.2),(28.3) and (28.7) and for the raising and lowering operators,  ˆ2

ˆ1 and J

J , employ the relations, Eqns.(27.2) and (27.4) and (27.5). We finally get, after some manipulations ( for details, see “Elementary Theory of Angular Momentum” by M E Rose) the recurrence relation for the Clebsch-Gordon coefficients, given by

jm m

m m j j m m j m

m j m j m j

jm m

m m j j m m j m

m j m j m

j

jm m m m j j m m m j

j j

j

1 ,

1 )]

( ) 1 (

) (

) 1 [(

1 ,

1 )]

( ) 1 (

) (

) 1 [(

, )]

( 2 ) 1 ( ) 1 ( ) 1 ( [

1 1

2 1 2 / 1 1 2

1 2

1 1 1

1

1 1

2 1 2 / 1 1 2

1 2

1 1 1

1

1 1

2 1 1 1

2 2 1

1









































(29.13) This is the recurrence relation for the C.G coefficients for fixed values of j1j2j and m^{in the}

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7

projection quantum numbers. Note that while the CG coefficient on l.h.s has, in its argument, the quantum number, , the first term on the r.h.s has and the second term has . 3.3.1 Evaluating the C.G coefficients for the =1

To see how this recurrence relation is employed to get the expressions of the CG coefficients, let us consider the specific when is coupled to to get the value of the total angular momentum, j= . Let us, for brevity call

j₁1;m, j₁1,m x_ , (29.14)

where mm1, is actually , which can take the values +1, 0, -1 for . Using Eq.(.9) and substituting the values, , j= ,we get three recurrence relations for 1,0,1:

μ= 1 : 2( j1m1)x1[2(j1m)(j1m1)]1/2x0 (29.15a) 2 1

/ )]1 ( 1

) 1 1

( 2 1 [ 2 / )]1 ( 1

) 1 1

( 2 0 [ 2 1 :

0         

 j x j m j m x j m j m x



(29.15b) 2 0

/ )]1 1 1

( 1 ) ( 2 1 [ ) 1 1

( 2 :

1 j m x  j m j m x



 (29.15c)

From these relations we can determine only the ratios, i.e.,

2 / 1 ) 1 1

( 2

1 0

1 2

/ 1 ) 1 1

( 2

1 ) ( 0

1 



 







 

 



 









 

m j

m j x

and x m

j m j x

x (29.16)

Now applying the orthgonality condition, i.e., Eq.(29.5), which , translated for the present case looks as

2 1 1 2 0 2

1 x x 

x (29.17) By substituting in this equation the values of x₁/x₀ and x_₁/x₀ from Eq.(29.16), we have:

) 1 2 ( ) 1 (

) 1 (

2

) ( ) 1 (

2

) 1 (

1

1 1

1 2 1

0

1 1 1

1 2

0



 



 





 



j j

m j m x j

gives This

m j

m j m

j m j x

(29.18)

We can determine all the ^xas far as the magnitude and relative phase is concerned. Since this equation together with the above ratios determines only the square of one of the coefficients, the overall phase still remains to be determined. Here we recall that in the preceding module, we adopted the convention that j1j2;j1j2 j1 j2,j1 j2 1. This choice means that positive square root is to

be taken, because .

Thus we write the first three elements of C.G coefficients as:

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8

2 / 1

1 1

1

2 / 1

1 1

0

2 / 1

1 1

1

) 2 2 ( ) 1 2 (

) 1 (

) , (

1 1 , 1 , 1

) 1 ( ) 1 2 (

) 1 (

) 1 , (

1 0 , 1

) 2 2 ( ) 1 2 (

) 1 (

) , (

1 1 , 1 1



 









 











 







 







 







 







 j j

m j m m j

j m

j x

j j

m j m m j

j m j x

j j

m j m m j

j m j x

(29.19)

