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/NBS monograph QC100 U556 V136;1973 C.2 NBS-PUB-C 1959 NBS MONOGRAPH 136

Graphical

Recouplingof

Angular Momenta
NATIONAL BUREAU OF STANDARDS

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Graphical Recoupling of Angular Momenta

National Bureau of Standards APR 2 9 1974 D. R. Lehman

Department of Physics

The George Washington University

Washington, D.C. 20006

and

J. S. O'Connell

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, D/recfor

Issued October 1973
Library of Congress Catalog Number: 73-600149

National Bureau of Standards Monograph 136

Nat. Bur. Stand. (U.S.), Monogr. 136, 18 pages (Oct. 1973) CODEN: NBSMA6

For sale by the Superintendent of Documents. U.S. Government Printing Office, Washington. D.C. 20402

(Order by SD Catalog No. C13.44:136)

Price 50 cents Stock Number 0303-01160
Graphical Recoupling of Angular Momenta

D. R. Lehman* and J. S. O'Connell

A diagrammatic method for solving angular momentum recoupling problems is presented. It is shown that a few graphical elements with a set of rules for their use lead to the solution of many I J pea of recoupling problems in an intuitive and systematic way. Several examples are given together with exercises to develop the reader's facility with the method.

Key words: Angular momentum; diagrams; graphs; quantum theory; recoupling; transformation theory.

1. Introduction

Many calculations in atomic, nuclear, and ele- mentary particle physics involve spin-angular inte- grals where the integrand is the product of three or more functions that possess definite angular mo- mentum. They arise, for example, in the computation

of matrix elements of the multipole operator with

many-electron wave functions in atomic physics and

in the determination of angular distributions in photonuclear physics. The evaluation of such integrals

by application of angular momentum recoupling al- gebra tends to be a tedious task. Recently, however,

M. Danos [1]

1 has presented a graphical method

whereby this complex process can be carried

through in a transparent way. Danos introduces a phase convention which eliminates minus signs from

the interior part of the calculation, thus greatly re- ducing the chance of computational error. This

graphical method resembles that of constructing a flowchart before writing a computer program, and

it is an excellent bookkeeping technique for recoupling algebra, just as Feynman's diagrams are in quantum electrodynamics.

Other diagrammatic methods [2, 3] have been de- veloped as aids to angular momentum calculations.

The reader can find additional references to the lit- erature in [2]. These methods usually start by assigning a graphical symbol to the vector-coupling coeffi- cient (3-7 or Clebsch Gordan) and compound the vector coupling to the 6-j or 9-j coefficients. Danos'

approach focuses on the recoupling aspect of angular momentum theory and uses four-element recoupling as its basic symbol. With the 9-j recoupling

as the central graphical element, separate graphical

elements need not be introduced for the lower symbols since they are already included at no loss of

'National Research Council Postdoctoral Research Associate. Present Address: Department of Physics. The George Washington University. Washington, D.C 20006.

1 Figures in brackets indicate the literature references at the end of this paper.

simplicity. For example, the 6-j recoupling is a 9-j

recoupling with one of the angular momenta equal

to zero. Moreover, the judicious choice of phase con- vention for the contrastandard elements, given in eq (1), removes the distinction between states and

operators, thus eliminating phases during the re- coupling. We feel these features give the method a power and elegance for recoupling problems not found in the previous schemes.

Our objective is to present a didactic exposition

of Danos' diagrammatic method for angular momentum recoupling algebra which will be useful for class- room use or self-learning. The reader or student is assumed to have a rudimentary knowledge of angular momentum theory comparable to that obtained

in a graduate quantum mechanics course at the level of Merzbacher [4], Messiah [5], or Schiff [6]. The

material discussed below is equivalent to or parallels that in Edmond's monograph [7]. In section 2, we introduce the fundamental elements of the dia- grammatic method and the rules for their application. Symbols are introduced for coupled elements,

normalization, insertion of a complete set of states, projection integrals, the contraction of two integral

angular momenta, grouping of momenta, and the

recoupling of two pairs of coupled momenta. The

tools developed in section 2 are then applied in section 3 to simple examples from the different types

of problems to which the method is applicable. These

include the computation of the multipole operator

matrix element for single-particle states in the j- representation, the derivation of a sum rule for the

6-j symbol, and the derivation of a general formula

for a (y, n) angular distribution. The main text then

closes in section 4 with several exercises, plus an- swers, by means of which the reader can develop

facility with the method and experience the fun in- volved in its application. A series of appendices gives

useful formulas and background material on the

basis of the method.

1
2. Elements and Rules

We first introduce the rotational wave functions

and discuss their conjugation, coupling, normalization, and closure properties. Spherical harmonics

are universally chosen to represent the rotational

properties of wave functions and operators in physical systems possessing spherical symmetry and in- tegral angular momenta. Danos has shown that

multiplying the ordinary spherical harmonics, as defined by Edmonds [7], by a phase factor (— i) 1 eliminates a mismatch between certain properties

of vector algebra and angular momentum coupling

theory. Therefore, the rotational part of a wave function for integral angular momentum is defined as

Y\!}(d,<p) = (-i)>Ylm (9, <p). (1)

Rotation functions with square bracketed superscripts are called contrastandard. Their Hermitian

conjugate (or costandard) forms are given by

Y^=WC= (-1 (-1)'+'»FL<L (2)

where the tilde means transposition (which has no

effect on spherical harmonics) and C means complex

conjugation. The two component spin-i wave function will be designated xf«i

2]

