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Combinatorial computation of Clebsch-Gordan coefficients.

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Ann. Physik 5 (1996) 103-120
Annalen
der Physik
Q Johann Ambrosius Barth 1996
Combinatorial computation of Clebsch-Gordan
coefficients
Klaus Schertler and Markus H. Thoma
Institut fur Theoretische Physik, Universitat Giessen, D-35392 Giessen, Germany
Received 18 October 1995, accepted 23 November 1995
Abstract. The addition of angular momenta can be reduced to elementary coupling processes of
spin-+particles. In this way, a method is developed which allows for a non-recursive, simultaneous
computation of all Clebsch-Gordan coefficients concerning the addition of two angular momenta.
The relevant equations can be interpreted easily, analogously to simple probabilistic considerations.
They provide an improved understanding of the addition of angular momenta as well as a practicable evaluation of Clebsch-Gordan coefficients in an easier way than within the well-known me&OdS.
Keywords: Clebsch-Gordan coefficients; Racah formula; coupling of angular momenta.
1 Introduction
I . I De$nition and properties of Clebsch-Gordan coeficients
A system of two quantum mechanical angular momenta is Thyactenzed by the angular momentum operators of the total ?gulp mpnentum J2, Jz and the operators of
the individual angular momenta J:, JI,;
J2z. These operators can be classified
into two groups in which the corresponding operators commute [l]:
Ji,
Hence the basis states can be chosen as
0
0
coupled eigenstates, denoted by IJI ,J2; J , M ) or IJ, M),
uncoupled eigenstates, denoted by IJII J2; MI M2) or I(MI,M2)).
These complete basis systems are related to each other via an unitary transformation:
104
Ann. Physik 5 (1996)
The amplitudes ( ( M I M2)
, IJ,M) in (1) are called Clebsch-Gordan (CG) coefficients.
General properties of the CG-coefficients are given e.g. in Ref. 1. Here only a few are
listed:
0
0
0
+
CG-coefficients vanish if M # M I M2.
The total angular momentum J satisfies IJ1 - 521 5 J 5 J1
CG-coefficients can be chosen as real.
+ 52.
I .2 Conventional methods
A standard method for calculating CG-coefficients (see e.g. Ref.2) is based on the
iterative application of the operator 5- = j , - iJy on the maximum state
where J = J I
by
+ 52 and M = M I + M2. The effect of the operator on I J , M ) is given
dqJ,M) = J(J+M)(J-M+
l ) I J , M - 1).
The CG-coefficients are obtained by projecting the resulting states onto the uncoupled product state ( ( M I,M2) I.
Furthermore, Racah gave an elaborate but explicit formula for the CG-coefficients
(see e.g. Ref. 2). Both the methods are very formal and render the evaluation even in
the case of “small“ angular momenta cumbersome. Thus CG-coefficients have been
listed in tables (see e.g. Ref. 3).
Here we propose an intuitive way allowing for an immediate computation of CGcoefficients. First we will consider the special case of coupling of two angular momenta to their maximum angular momentum, before we will turn to the general case
in section 3. Based on these investigations we will provide a very simple way of calculating CG-coefficients in section 4. In the appendix we will prove the equivalence
of our method with the Racah formula.
2 Addition of two angular momenta to their maximum angular momentum
By speaking of a spin-1-particle in the following we denote any object with angular
momentum J, e.g. also orbital angular momenta, in order to avoid the inconvenient
expression “objects with angular momentum J”.
2. I Decomposition of a particle into spin-;-particles
According to Schwinger’s oscillator model of angular momentum [ 1,4] every spin-Jparticle can be considered as a composition of 21 spin-&particles. Then a particle described by the state IJ,M) with angular momentum J and z-component M consists of
K. Schertler and M.H. Thoma. Combinatorial computation of Clebsch-Gordan coefficients
105
Table 1 Constituent states of a particle described by the state ( J ,M )
M=
-1
-5
I
I
2
0
7
\
J-L
-2
1
\t
J= 1
I
spin---particles,
2
u(J,M ) G J + M spin-up particles /,
d ( J ,M ) J - M spin-down particles \ .
j(J) 2.l
This idea relies on the fact that the 25 spin-$particles satisfy the same transformation
relations under rotation as a spin-J-particle. Thus, if we are interested in the angular
momentum of a spin-1-particle, i.e. its transformation properties under rotation, we
might consider j = W spin-&-particlesas well. Only in this sense we might speak of
a spin-J-particles consisting of spin-$particles. In order to distinguish real spin+paticles from the constituent spin-+particles we will call the latter constituent spln-iparticles.
In the following the question, how many possibilities of decomposing a particle
described by the state IJ,M) into constituent SpinSparticles exist (see Table I), will
play a crucial role. We know that there are u = 1 M constituent particles of the j
particles after the decomposition in the state 1.Then the number of possibilities is given by
+
s(j,u)
j!
= (:>= u!d!
‘
Table 2 Number of the constituent states of a particle described by the state 15,M )
M=
s (JN)
-2
_ -32
-I
0
1
J=;
1
4
I
3
T
2
I
1
I
3
3
1
J=$
I
I
2
1
J= 1
J= 2
1
-2
6
4
1
106
Ann. Physik 5 (1996)
We say, the state IJ,M) consists of s constituent states. As we will see later on, the
CG-coefficients depend on the number of the constituent states belonging to the particles under consideration. This number is given by the U t h row of the binominal expansion (Pascal’s triangle) (see Table 2) for all (for the spin-J-particle) possible M values.
2.2 Computation of the CG-coeficients for J = J1
The matrix
+ Jz
no:
We will now investigate the special case of the coupling of two particles with angular momenta J I and 52 to a system with the maximum angular momentum
J =J,
+J2,
j =j, +j2.
As we have seen in (I), the coupled state IJ, M ) is composed of all uncoupled states
[ ( M I, M 2 ) ) for which M = M I M2 holds. Since there must be no interference between the uncoupled states - this would lead to J < J1 52 - , we expect real and
positive CG-coefficients (according to the usual phase convention). This enables us
to determine the CG-coefficients from square roots of probabilities. The square of the
, 2 ) IJ, M ) corresponds to the probability of finding the indiviCG-coefficient ( ( M I M
dual particles in the state I ( M I ,M 2 ) ) at the instance of measuring the total angular
momentum. This probability follows by decomposing the coupled and uncoupled
states according to section 2.1 into their constituent spin+-particles and counting the
number of constituent states generated in this way. We denote the number of constituent states of the coupled system by s, and the one of the uncoupled system by sue,
respectively.
+
+
The quantity su, is given by the product o t e number of the constituent states of the
states of particle 1 can generate a new
individual particles, be a se each of the
state with each of the
states of partic e . The fact that the coupled state and the
sum of all uncoupled t tes with M = M I M2 describe the same physical system
suggests that both should have the same number of constituent states, leading to
k)
+
+
where the sums extends over all U I ,u2 with U I u2 = u, corresponding to
M I M2 = M . Eiq. ( 3 ) is known as the addition theorem for binomial coefficients
(see e.g. Ref. 5).
The desired probabilities, i.e. the squares of the CG-coefficients, will be obtained
conby the simple assumption that the combined system is found in each of the
stituent states with equal probability.
+
c)
K. Schertler and M. H. Thoma, Combinatorial computation of Clebsch-Gordan coefficients
107
If we could determine the constituent state of the system, we would obtain a state
after ,I
of
measurements in the average, which according to (3) originates
112
uniquely from the decomposition of the uncoupled state I ( M l , M 2 ) ) . Hence the desired propability of finding the system in the state I(M1,Mz)) is given by
In this way we obtain an expression for the CG-coefficient in the
case J = J1 52:
p) (j2) c)
(Iu:) @) c)-’.
+
(4)
Therefore, in this special case the CG-coefficients can be understood in an elementary way. Nature chooses the simplest way, not distinguishing any of the s, constituent states. The u = u1 u2 /-states and the d = d! d2 \-states are coupled with
equal probabilities to all possible constituent states. Some of these states (s,,,), however, are interpreted as an uncoupled state. Thus the coupled system consists according to the ratio s , , / ~ ~ of this uncoupled state.
Expressing (4) by J i , M i and using the definition of the binomial coefficients, we
recover the Racah formula for J = J I J2,
+
+
+
X
(JI
+
+
(J M ) ! ( J - M)!
Ml)!(JI - MI)!(J2 M2)!(J2- M2)!’
+
as it can be found e.g. in Ref.2.
In order to simplify the explicit calculation of CG-coefficients and the generalization to arbitrary total angular momenta, we define a matrix 510 containing all essential
information of the r.h.s. of (4) for all ui ,u2. The crucial quantities are the values of
s.,
The values of s, only take care for the correct normalization of the CG-coefficients and depend according to (3) on suc. We define the matrix IRo via their components
Here u1 extends from 0 (first TOW) t o j i and u2 from 0 (first column) to j 2 , respectively. Qo is a U ,+ I x U2+ I-matrix. Sometimes we will use the angular momenta J I , J 2 as the arguments of 510 = Ro ( J I , ~ ) .
Example: Addition of two spin-1-particles to spin 2
One of the essential advantages of (4) compared with the Racah formula ( 5 ) consists
in the possibility of extracting all results Of (4) easily by means of the matrix
AS
a.
108
Ann. Physik 5 (1996)
Table 3 M and u-values of
M2
-1
1
0
~
MI
UlkZ
-1
0
~~
1
2
0
1
2
0
1
4
I
2
I
2
2
I
2
1
a first example we will calculate all CG-coefficients corresponding to the coupling of
two spin- 1 -particles to a spin-2-state according to (4). l2o follows according to (6) by
multiplying two J = 1-rows of Pascal's triangle, denoted in the following way
1 1 2 1
11121 =(i ; i).
2 1 2 4 2
1 1 1 2 1
Each of the 3 x 3 components of this matrix reproduces the number of the constituent states of the state I(M,,M2)). The connection between the position in the matrix
and the corresponding M or U-value is shown in Table 3. The maximum state I( 1 , l ) )
is found down-right. The diagonals (raising from left to right) correspond to a given
M I M2 = M. The M-values of the diagonals increase from M = 41
- J2 (up-left)
appearing in (4) is extracted according
to M = J I J2 (down-right). The quantity
to (3) by summing up the diagonal elements. In our example these sums give
{ 1,4,6,4,I}, corresponding to the J = 2-row of Pascal's triangle, as expected from
(3). The CG-coefficients are read off from n
o by dividing the components (h)
u,
u2
+
c)
+
c)
(">
and extracting the square root. This proceby the sum of the diagonal elements
dure yields the following CG-coefficients, as can also be found in Ref. 3:
12, -2) = l(-l, -1)).
K. Schertler and M. H. Thoma, Combinatorial computation of Clebsch-Gordan coefficients
109
Utilizing the &-matrices we are able to determine all non-vanishing CG-coefficients
with J = J1 + 52 immediately. As a further example we present the coupling of a
J1 = $particle with a 52 = 1-particle to the J = $state:
-
1 2 1
1
1 2 1
3
3 6 3
3
3 6 3
1
1 2 1
1 2 1
3 General addition of two angular momenta
Next we will consider the general case in which two particles with J1 and J2 may
couple to a total angular momentum J C Ji 4-52. This problem can also be treated
by means of constituent spin+particles. Assuming that two of these constituent particles couple to spin 0, they do not contribute to the total angular momentum J anymore. If there are n spin-0-particles in a System of constituent spin-+particles, the total angular momentum is reduced to J = Ji 52 - n. In this way the general addition of two angular momenta can be described by two elementary processes of constituent spin+particles, namely the coupling of some constituent particles to spin o
and the coupling of the remaining particles to their maximum angular momentum according to section 2. First we will consider the coupling of two spin+particles to
spin 0, before we will combine both processes.
+
Coupling of two constituent spin+particIes to spin 0:
From
we obtain
From the requirement ((0,O)
I( 1 0)) = 0 we find
110
Ann. Physik 5 (1996)
We indicate the state l0,O) as
/”\-\/”.
(7)
3.1 The matrix anof the coupling
to
J = J1
+ JZ - n
+
52 - n can also be constructed from a matrix, called Qn from which we can read off the CG-coefficient, similar to n, directly. However, we cannot demand positiveness of the CG-coefficients any longer.
Negative CG-coefficients will be identified from R, simply by a minus sign in front
of the corresponding component.
In order to exemplify this prescription we will show Q($, 1) here. The details of
its calculation are given below. It describes the coupling of a spin3-particle with a
spin- 1-particle to spin $:
As we will see, the general case J = J1
nl(Z”)=
3
(I
0 -3
-6
.);
From this we can read off the CG-coefficients (from down-left to up-right), e.g. from
the 4th diagonal corresponding to M =
4:
The minus signs in 01 just indicate that the corresponding CG-coefficient is negative. Calculating the normalization, these signs must not be considered in the sum of
the diagonal elements, here 15.
We will regard two R, matrices as equivalent, if they differ only by a factor a, denoted by aR,32,. Also two matrices are equivalent, if their diagonals, raising from left
to right, deviate only by a factor. In both cases we find the same CG-coefficients.
The matrix
R, follows from the requirements that
1. among the constituent spin-&particlesn Spin-0-particles arise and
2. the remaining constituent spin-$-particles couple to their maximum spin.
Spin-0-particles arise according to (7), if the constituent spin-$particles generate /\or \/-states.
The components of n, play the role of non-normalized probabilities, analogously
to the interpretation of Ro. They are composed from the probabilities of the processes
1. and 2.
111
K. Schertler and M. H. Thoma, Combinatonal computation of Clebsch-Gordan coefficients
~
Defining
0 A as the matrix describing the probabilities for the state I
\
,
0 y as the matrix describing the probabilities for the state \I, and
0 IR, as the matrix describing the probabilities for the coupling of the remaining
constituent particles to their maximum spin,
the above requirements translates directly to the matrix equation
where the products (and), denoted by 0, and the subtractions (or) between the matrices have to be performed for each single component. When we talk of probabilities
in the following, we may not refer to normalized probabilities, as the product by
components in (8) allows for arbitrary factors (normalization) in A - V and fin.
3.2 The matrix (A - ,)In1
of n spin-0-particles
For calculating (A - V)["'
we need the probabilities of creating /\-or \/-states from constituent spin-12-particles. The probability of creating /\ is proportional to the probability of finding /F at particle 1 and \ at particle 2 at the Same time. This in turn is proportional to the
product of the number of the constituent particles in the corresponding states. An analogous
statement holds for \/. Therefore we define
leading to
The expression uld2 - dlu2 can be interpreted (if it is positive) as the surplus of I\states of an uncoupled state. If it is negative it yields the (negative) excess of \/states. This pays attention to the fact that according to (1) interference can occur
only between uncoupled states but not within an uncoupled state itself.
Performing the exponentiation [n]one has to take into account that the probabilities A - V depend on the number of the spin-0-particles already generated. If e.g.
already one /\-state has been built, there are only U I - 1 and d2 - 1 particles available. Therefore the exponentiation means
where
112
Ann. Physik 5 (1996)
=
(uld2)[n-k1
n-k-1
fl
(u1 - i)(& - i),
i=O
=
k- 1
(d*u2yk1 n ( d 1 - i)(u2 - i).
i=O
Example: (A - V)In1for 51 = and J2 = 1
As an example we will calculate A - V and (A - V)[21for the coupling of J1 =
and J2 = 1, already mentioned above. The A and V matrices in this case read
-
A=
2 1 0
-
0 1 2
3
0 3 6
2
0 2 4
4 2 0
1
0 1 2
6 3 0
0
0 0 0
0
0 0 0
1
2 1 0
2
3
, v =
leading to
/O
-3
-6\
The matrix (A - V)121
is given according to (10) and (1 1 ) by
(A - V)[’]= A o A‘ - 2A o V -I-V o V‘,
where
A’I,,,,* = (u1 - w
2
- 1)
and
V’lul,a= (dl - 1)(w - 1)
read
5
K. Schertler and M. H. Thoma, Combinatonal computation of Clebsch-Gordan coefficients
;'"
+%
0 1
I
0 0 0
1 1 0 0 1
,v'=
1 / 1 0 0
0 1 0 0 0
2 1 2 0 0 ,
0 1 0 0 0
113
From this we get
(A - V)/2j=
/o
0 0
0
4
\12
0 0
I
0 0
0 0 12
-2
=(i;
i)
O": ;(+)"O
0 0
2 0
0 0
0
12
0
;)A(:
3
0
3
0
0
.).
r:
All higher exponents (n = 3,4, ...I of - V vanish in accordance with the rule
IJl - J21 5 J 5 Jl J2. In-order te cabdate the CG-coefficients of the above example, we need the matrices Rt and a2 of the remaining system in addition. Their computation is the topic of the next section.
+
3.3 The matrix fl,,of the remaining system
The matrix finfollows directly from n
o by removing the constituent spin+particles,
contained in (A - V)In1.Here removing a 7 from the particle i means that the total
number of constituent spin+particles and the number of 7 is reduced by one; i.e.,
we replace
Analogously a \ is removed by
For constructing fi, a spin-0-particle has to be removed from In,; i.e., according to
(7)a 7 will be removed from particle 1 and a \ from particle 2 at the same time or
a from particle I and a /" from particle 2. Together with
114
Ann. Physik 5 (1996)
Qolu,,u2
=
(f) (t)
we get
where the removing of a
\
has been performed via
The binomial coefficients vanish for all unphysical values, e.g. if ji - 1 < ui or
Ui - 1 < 0. This corresponds to fhe fact that there is no state with uj < 0 or di < 0.
The matrix R2 is obtained from f21 by removing an additional spin-0-particle:
-2(jl-2)(j2-2)
u1 - 1
u2-1
+(jl-J(h-2)
u2-2
In general f i n is given by
fin
can be calculated easily, as they consist essentially out of
J2 - n can be
imagined as originated from coupling of spin Ji - 3 and spin 52 - 5 to their maximum spin. This will be illuminated in the following example.
The matrices
Ro (51 - 5, J2 - 5). This reflects the fact that a system with J = J1
-
4
Example: $2, for J1 = and
J2 = I
fi, and f& in the case of the coupling of
First, however, we will repeat
We compute
+
a:
Jl
= $ and J2 = 1 from (12).
&):
For physical values of u1 and u2 the expression (Ul!(nlk)) in (12) leads to the
smaller matrix no(J~- 5 , J2 - 5), in our example
K. Schertler and M. H. Thoma, Combinatorial computation of Clebsch-Gorclan coefficients
115
ifn=land
if n = 2. For all other values of U I and u2 the expression is identical to zero. The value of k fixes the positjon of Lhese matrices: k=O corresponds to down-left, k=n to
and R2 thus are found easily:
up-right. The matrices
:j-(: ;)=(; : ;j7
0 0 0
Q
- =(:
;
0 1 1
f
1 1 0
"-(p :
0 0 0
0 0 0
:)-2(:
; :)+(:
: A)=(;
0 0 0
0 0 1
0
0
0 0 0
0 0 0
1
10;
1
.)A
0
1 0 0
3.4 Results of the example J1 = and
J2
=1
We are now able to give all CG-coefficients in form of the a-matrices
ducing the well-known results [31.
0
0
noto a2,pro-
3
Coupling of J , = and 52 = 1 to J = ;:
3
3
Coupling ofJ , = and J2 = 1 to J = 2:
From (8) together with the results of the sections 3.2 and 3.3 we end up with
RI= ( A - V ) o G l
0 -3
6
3
-6
0 -1
-1
0
-3
6
3
-6
116
Ann. Phvsik 5 (1996)
It should be noticed once more that a minus sign in front of a component only indicates that the corresponding CG-coefficient is negative. Thus a minus sign is transferred to the product matrix, although if both the components which are multiplied
are negative. (As we will see in the next section in (13), the components which are
multiplied have the same sign.) The matrix R Ihas been discussed already in section
3.1, where the extraction of the CG-coefficients from it has been demonstrated.
0
1
Coupling of J1 = and 52 = 1 to J = 3:
02 = (A- V)[*]o 62
0 0 3
=(! f:
;)o(:
0
;;)=(:
;;)*
0
1
0
0
3
4 Alternative formulation
So far we are able to compute all CG-coefficients from (8)
a,, = (A- V)"" h,,.
0
Comparing the definition (10) of (A - V)["]
with the definition (12) of
lowing relation between both the matrices can be recognized:
hfl,the fol-
This relation offers the possibility of finding an alternative formulation that requires
only the knowledge of one of the both matrices and G.It can be shown directly by
with 00.
For this purpose the terms (11) in (10) are written
multiplying ( A- V)["]
as
ul!d2!
( U I- (n - k))!(d2- (n - k ) ) !'
(~ld2)'~-~]
=
(dlu2)'" =
d1 !UZ!
(dl - k)!(~2- k)!
'
Multiplying these terms with the components of
yields
K. Schertler and M.H. Thoma, Combinatorial computation of Clebsch-Gordan coefficients
h!
il !
(u1
117
- (n - k))!(dl- k)! (u2 - k)!(d*- (n - k ) ) !*
This expression is proportional to
(
)c+J
j1-n
(n- k)
U ]-
-
( h - n)!
(j2
- n)!
(u, - (n - k))!(dl- k)! (u2 - k)!(d2- (n - k))! '
showing up in (12). Thus (13) is proven, from which the following equations,
equivalent to (8), result:
where fit fino fin etc. The matrix 0,' contains the inverse components of Ro (or
a multiple of them).
As we will show in the appendix, (14) and therefore also (8) are equivalent to the
Racah formula.
0
.-(;
1
0
1
-2
01 ) +
0
0
0
1
0
1
-4
01)
0
0
Furthermore,
1 2 1
6 3 6
6 3 6
The final result reads
*
118
Ann. Physik 5 (1996)
which is equivalent to the result of section 3.4 up to a factor. In conclusion, (14) provides a Xery simple method for a practicable computation of CG-coefficients. For obtaining
and R,' only the matrices Ro(Jl,J2) and no(J1 - ,Jz - 5) are needed.
Actually, the calculation can be simplified further by considering the reflection symmetry of Szo and a, at their center, if n is even. If n is odd, the signs of the reflectedcomponents have to be chan ed. This property of f20 and R, corresponds to the
M 2 ) JJ,M ) = (-)
-J ((-MI,
-M2) IJ, - M ) (see Ref. 2). Because of
relation ((MI,
the component-like multiplication and subtraction of the matrices yielding R,, only
half of the components of R ~ ( J I , Jand
~ ) no(J1 - 4 , J2 - 5) has to be computed.
AJ,
5 Conclusions
In this paper we have demonstrated how the coupling of two angular momenta, J1
and J2, can be understood easily by decomposing each of the angular momenta into
constituent spin-&particles. Counting the possibilities of decomposing the angular
momenta Ji into these constituent particles by simple combinatorial manipulations
(binomial coefficients), leads to the CG-coefficients in the case of coupling to maximum angular momentum J = J1 J2. Here the basic assumption of equally distributed probabilities for each of the decompositions has been adopted. For extracting the
CG-coefficients, it is convenient to introduce the matrix no, defined in (6). The CGcoefficients can be read off directly from the components of this matrix divided by
the sum of the diagonal elements and extracting the square root.
The general case, J = J1 + J2 - n, can be considered as the coupling of 2n constituent spin+particles to spin 0 and the coupling of the remaining constituent particles
to maximum spin. In this way the matrix R,, defined in (8), results, from which the
CG-coefficients can be read off in the same way as from
apart from the fact that
minus signs in front of the components simply indicate negative CG-coefficients. Utilizing a relation between the matrix describing the coupling of the constituent particles to spin 0 and the one describing the coupling of the remaining system to
maximum spin, the matrix a,, can be found easily by (14). This method, shown to
be equivalent to the Racah formula, requires much less computational effort than the
latter one.
The new, probabilistic interpretation of the addition of angular momenta, presented
here, provides an elegant and practicable way of computing all CG-coefficients for
the coupling of two arbitrary angular momenta.
+
a,
Appendix: Equivalence to the Racah formula
Here (14), R,;@
o
shall be derived from the Racah formula (see e.g. Ref. 2 )
K. Schefller a d M. H.Thoma, Combinatonal computation of Clebsch-Gordan coefficients
119
+ J2 - J ) ! ( J , - J2 + J)!(J2 - J1 + J ) !
(51
X
(5,
+ Ml)!(J2+ M2)!(J+ M)!(Jl- M,)!(J2
- M2)!(J- M)!
+ M I + k)!2(J - J I - M2 + k)!2(J1+ 52 - J - k)!2
( J - 52
'
(A. 1)
For this purpose we express the angular momenta showing up in the Racah formula
by the number of constituent particles, substituting
Then we find
u = UI
+ u2 - n,
The components R,},l,,lof the matrix R, follow by squaring the CG-coefficients, taking care of their sign (indicated by the exponent 121). The first square root in (A2)
can be omitted, &cause it is constant within the diagonals of the matrix. T ~ U Swe
get
(jl - n>!(j2- n)!ul!dl!u2!d2!
12'.
nflIw,u, =
/(dl - k)!2(~~
- n k)!2(d2- n + k)!2(u2- k)!2
(T(-Ik(p>
+
)
(A.3 )
Multiplying (A3) with the components
of the matrix
leads to (neglecting the constant factor jl !j2!)
120
The 5h.s is identical with the square of the components of
Q,32: o SZ;' has been derived from the Racah formula.
Ann. Physik 5 (1996)
fin in
(12). Hence
References
J.J. Sakurai, Modem Quantum Mechanics, Addison-Wesley, Reading 1994
A. Messiah, Quantum mechanics, Volume 11, North Holland, Amsterdam, 1963
Particle data group, Phys. Rev. D 50 (1994) 1173
J. Schwinger, in: Quantum Theory of Angular Momentum, edited by L.C.Biedenharn and H. Van
Dam, Academic Press, New York 1965, p. 229
[ 5 ] M. Abrarnowitz, LA. Stegun, Handbook of Mathematical Functions, National Bureau of Standards,
Washington 1964
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