Before writing the expressions for these C.G coefficients, (which are given in the Tabular form below), we should mention that similar procedure is to be adopted to evaluate the C.G coefficients for the

cases j= , and . The important point to note here is that in the

coupling of angular momentum with , the total angular momentum j can have the values and for each value of j, has the values +1,0,-1. Thus, as a result, in this

example, we have to write the 9 elements in the form of a matrix as given below:

Table 29.1 j11m, jm

J

1/2

) 1 2 2 ( ) 1 1 2 (

) 1 1

( 1 )

( 



 







j j

m j m

j 1/2

) 1 1 ( ) 1 1 2 (

) 1 1

( ) 1 1

( 



 









j j

m j m

j 1/2

) 1 2 2 ( ) 1 1 2 (

) 1 1

( 1 )

( 



 









j j

m j m j

-

2 / 1 ) 1 1 1( 2

) 1 1

( 1 )

( 



 









 j j

m j m

j

2 / 1 ) 1 1 1(

2













 j j

m

2 / 1 ) 1 1 1( 2

) 1 1

( 1 )

( 



 







 j j

m j m j

2 / 1 ) 1 1 2 1( 2

) 1 1

( 1 )

( 



 









 j j

m j m

j -

2 / 1 ) 1 1 2 1(

1 ) ( 1 )

( 



 







 j j

m j m

j 1/2

) 1 1 2 1( 2

1 ) ( ) 1 1

( 



 







 j j

m j m j

In obtaining the second and third rows, ambiguity in the overall sign of each row is once more found. The rule applied for the first row above does not apply here. Note that these questions of phase convention are a general characteristic of unitary transformation. A change in overall phase of any row or column of a unitary matrix, i.e., multiplying by a common factor, exp(iδ), where δ is real does not alter the unitary character. This can be readily seen by inspection of the orthogonality relations Eqs.(29.5) and (29.9). We here adopt the convention, in an ad hoc manner, by specifying that C- coefficients for shall be non-negative. This explains why in the third row, the first element is to be taken as positive and as a result the second has a negative sign.

You can easily ensure by writing the sum of the square of the elements in each row that the orthonormality relation given by Eq.(29.5) is built in.

3.3.2 Evaluating C G Coefficients for the case

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9

Let us apply the above recurrence relation to a simpler case of . Here, since , we have to calculate the values of four C.G coefficients only. To calculate the C.G coefficients for the first row, i.e., for j= j₁1/2, here too we introduce the symbol:

j₁1/2;m, j₁1/2,m  x_ , where 1/2

(29.20) ,

Thus for the case , , using the recurrence relation, Eq.(29.13),

we get:

 

2 / 1 2 / 1

1 1 2 / 1

2 / 1 2 / 1 1

1 2 / 1 1

) 2 / 1 (





 











 













m x j

or

x m

j m

j x

m j

(29.21)

Again, using the orthogonalty relation, i.e., x₁_/₂ ²  x_₁_/₂ ² 1 we get

) 2 / 1 (

) 1 2 ( 1

1 1 2

2 /

1  

 

 j m

j x

Choosing the proper convention , we get

2 / 1

1 1 1

1 2 /

1 2 1

2 / , 1

2 / 1 2 / 1 , 2 / 1

; 2 /

1 



 







 





 j

m m j

j m

j

x (29.22)

and

2 / 1

1 1 1

1 2 /

1 2 1

2 / , 1

2 / 1 2 / 1 , 2 / 1

; 2 /

1 



 







 







 

j m m j

j m

j x

(29.23)

To get the two other values for the case, j= we again use the recurrence relation.

This gives:

2 / 1

1 1 2

/ 1

2 / 1

) 2 / 1 (

) 2 / 1

( 



 







 



 j m

m j x

x (29.24)

Using the orthogonality relation,i.e.,

x₁_/₂ ²  x₁_/₂ ² 1 (29.25) Using Eq.(29.24) in (29.25), we get

) 26 . 29 1 (

2

2 / 2 1

/ 1 2 / 1 , 2 / 1 ,

; 2 / 1

) 26 . 29 1 (

2

2 / , 1

2 / 1 2 / 1 , 2 / 1

; 2 / 1

2 / 1

1 1 1

1 2 / 1

2 / 1

1 1 1

1 2 / 1

j b m m j

j m

j x

j a m m j

j m

j x



 







 









 







 









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10

Note that in conformity with the sign convention used, we have ensured that the C.G

coefficient must have a positive sign for . That is why the third element in the 2x2 matrix corresponding to is provided with the negative sign. The table for this matrix is given in the next module.