- These functions and

their representations are discussed in appendix B. A general spin-angle wave function for a particle

(with either integer or half-integer angular momentum) will be written as

iM/; ] (H) with $w'=(-iy+m$w (3) The coupling of the angular momenta of two particles to a total angular momentum J is given by

m 1 m2 - [^J(fi1 )x^j(n2 )]w (4)

and this equation is to be read as ji coupled with

j-i to J. The resultant J can take on values from

\ji—h I tOJi+j2. (jimij2 m2 \JM) is a vector-coupling

coefficient |_7]. This coupling is represented graphically in figure 1. A common single-particle coupling is that of a

particle's orbital and spin momenta

W= %{lrnhhms \jm)Y\k}x^

= [yt']x x

u/2]]U]. (5)

The normalization condition for wave functions

like those in eqs (4) and (5) is written

j^]t ^

] =l< (6)

r, . DO Art

(fl 2 ) J m

J2

FIGURE 1. Coupled angular elements.

where jis defined to mean that all spin sums and

angular integrations are to be performed. The product of wave functions in eq (6) can be expanded in terms of all possible coupled pairs

m

since the vector-coupling coefficients satisfy ^ (j — mjm\J0) {j—m'jm' \J0) = 8mm>. j

If i//^ were an ordinary spherical harmonic, the

coupled pair in eq (7) could be replaced by a single

spherical harmonic of angular momentum J through

use of the addition theorem for spherical harmonics

[7]. After integration over the angular coordinates,

only the J — 0 term survives, because

Likewise, only the 7 = 0 term survives the spin-angle

integration when eq (7) is substituted into eq (6). This leads to

j

[^x^][»l=(2j+l)i/2 (8)

where we have used (j— mjm\00) = (— l)j +m/j.

Equation (8) is the contrastandard element's analog

of the normalization equation. A spin-angle integral over a zero coupling will be called a "projection

integral," and will be represented graphically by a box as shown in figure 2a (also see appendix A).

Thus, by eq (8), two lines terminating in a box gives

the factor j.
Tfl] [j]

7 j

< ) < )

DO

x V//

< > ( )

k k

(a)

T

K J2

< > ( » i >

(b)

(c)

FIGURE 2. (a) Two-element projection integral; {b) insertion of intermediate states; (c) projection integral of the invariant

triple product.

which becomes

[vd/2] x x

ll/2l ][ °l = I. (13)

In general, the closure relation for constrastandard

elements is^ (-Dai[^xjiW] [»] = 1 (14)

where a represents all other quantum numbers,

e.g., radial, and H is the unity operator in the space

of the functions. For the usual spin-angle functions

of eq (5), the unity operator is 8(0 — ft') I. A graph for eq (14) can be constructed by using

figure 2b followed by a recoupling of the fines to move the element with the tilde to the right. This

recoupling (as will be shown) introduces the re- quired phase ( — l) 2k

. The summation is automatic

in any diagrammatic element that introduces new

quantum numbers.

We next consider the coupling that arises in the

evaluation of the matrix element of an operator T

between final and initial quantum states. If the

states and operator have been expanded in terms

of angular momentum eigenfunctions, the general

matrix element will be of the form (see appendix A)

Similarly, two fines taken out of a box, as in figure

2b, is the function A

: [i// [fc] Xi//M]M. This function is useful as part of the representation for inserting a complete set of states into a recoupling problem by means of the closure relation for orthonormal functions. For integral momenta, the closure relation is 2 mi(ft) y®* (ft' ) = § ( ft - ft - ) . ( 9)

Im

In order to write eq (9) in the form of an angular mo- mentum coupling, we use

[Fm X rW]M=2 (l-mlm\0O)Y^l FJM

m

l m

then eq (9) can be written as y /[yw(ft)xrt'J(ft')] [0] =8(ft-ft').

i (ii)

Similarly, for the spin function

ExLfx^-^)-! (12)

j

[fori* [IWX.$W]I"1]W.

Note that, just as in the normalization integral, the

total coupling is to zero since any other coupling

will vanish in the integration. The internal brackets

may be removed because the net result will be the same no matter which pair of functions is coupled

first. The integral over the zero-coupled triple

product is a projection integral and since it occurs

frequently is given a special notation

[^[^]|7W|<j>Ua]] =

j

[^,[J>] x T[K] X 3>U*]][o]_

(15)

Its graphical representation is shown in figure 2c. The projection integral of a triple product is closely

related to the usual reduced matrix element (see

appendix A),

(16) Two coupled integral momenta of the same

argument can be contracted by the formula

[yM(ft) xyt^(ft)]w=<?W

2yta(ft),

3
Q

[i]

(a)

(b)

FIGURE 3. (a) Contraction with a Q-box, (b) temporary

grouping.

where

OR = 0-1;

<1<2

(/1 + /2 + 0/2. /./2 //l /'2 Z

(47T) 1 / 2 \0 0 0

(17)

which we recognize as the addition theorem for spherical harmonics [7]. ^ m3 )

is tlle Wi§ner

3-7 symbol.

The graphical symbol, figure 3a, called a Q-box,

is used to represent this operation. On the other

hand, any two coupled functions can be temporarily

labeled by a single line, as in figure 3b. This is just a convenience and the two functions must eventually

re-emerge from the grouping in the same order as they entered.

Finally, we consider the fundamental recoupling

operation upon which the present system is based.

All recouplings will be expressed as combinations

of the two-pair recoupling transformation

[ [>M X <pM ] H X |>M X ,pM ]

[/) ] M

[ [<pM X ^M] to! X [<pM X <pM] W] I 29ft

a b c d e f

-g h i.

(18)

where the boxed coefficient is related to the usual

9-7 coefficient by

a b c d e f

-g h i_

(19)

with j = (2/+ 1)

1/2

. Equation (18) is expressed in words as follows: A system of four wave functions

[, nN

c

a b

)c C

d e

)f Xa )g C

d

— ~~<

lb

>h

e

hll

FIGURE 4. Two-pair recoupling box.

with angular momenta a, b, d, e is originally coupled

ab to c, and de to /. The intermediate pair momenta

c and / are coupled to the total system momentum

i. If the pairs are recoupled ad to g, and be to h,

the original system wave function can be expanded

in terms of the new system wave functions with in- termediate momenta g and h. The expansion coefficients are the boxed coefficients. The recoupling

transformation is represented graphically by the

diagram in figure 4. Note the correspondence be- tween the labeling of the line elements and loops

in figure 4 and the boxed coefficient. The diagram

is to be read from top to bottom. The values of a through / and i are given; the intermediate momenta

g and h can take on the range of values

\a — d\ ^g^a + d

\b — e\ =£ b + e

(20)

subject to g, h, and i satisfying triangularity.

Now we have sufficient tools to solve a variety of angular momentum recoupling problems. Symbols

have been assigned to coupled elements (fig. 1), normalization (fig. 2a), insertion of states (fig. 2b),

projection integrals (fig. 2c), the contraction of two

integral angular momenta (fig. 3a), grouping of momenta (fig. 3b), and the recoupling of two-pairs

of coupled momenta (fig. 4). These graphical ele- ments are to be combined according to the follow- ing rules in problem solving:

Rule 1. Only coupled elements can be recoupled.

Rule 2. Lines are assigned to each distinct ele- ment in a coupled expression preserving

the order. Rule 3. Lines can cross only in a recoupling box

as shown in figure 4. Mock zeros (dashed

fines) can be added anytime to achieve

four lines in a box.

4
Rule 4. The initial coupling equals the final coupling times the product of all re- coupling boxes and (?-boxes summed

over any unspecified intermediate

quantum numbers.

3. Examples and Applications

3.1. The Projection Integral for Three Inte- gral Angular Momenta

A physical problem in which the initial and final state wave functions and the transition operators

are expanded in spherical harmonics leads to the

projection integral

[YM | y[«2]

| y[/3]] =

j

dn[Fi''] x F/2 J x Y^Y®.

(21) The invariant triple product can be evaluated by

the successive contraction of angular elements by

(?-boxes as shown in figure 5. The first Q\'% replaces

[yv>] x FM]M by Y[h\ The second contracts the two

Ff /3 )'s to Y^°\ Therefore, we have

/1/2/3 / 1\ I'Z I'i

(47T) 1 / 2 \0 0 0

(22)

This result will be very useful in what follows.

i, h

3.2. A Multipole Operator in j-Representation

The expansion of the plane wave in angular

elements

iq • r (47r) i/2 £ {-\YLjL {qr)Yf\ar), (23)

where jt is a spherical Bessel function and the di- rection of q is taken along the z-axis, gives the often

encountered projection integral of the multipole

operator ji¥^ L\ For single particle states of eq (5), the Coulomb and other particle momentum and

spin-independent operators have projection integrals of the form

M,Xq) - [Rr(r)[Y^

X x

lll2] ] 1/1 \jiXqr)Y^ \R,(r) [F'J X tfM] W], (24)

where Ri(r) are the radial functions with their radial quantum number, n, suppressed. Figure 6

shows the graphical recoupling involved to separate

the radial, angular, and spin integrals. The box at the top of the figure is the projection integral to be evaluated (Rule 1); those at the bottom are standard projection integrals whose values we

know. The recoupling boxes between them relate the two. Starting at the top, lines are assigned to each distinct element (Rule 2); in this case, the two

orbital momenta / and /', the two spin-2 functions,

R/tYWxX^lj^lR^YW.xM] 111

]

IV j <"

* 2

1

L 0 L

./' | i .

FIGURE 5. Two contractions. FIGURE 6. Flowchart for evaluating a multipole operator.
and the operator of rank L. The first recoupling

separates the spin function from the orbital function

in the final (primed) state. A zero angular momentum

(the dashed fine) is added to make up the standard

box (Rule 3). The fact that L and V couple to / is not obvious at this point. The whole box has angular

momentum j since the total problem is coupled to 0 and the pair of fines excluded from the first box

are coupled to j. The second box uncouples the

orbital and spin functions of the initial (unprimed)

state and couples all the spatial functions together

and all the spin functions together. The final two

loops must be zero because projection integrals are made over the final products. The invariant triple

product among the /, I' , and L now show that L and

I' must be coupled to / in the intermediate loop

between boxes. In general, after a diagram is drawn,

one looks at any apparently arbitrary intermediate

couplings to see if they can in fact be specified by

relationships elsewhere in the diagram. By Rule 4

then

\v i r \i * n

ML {q) = L 0 L

J i j. _0 0 0_

[^fi/2]

| x

[i/2]] [y[/<]| yr/.j {Rl,

\ jl {qr ) \ Rl y (25)

The radial factor is the integral

(Rr\j,.(qr)\R,)

Jo (r)j L (qr)Ri(r)r2dr.

(26)

The spin projection integral is discussed in appendix

B and has the value

[£[l/2]| x[l/2]] =+ i = V2 (27)

The other factors on the right-hand side of eq (25) are given by eqs (19) and (22). Useful relationships

for reducing the 9-j coefficients when they contain

zeros are given in appendix C. The result is

Mi.(q)=(iY ,+ '-''K-l) s

' +11 I'lj'jL

47T

/' r » V L I 0 0 0

(Rv \j,.(qr) \R,). (28)

Recoupling applied to other electromagnetic

multipole operators is discussed in [8].

3.3. A Sum Rule for the 6-j Symbol

Identities and sum rules among the 6-j and 9-j coefficients can be verified with the recoupling diagrams. If a recoupling can be accomplished in two

different ways (i.e., by two different flowcharts), the

two transformations must be equivalent.

Before we begin an example, let us note the useful simplification occurring in a four-element re- coupling box when the two outside lines are mock

zeros. Figure 7a is a crossing of the nonzero lines and results in a phase factor for elements that commute:

0 a a bob

b a c = (-!)' (29)

Figures 7b and 7c show schemes in which ab

and cd are recoupled to ac and bd. Figure 7b does

it with one 9-j box while figure 7c does it with three

9-j boxes. Since the transformations are equivalent,

independent of wave functions, one has the relation

(30)

a b e b 0 b a 0 a

c d e C d e c b x -f f 0- X _x d a_ -fb d.

which reduces to the indentity

3.4. A Product Operator

c b x~\ d a f\'

(31)

This example involves introducing a complete

set of intermediate angular states through the

(a)

FIGURE 7. Flowcharts for proving a 6-j identity.

6
closure identity, eq (14). A projection integral that has two coupled operators is evaluated by uncoupling the operators and inserting

2 (-iH[f)xei][«)= (32)

between them. The graph for equation (32) is made

by combining figure 2b and figure 7a. Figure 8

shows how the operators iA" and are recoupled

to produce the product of the projection intergrals of each operator. The labeling of the intermediate

loops as ji and ji is inferred from the final zero coupling of (jz L k) and (k J ji). The flowchart

produces the equation

[^U*][.[^XFlJ ]]W.|.|^it]]

^k(-iy

k k 0

0 J J

\_ k h JJ

L 0 L

k J

7« Ji K

0 ji ji

U-2 o ji

[^2] if/['-]|^] ][e-] i^ ;l i^' ,] ] (33)

- \l J k)

.[^]l^[/-]l^]][^IFt^l^--]].

Examples of such product operators are the

orbital angular momentum

L=rXp= V2 [^XpU]W (34)

and the electromagnetic current multipole operator

[j L (qr)^XpM][J\

M [ J

J

L J

r..i I'M U x V

r

J

FIGURE 8. Flowchart for separating a product operator.

3.5. An Angular Distribution

Angular distributions [9, 10] play an important

role in atomic, nuclear and elementary particle

physics, since spins and parities of final states and

the nature of intermediate states can be ascertained. The derivation of the appropriate formula

is a recoupling problem.

We consider as a specific example a general (y,

N) angular distribution as depicted in figure 9. The

transition is regarded as a two-step process through

the intermediate state, Ik. The first step is the transition from the ground state, /o, to the state, h,

by absorption of a photon of multipolarity, L. The

decay of the state, h, through particle emission

involves coupling the emitted particle spin, s, and

orbital angular momentum, /. We do this in the

./-representation (see fig. 9) as opposed to the

channel-spin representation where I/+s = S and

S + l=Ifc; one representation can be obtained from

the other simply by a recoupling.

FIGURE 9. Photonuclear angular distribution.

For the initial state, we can write

^inmai =^>^]e

ifc2eW, (35)

where "J?^"1 represents the ground state and e**zeM

the photon incident along the z axis. e|*j is the

photon spin function which has two components,

mi. = ±\ (transverse — circularly polarized). The

505-690 O-LF- 73-2
product e

ikze$ can be written as the familiar

multipole expansion in terms of vector spherical

harmonics (appendix C):

- 2 (-l) LLC,Akr)Vy;l(Cl r ).

(36)

(37)

If eq (37) is substituted into eq (35) and the result coupled to states of total angular momentum, Ik , eq (35) becomes

&„itiai=2 {-WLCdkr)

uk (LmJ0mo\Ikmk ) [¥W X*^]^. (38)

Since total angular momentum and parity are con- served, the final state can be written as

t/>finai=2 (- 1)

LLA

,

k (LmJomo\lkmk ) H

]

, (39) uk

with At,ik = CiMi/c, where Mikis the amplitude for the photoabsorption through the state h. The state EJ'^ can now be resolved into its resultant con- mk figuration, which in the coordinate representation is

H=(i) I [iywW)X-XM] W X0W*1

. ( 40) mk

where the phase (t)' comes from using contra- standard spherical harmonics. The angular dis- tribution of the outgoing particle with respect to the incident photon direction (the z axis) is

ftnnai '/'nnai averaged over initial spins. The

/

means do all spin-coordinate integrations, except

those over the nucleon's angular coordinates.

Explicitly, we have

W{6)

^

2 2 (-i)/-+w)/2 £i [

0 LL'hl'

k m0 mL = ±l J

• LL''A\< ,ikAi,i (L'mJomo | I'

k mk ){LmiJ0 mo | Ikmk )

• [[y[i']x x

[sl p']

x §fjJ]fJti c [[yw x x

m]uj x ew*]. (4i) mk mk

'

The primed variables allow for interference. Using

the conjugation property, eq (A~2), brings in a phase (— iyic +mk, and then we couple the last two

factors to angular momentum A.:

f^(0) ^ ^

(-1)L+ L' + U-1')I2 +/;+m, f 2

'

o LL'hI'k \ mmi=±l J

• LL'Al^A, , k {L'mJomo \ rkmk )(LmJ0 m0 \ Ik mk )

• {rk -mkIkmk | X.0)[[ . . . ptfx [ . . . ][/ fcl]W.

(42) The problem now amounts to simplifying eq (42). The simplification of eq (42) is done in two steps:

(1) Removal of the spin functions and (2) Recombination of the three vector-coupling coefficients.

k

s 1 s ><

t> <

FIGURE 10. Recoupling the angular distribution formula.

Removal of the spin functions can be achieved

with the flowchart in figure 10. Using the rules established above, we get

/

[[. . .] wgx [. . .] t'AJ ]W
•t J If /"

I' s f

j h /* I s j

K 0 X_ _X 0 A_

IfsQMYl*(nN), (43)

which reduces to = (-1 )J'+J+ S+// +'* + « - ''"2 //' lj'k\i I

'

(44)

j f x) y v k)/i v x\ 1

0 0 0/ 477

I'k Ik If) U J s Px(cos 0),

where we have used YTM(SlN ) = (- l)-x/2 X/Mcos 6)1

(4,77)1/2 jf g pj n Qf t he ej ected particle is i, eq

(44) can be simplified to = (_ iy+j-ii2 +, f

+ 'K^^j]'lJ'k\

'

A '.f/XUpk (cosS), ,45,

with the identity

a v kui v x\ __L(J j' K

)

If j il \o 00/ r/'\i -i 0/

(46)

Recombination of the vector-coupling coefficients can be achieved in several ways. We could write

them as 3-7 coefficients and manipulate the order

of their arguments until use can be made of the

identity which relates a sum over the product of

three S-j's to the product of a 6-j and a 3-j (See

Edmond's eq (6.2.8, p. 95) [7]). However, one can

also recombine them with the aid of a diagram.

This combination of vector-coupling coefficients is like that which occurs in the coupling (with arbitrary

functions)

[[$['/] x tp°]].r'fc] x [ xw x si ">]]['*]] w

- V (L'm'L I0 m'0 I I'k - mk ){LmJom0 \Ik mk)

(4Hmto|X0)^*gxSS^),

which can be recoupled to

(47)

= 2

u h rk L lo h

r j x

[

[<!>['-'] x X

W]^'] x [<//[">] x ]

['i ]

t ^

(48)

L /o //, / J X X (y'm.'Jm,|X0)(/>'m?Xm/,|y'm.',)

mjfnj

(49)

ll Io L Io

XC

• > c

J'c XC

-— —

>

.—

L' L Io I(

Figure 11. Recoupling graph to obtain a vector-coupling

identity.

with the diagram in figure 11. Since the functions

Xi *K anQl 2 are arbitrary and the sums over mt,

m[, mo, and m0 appear on both sides, we have the

identity ^ (L'm'J0m'0 \Ik — mk ) (LmJ0mo\Ikmk )(Ik mk

= 2jj'

mjtn'j

v io rk L Io h

r j x

— 7»fc/fcOTfc|X0)

(50) Wm'jjmj\K0) (L' m'

LLmL \J' m'j)

(IomoIom0 \Jmj) •

Equation (50) is put into the form we want by four

steps: (1) Let m'0 = — m0 which implies m'L — — mL and mj=m'j=0. (2) Multiply both sides by (/<>—

mj0m0 \0 0) = (— l)'o+ mo//0 and sum over m0 . (3) On

the right-hand side use ^ (7( , — mj0 m0 \0 0){I0 —

molomo\j 0) = 8v(i, which, since 7 = 0, implies J' = X.

(4) Sum over mL — ±l and simplify the right-hand

side. We then have

9
2 (-iy^(h)-HLf -mJ0 -m4l'k -mK ) m km0 mL =±\

(LmiJ0 m{ )\I kmk ) (I'k — mkIkmk \\0)

= (—l) L'+>»+'k 2Ikrkk )L L' kUL V k\

7o (7* h /of \— 1 1 oj. (51)

Therefore, the combination of vector-coupling coefficients appearing in eq (42) can be rewritten by use

of eq (51) and the facts that

(L' — mJo — mb\I'k — mk ) = {-l) L' + l»-'k{L'mJ0m0 \rkmk )

and (— 1) /«/, = — 1.

The result is

2 (-l) mk(L'mJom0 \rkmk ) mL~ ± 1

(LmJom0 \I kmk )(rk — mkIkmk \kO)

(L V \\ IL U k\ = (-l)Ia+Ik-Ifrl2IkJ'k\

[rk h h) \- 1 1 0/. (52)

Substituting eqs (45) and (52) into eq (42), as an example, we get the y-nucleon angular distribution

formula

where overall factors have been dropped and we

used the fact that L + L' + A = even integer in writing the phases. The photoproduction angular

distribution of pions from nucleons [11] can be

obtained in exactly the same manner by simply

specializing to /o =//=£ and 5 = 0.

4. Exercises

At this point the reader might like to try to solve a recoupling problem with the aid of diagrams.

These exercises will develop facility with the

method and display its power and beauty. Appendix

C provides a collection of relations useful for solving these problems.

4.1. The jj; to LS Coupling Transformation

There are two ways to couple spin 1/2 electrons

in atomic orbits l\ and I2 to a total angular momentum J. First, an individual / is coupled to the spin

to give a total particle momentum j and then ji and

J2 are coupled to the total pair momentum J. Second,

the two orbitals l\ and h are coupled to a total orbital L and the two spins to S. Then L and S are combined

to form J. The exercise is to draw and label the 9-j box cor- responding to this recoupling and show that the

transformation is given by:

[[^"l(l)x^i(l)]Wx [>M(2) x x

[i/2i(2)]i^]

LS X[xW2](l)X XW2l(2)p)]^]

W{e) = ^Al,rkAL , lk

L l k V Ik

(53)

, , (l V \] (h U k\

(-l)

IM

XLL'hI'k \ if

(/' Ik /oj \-l 1 0/

(-iyk -'f-j-r + ^ik rjj'k\Jri

k

f

k Uft lk If

(54)

Note that the same coefficient summed over jx and ji applies to the inverse transformation. The 9-j diagram read down to up gives the same coefficient because of the transposition symmetry of the 9-j coefficient and 9-j box.

4.2. A Spin Multipole Operator

In analogy to Example B, construct the projection

integral for the spin multipole operator y/ [F|/] X

o-[i]]W for t he wave functions of eqs (5). The spin

projection integral is given by

7 A /\(cos 9),

[ x

ti/2]

I fj-m

I x

W2]] = i V6. (55)

10
Obtain the result

= (—1 ) i-J-i'+ ( r + + 1 + 0/2

4-7T

1/2y//y/7 (56)

V; j J)

4.3. The Normalization Condition for the 9-j

and 6-j Coefficients

Prove the equation

£ (2713+D (2724+D <7a h 734

( ' 7l3 724 7 '

(27.2+d (2734 + 1: (57)

by recoupling the two pairs (ji, jz) and Qa r fa) from

7i2 and 734 to7i3 and 724, respectively, with one box,

and then recouple with a second box back to the

original wave function with intermediate momenta

J'l2 and J'3i - The total angular momentum is taken

as J. When 7i2 = 7i2 and 734 = 734, the transformation

must equal unity since it reproduces the original wave function.

Show by recoupling that the 6-j coefficient normalization is

2(27i + Dj-H2 i'T-wrn J*hh) 272 + i

" i

where 7i and 72 are the pair momenta.

4.4 A (y, y')(A — 1) +N Angular Distribution

A problem which is of current experimental in- terest is that of photon absorption with subsequent

photon + nucleon emission. The process can be

visualized as taking place in three steps as depicted

in figure 12. Experimentally, the emitted nucleon

goes undetected and the angular distribution of the

emitted photon is measured. Show by methods

similar to those in Example D that, after integrating over the nucleon coordinates and removing its spin

functions, the emitted-photon angular distribution is given by

(- 1 )/*-/<»+ ixLL 7 /:.

(-1)/'+;+/^

"hi it j).

Pa (cos d), (59)

where 9 is the angle between the incident and

emitted photons. (Hint: the following identity is useful in handling the emitted-photon step

& 1 \o o 0/ \i — 1 o>

(60)

4.5. A Two-Body Matrix Element

Draw the recoupling flowchart that separates a two-body operator sandwiched between two-body wave functions into a product of one-body matrix

elements, i.e., show

[r>^](i) x^i(2)]i "|

XT^(2)Y'^\ [<p

[kl] (l) x^f**l(2)]I*l]

\h h 71 = jtk\h k l [<^-'](i)|ri"i(i)

[kt h K)

t^0?J(2) I rt'*V(2) (61)

* + s ; T+ 1£

= I

k

Io

If

Figure 12. The A(y, y')(A-l) + N process.

11
All recoupling identities using conventional ele- ments (e.g., tyjm) are also true for the contrastandard

elements because the difference in phase

conventions appears on both sides of the identities and therefore cancels. To check reduced matrix

element identities with recoupling diagrams, one must remember to convert reduced matrix elements

to projection integrals using eq (16).

4.6. A 6-j Sum Rule

Derive the sum rule 2 (-1)"(2*+1){: J

(62)

by recoupling from

\U[J i]TtK]tf)[ J n] = V

-Mi m M2 £a

l,L

(Ji- M, L m +M2 \ Im , ) ( Km ./, /!/•> \Lm + M2 )

(A-4)

which reduces to only the /=0 term when the

angular integration and spin algebra have been

completed. The coefficient of this term can be

written in terms of the Wigner 3-7 coefficient as [14]

J{= (_l)J.+Mi(_l)2X(_l)Ji+Jf+J2

[[aX fl ][°Jx[ix6][«]]i»] (63)

to [[oX6]Wx[oX6]M]M (64)

by two different flowcharts.

5. Appendix A: The Projection

Integral

The projection integral introduced by Danos is the quantity to which the tools of Racah algebra

are directly applicable. It is related to a general

matrix element in a manner similar to the reduced

matrix element, but with essential differences.

Consider the matrix element

J Mi m M2

v '

where, in general, i/> and T represent many-body wave functions and operators, respectively. The

angular momentum notation is that of Fano and

Racah [12] with t meaning Hermitian conjugate, i.e., i|/t=t// c '. The transpose operation, ~, which affects only the half-integral angular momentum, will be

discussed in appendix B. Conjugation [13] is defined

as

0jj] c =(_l)/+M0W, (A_

2)

for both integral and half-integral values of J. Equation (A- 1) can then be written as

(A-3)

The product of three contrastandard tensor components under the integral can be expanded in terms of angular momentum eigenfunctions

/ Ji K J2 \

\—Mi m M-J

•j ^J'lxl'Wx^l]TM. (A-5)

The integral of the invariant triple product is a pro- jection integral denoted by

[^Ui]|TM|$[^l]=j ^iJx rlA'l X$W]to].

(A-6)

The same matrix element expressed in terms of the reduced matrix element as defined by Edmonds,

Racah and Wigner is (see Edmonds, p. 88) [7]

V {- Mi m MJ

|| TW || #[/*]), (A-7)

thus

= r_i)ji+jr-/8^Wf[/i]'||7, [x]||$[J»]).

(A-8)

The main differences between these two quantities are the fact that the projection integral can be used

directly in recoupling algebra without phases to worry about, and, in the projection integral, operators and state functions are treated on equal footing.

6. Appendix B: Half-Integral Angular

Momentum

The case of half-integral angular momentum

deserves special treatment, since it illustrates the

meaning of conjugation, eq (A~2), and permits us

12
to derive a relationship between ordinary spin-i

eigenstates and the contrastandard ones. We begin

by considering the conjugation property.

The conjugation property defines the costandard

sets in terms of the contrastandard sets (see FanoRacah,p. 23) [12]

n/ n/ (-i) j+M (B-l)

where super ( ) denote costandard. For our pur- poses, it is the rule for replacing all costandard

elements involved in recoupling by contrastandard

forms. We know that the integral angular momentum eigenfunctions (17) , satisfy eq (B— 1), if the conjugation operation means complex conjugation. However, this does not apply to the half- integral angular momentum case.

Consider conjugation to be an operator, U, such

that

and

ty[J} c =U^= ./rW (B-2)

U^lf s £/2|//W= ^j). (B-3)

Only for J an integer can U be interpreted as complex conjugation. When J is a half-integer, U must

have another meaning compatible with eq (B~3).

We uncover the meaning of U for half-integral

angular momentum by specializing to the spin4

case. We choose the representation in which the

third Pauli matrix is diagonal and thus

Y [l/2] = *l/2 v [l/2] = (B-4)

Then by the conjugation property eq (B~l), we get

v[l/2] (' = _ V

[1/2]C = A-l/2 (B-5)

If we impose orthonormality, x£\V — by use

of the ordinary definition of matrix transpose, a contradiction arises. One way out of this impasse

is to introduce more freedom into the problem, for example, let the transposed quantities corresponding to eq (B—4) be

X!H

2] =(« 0); XL^=(7 §), (B-6)

with a, j6, y, 8 real constants. If we assume the

transposed quantities conjugate by eq (B—

1), orthonormality now gives

a = 0

0 = 1 y =-l

8 = 0.

(BExplicitly,

XSK

21 =(0 1); xL^J=(-i 0)

x\p

r =(i o); xfifl8 c -(o i).

(B-8)

(B-9)

The meaning of the conjugation operator for this representation is clear:

U — i<j2 =

0 1

1 0

(B-10)

where cr2 is the second Pauli matrix. In general,

U is the Fano-Racah U matrix, which corresponds

to a rotation about the y-axis by tt radians [15]. Specifically, U is the operation

The main point of the preceding paragraph is that no phase is needed when relating the contra- standard spinors to the ordinary spin4 spinors. In actual practice an explicit representation is not needed. We want simply

for half-integral j . We define

and

From appendix A, eq (B~13) becomes

1

and thus

< X

fi]|lIo]| x

b-] )=, [x[J ]| x

[j ] ]8MV,

(B-ll)

(B-12)

(B-13)

(B-14)

(B-15)

where j = (2jf+l) 1/2, The projection integral,

eq (B—

15), is the important quantity in recoupling,

not the explicit form of x

[j J

-

7. Appendix C: Useful Relations

We collect a number of standard formulas needed

to work the exercises and to make this article self- contained.

9-j Coefficients The 9-j coefficients are invariant under an even number of permutations of rows or columns and

under transposition about one of the diagonals. An

odd number of permutations of rows or columns

introduces a phase of minus one to the sum of all nine indices. A 9-j coefficient with a zero as one

13
element can be reduced to a 6-j coefficient:

I U h h \ = —

7s J4 Ja

(C-l)

A 9-j with two or three zeroes reduces to one of the

following cases:

fa b c (C-2)

(C-3)

L0 6 0

(0 a a] a 0 a =

la a oJ

(_!)»«

(C-4)

6-j Coefficients

The 6-7 coefficients are invariant under any per- mutation of columns or against interchange of

upper and lower elements in each of any two col- umns. A 6-j with a zero as one element can be

reduced to

71 J2 J3 h h 0

1=1)Jl +)2 +J3 (C-5)

J1J2

3-j Coefficients The 3-j coefficients are invariant under an even number of permutations of columns. An odd number of permutations of columns or changing the

sign of all three projection quantum numbers in- troduces a phase of minus one to the sum of the

three angular momenta involved. The 3-j symbol is defined in terms of the vector-coupling coefficient as

h h J

mi m2 —M

and when J = 0

(— 1) J1-J2+J1 (jimJ2 m>t\JM)

(C-6)

; ; 0

m — m 0

(-1 j-m

(C-7)

Phases

Often there are a number of equivalent ways' of

expressing a given phase factor. Some useful

identities for manipulating half-integral phases

follow:

(1) Phases are invariant modulo 4j, i.e.,

( _ 1))lj= (-i)(»±4)j.. (Oft)

(2) 2/= odd integer and 2(71 +j2 ) = even integer.

(3) Any three angular momenta which satisfy

triangularity (from a 3-;, 6-j, or 9-j coefficient) sum

to an integer.

We always use the standard convention that i=V:r l, so that (i)'= (-D'/2

, (-»')'= (-1)'.

(- 1

) V« = (- 1

) -W , and (

i )"'=(- 1 ) "'/2

. The reader

is reminded that complex conjugation is equivalent

to computing the inverse of a phase, i.e.,

(-1)

y tin (-1)

Spherical Harmonics

Formulas involving spherical harmonics with a sum over the projection quantum numbers arise frequently in physical problems. The following

are examples:

2 ra2 =s \Y»n \ 2 =^ (C"9)

(C-10)

^ m

2 |yMl 2 =

/(/+1 ^2/+1)

sin2fl (C-ll)

m £ Y\$*(l)Yl$(2) = tlYW(l) xYM (2)][°] m

2/ + 1

4-7T Pi(cosdn). (C-12)

Sectors On numerous occasions it is useful to express

vector operations as angular momentum couplings.

For example,

A-B = V3[#]xfiW]M (C-13)

A x B = V2 [A W x ] Hi (C_ 14)

and the coupling of a unit vector with a spherical

harmonic to form a vector spherical harmonic:

,[1] 'X = ^(/ml\|;M)i-We

»( , A = (_ 1

)(/+i)/2 ^ (fml\[yilf>yIm e,

= (-l) ( ' +ll'2 ^,(

(C-15)

(C-16)

A

(C-17)

(C-18)

14
Gradient Operator

The projection integrals for the gradient operator

are

[Ri +1 Yl l +V\VM\RiYM]

(C-19) = (Z+l)i/s (R 1+1 \y --\Ri)

or r

(C-20) = /i/.(H I . 1 |f + i±l|HI >. or r

The authors wish to express their appreciation to Michael Danos for teaching them the diagrammatic

recoupling method and for a careful reading of the

manuscript. John Lightbody, Jr. and James Murphy,

II helped us by serving as test cases for the pedagogical aspects of this article. Finally, we thank

Theresa Dunn for a fine job in preparing the

manuscript.

8. References and Notes

[1] Danos, M., Ann. Phys. (N.Y.) 63, 319 (1971).

[2] Briggs, J. S., Rev. Mod. Phys. 43, 189 (1971).

[3] El-Baz, E., and Castel, B., Am. J. Phys. 39, 868 (1971).

[4] Merzbacher, E., Quantum Mechanics (John Wiley & Sons,

Inc., New York, 1961).

[51 Messiah, A., Quantum Mechanics, Vol. 2. (John Wiley &

Sons, Inc., New York, 1961).

[6] Schiff, L. I., Quantum Mechanics, 3rd ed. (McGraw-Hill,

New York, 1968).

[7] Edmonds, A. R., Angular Momentum in Quantum Mechanics

(Princeton University Press, Princeton, N.J., I960).

[8] O'Connell, J. S., Electromagnetic Multipole Transitions in the Recoupling Picture, Mat. Bur. Stand. (U.S.I, Tech. Note 713, 23 pages (1972).

[9] Goldfarb, L. J. B., Nuclear Reactions I, edited by P. M. Endt

and Demeur, M. (North-Holland Publishing Co., Amster- dam, 1959), pp. 159-214.

[10] Hayward, E., Photonuclear Reactions, Nat. Bur. Stand.

(U.S.), Monogr. 118, 47 pages (1970).

[11] Kallen, G., Elementary Particle Physics (Addison-Wesley

Publishing Co., Inc., USA, 1964), pp. 152-156.

[12] Fano, U., and Racah, G., Irreducible Tensorial Sets (Aca- demic Press Inc., Publishers, N.Y., 1959), p. 23.

[13] In Danos' paper, conjugation is defined as ty\^ c = (— l)'-*

^LjJ. Danos now suggests that eq (A-2) be used for con- formity with Wigner and Fano-Racah.

[14] The coefficient (-1 )',«+/,(

K

{

2 ) is the F-coeffi- V— Mi m Mil

.-./ Ji K J2 \ cient, V , of Fano-Racah. Note that it V— Mi m Mil

arises naturally in this discussion.

[15] Note that x^1 '21 '*' - xi"

2]C = xjt1/2]7', X

lJ

l2]C = Uxllia = (-1)"2+ '1 XTM. X1

1,2JC = X'

,/21 ^ and x£«=-6tfW*)

r

, ^re T

means ordinary matrix transpose.

15
FORM NPS-114A (1-71) U.S. dept. of comm. 1 . PU BLIC ATION OR RE PORT NO. 2. Gov't Accession

BIBL'OGRAPHIC DATA NBS-MONOGR 140 No- SH E ET

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Graphical Recoupling of Angular Momenta

5. Publication Date October 1973

6. Performing Organization Code

7. AUTHOR(S) ^ , . , „

D. R. Lehman and J. S. O'Connell

8. Performing Organization

9, PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2400104 and 2410231

11. Contract/Grant No.

12. Sponsoring Organization Name and Address Center for Radiation Research

RadP, B102

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

A diagrammatic method for solving angular momentum recoupling

problems is presented. It is shown that a few graphical elements with a set of rules for their use lead to the solution of many types of recoupling

problems in an intuitive and systematic way. Several examples are given

together with exercises to develop the reader's facility with the method.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Angular momentum; diagrams; graphs; quantum theory; recoupling;

transformation theory.

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[XJ UNLIMITED.

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