clebsch gordan coefficients

8/12/2019 Clebsch Gordan Coefficients

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APPENDIX BCLEBSCH-GORDAN COEFFICIENTS

Clebsch-Gordan coefficients arise when two angular momenta are combined into atotal angular momentum. This will occur when the angular momentum of a systemis found as the combination of the angular momenta of two subsystems or when twotypes of angular m omenta relating to the same particle are com bined to find the totalangular momentum for that particle, as in the addition of orbital and spin angularmom enta to obtain a total angular mo mentum for the particle.

Let us find the total angular mom entum jas a sum of the mom enta ji and j 2 Thewave function for the total momentum can be written as(JU2, \ 2 \jm) ip hm i ip hm 2, (B.I)

YYl\ ,TYli

where the indices on the wave functions characterize the angular momentum and itsprojection onto a fixed axis. The coefficients in this expansion are called Clebsch-Gordan coefficients. We now examine their properties.

B.I PROPERTIES OF CLEB SCH-GO RDA N COEFFICIENTSB.I .I Condition for Addition of Ang ular M om entum ProjectionsIt follows from the conservation law for the sum of angular momentum projectionsthat

( J i J2> mim2 \J m) 0 if m \ + m2 ^ m- (B.2)Radiative P rocesses in Atomic Physics.V. P. Krainov, H. R. Reiss, B. M. Smirnov 2 35Copyright 1997 by John Wiley & Sons, Inc.ISBN: 0-471-12533-4


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23 6 CLEBSCH-GORDAN COEFFICIENTS

B.1.2 Orthog onality Cond itionOrthogonality properties of the wave functions are expressed as

This leads to the orthogonality condition for the Cleb sch-G ordan coefficients, whichis2 _]Oi72,m \m 2 \jm) (j\j2 , m\m 2 jm f) = 8 mm >, (B.3)

where8mm> is the K ronecker delta sym bol defined by1, m m 10, mi- m 1

B.1.3 Inversion PropertyIn the inversion transform ation of the radius vector,r> r, the wave functions willchan ge sign or not, depending on their parity. That is, the wave functions transformas

\\r. v i\i~m \\r .This leads to

so that we obtainO"i7*2,-m h -m 2 \j, ~m) = (-l)J~ J l~ j2 ( j iJ2,m\m 2 \jm). (B.4)

B.1.4 Perm utation PropertiesIt follows from the rules for the construction of the Clebsch-Gordan coefficients andthe rules for the addition of two ang ular mom enta into a zero total angular mom entumthat

{j\h,m xm 2 \jm) = (-l)J l m' xj ( jljfm lt ~m\j2 - m 2) (B.5)

72, m,m 2 \ j\ mi).


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EVALUATION OF CERTAIN CLE BSC H-GO RDA N COEFFICIENTS 23 7

B.2 EVALUATION OF CERTAIN CLE BSC H-G OR DA NCOEFFICIENTSWe consider first the evaluation of Clebsch-Gordan coefficients for the frequentlyoccurring case h ~ \- We begin by obtaining a relation for Clebsch-Gordan coef-ficients that is valid for any angular momenta. From the addition properties used toform j = ji + J 2 ,we obtain the condition

j 2 ~ j i ~ J2 = 2ji j 2= 2 juJ 2 z + 7*1+72- + 71-7*2+.When this operator acts on the wave function in Eq. (B .I), the result is

J U + 1) ~ 7i0*1 + 1) ~ 72(72 + 1) = y^ 0'i72 >raim217m)p 72 , m[m 2 \jm)X {^hm^hm^hzhz + 7*1 + 7*2-+ 7*1-7*2+l^ ,m> ;2m 2).

For the given value of m, the conservation rule for angular momentum projectionsgivesm\ = m 1712, m[ = m m 2 .

W ith the help ofthemomentum operator eigenvalues in Eqs. (A.2) and (A.9), we findthat7(7 + D - 7 i ( 7 i + l ) - 72(72 + D

m2- mQQ'i + mi + \)(j2 - m 2 + l)(j2 + rn 2)

m 2x {j \J2,m xm 2 \jm)(j iJ2,mi + 1,m 2 - l\jm)^ 72 - m 2)( j2 + m 2 + 1)

m2x O'i7*2, rn xm 2 \jm){jxJ2y mi - 1,m 2 + \\jm). (B.6)

Now we apply the above results to the particular case wherej2 = \ . For the givenvalues 7, 7i, there are two nonzero Clebsch-Gordan coefficients that we shall denoteasX={h\,m-\,\\jm),


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238 CLEBSCH-GORDAN COEFFICIENTSUsing this notation, Eq. (B.6) for j2 = \ becomes

J U + 1) " 7i O'l + 1) " I = (m - \)X 2 - [m + i ) Y

The normalization condition for the Clebsch-Gordan coefficients given in Eq. (B.3)is of the form

X 2 + Y 2 = 1.When this is inserted into the preceding expression, we obtain

U - h ) U + h + 1) - 3 = m (X 2 - Y 2) + 2\l(jx + | ) 2 -m*XY.With the no tation

we can rewrite our results as the system of equationst( X 2 - Y 2) + 2 \ / l f-XY = 1 , X 2 + Y 2 = \.

The ambiguous sign is such that the upper sign corresponds to j = j \ + \, whilethe lower sign is associated with j = jx \.The solution of the equations we have obtained is

for7 = 7i + 5, and for j = j \ ~ \ it is

These results are summarized in Table B.I.TABLE B.I . ( j \ \ , m - V 2 y , +/ .( + m

i /

+" \1+ i


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EVALUATION OF CERTAIN CLEBSCH-GORDAN COEFFICIENTS 239

Other values for Clebsch-Gordan coefficients are generally more complicated toexpress. Thorough treatments of the relations between C lebsch -Go rdan coefficientsand tables of values are available in the literature. See, for example, Refs.9or 10.Another relatively simple form taken by the Clebsch-Gordan coefficients occurswhen the projection of the angular momen tum co incides with this mom entum. In thatcase we have

0 . . . . v O/'i +m 2 ,m\j2,J\m2\jm)=y/U\ +72C/2~W2) (7+m)\ 2j + 1) (271 ) (7 71+72)C/2+m 2)l j ~m)\ (7 - 72+7) (7 ,+j 2- 7 ) '

(B .7)

V(7i +72+7x /(7i mi) j+m)\ (2j+1) 2j2)\ (J i~ h+7)

{]\j2,mxj2\jm)=

(7i+m x)\ j- m)\ j- j +j2)\ (J i+72~7 ) '(B.8)


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2 4 0 CLEBSC H-GORD AN COEFFICIENTS

B.3 WIGNER 3j SYMBOLSA quan tity closely related to the Cleb sch-G orda n coefficient is the Wigner 3 7 coef-ficient, designed to achieve maximum symm etry. It can be defined as

^ ) = (-l)j]~ J2~m3( J i J2>mim2 \j 3, -m 3). (B . l l )The 3 7 symbol has the property that

= 0 if mi + m 2 + m 3 = 0,mx m2 - \ i i >in place of Eq. (B.2).

We list the principal sym metry and orthogonality properties. Even perm utation ofthe columns leaves the 3j symbol unchanged, or7i h h \ = ( h h h \ = ( h h hmi m 2 m 3 I V m 2 m-x nt\ J V m 3 mi m 2

Odd permutation of the columns, on the other hand, is equivalent to multiplicationb y ( - i y i + 7 2 + j 3 , s o t h a t h h \= h h hmi m 2 m 3J \m 2 mi m 3

= h h h = Ih h J\\ mi m 3 m 2I m 3 m 2 miOrthogonality properties of the Wigner 3j symbols are

h h h \ ( J \ h h \ _ o cjI mi m 2 m 3J \m[ m 2 m 3h h h \ ( h h ^ \ = hj fnvnjSUUlh)mi m 2 m 3 J \mi m 2 m \ j 2/^ + 1

where 8 (jij2 j3) in Eq. (B.12) is a quantity defined as8 (71/2/3)= {I [f[jl .hl ~ h - h + h10 otherwise

The statement in Eq. (B.13) is called the triangular condition.

clebsch gordan coefficients

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