TABLE OF CONTENTS 4. Learning Outcomes 5. Introduction

6. Clebsch-Gordon Coefficients

3.1 Orthonormality Properties

3.2 Symmetry Relations of the Clebsch-Gordon Coefficients 3.3 Derivation of Recurrence relations to determine the Clebsch-Gordon

Coefficients

3.3.1 Evaluating the C.G coefficients for 3.3.2 Evaluating C G Coefficients for the case

4. Summary

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11

1. Learning Outcomes

After studying this module, you shall be able to

 Know how to derive the orthonormal properties of the C.G coefficients

 Learn some useful symmetry relations between the C.G Coefficients

 Learn how with the help of the recurrence relations, one can derive the expressions for the C.G coefficients

 Know a few examples to evaluate the C.G coefficients 2. Introduction

Having introduced the Clebsch-Gordon(C.G) coefficients, the triangular relations between and the m-sum rule in the preceding module, we proceed to study here the ortho-normal properties and certain symmetry

relations between the C-G coefficients. We shall also indicate briefly a method based on deriving a recurrence relation in the projection quantum numbers for fixed values of and apply it to show how this relation can be used to determine the expressions for the C-G coefficients in a few cases. We present a couple of tables for some typical cases.

3. Clebsch-Gordon Coefficients 3.1 Orthogonality Relations

Let us rewrite Eq. (28.9) in the form











1 2 1

1 2 1 1 1

1 2 1

1 2 1 1 2

1 2 1

m m m

m m j m j jm m m m j j

m m j m j jm m m j j jm

(29.1) where from now onwards, we denote the Clebsch-Gordon coefficients,

j j j

m m Cm¹ ²

2

1 ,

as ^j¹^j²^m¹^m² ^jm ). Further, in writing the latter part of the right hand side, we observe that since are not independent but are given by the relation m= , we use it to take care of the summation over and writing for

wherever it appears.

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12

As stated in the preceding module, the C.G coefficients are elements of unitary transformation and therefore must satisfy the orthonormal properties. These can be shown in a straight forward way, using the fact that the eigenstates are ortho - normal, i.e.,

^jm ^j^^m^ ^^^j^j^^^m^m^ (29.2)

With the standard phase convention chosen such that the C.G coefficients are real, we substitute Eq.(29.1) in Eq. (29.2) to have

¹ ¹ ¹ ¹ ² ¹ ² ¹

,

, 1 , 1 1 2 1 1

1 2 1

1 1

m m j m m j m j m j m m

m j m m m m j j jm m m m j m j

m j m j

j jm

 

 



 



 



 



 ^ ^



(29.3) Using the fact that the eigen states are ortho normal, viz.,

^j¹^m¹ ^j¹^m¹^ ^j²^m^^m¹ ^j²^m^^^m¹^ ^^^m¹^m¹^^^m^m^ (29.4) Eq.(29.3) reduces to

j j m

j     



^

1

, 1 1 2 1 1

1 2 1

(29.5) With this orthogonality relation, one can obtain the inverse expansion of (29.1):



^





j

jm jm m m m j j m

m j m

j₁ ₁ ₂ ₁ ₁ ₂ ₁ ₁

(29.6) Thus multiplying both sides of this equation by ^j¹^j²^m¹^m^^m¹ ^j^^m and summing over ,

we get:

 

^ ^ ^ ^



^ ^ ^

j m m

m m j m j m j m m m j j jm

m m m j j m j m m m j j jm

1 1

1 2 1 1 1

1 2 1 1

1 2

1 )

(

(29.7)

The quantity in the parenthesis on the left hand side is just ^^j^j^, according to Eq.(29.5). Thus we have: