ANGULAR MOMENTUM THEORY

Paul E.S. WormerInstitute of Theoretical Chemistry, University of Nijmegen

Toernooiveld, 6525 ED Nijmegen, The Netherlands

E-mail: pwormer@theochem.kun.nl

I. INTRODUCTION

The best known example of an angular momentum operator is the orbital angular mo-mentum l associated with a vector r,

l ≡ −ir ×∇r, (1)

also written as

li = −i∑

jk

εijkrj∂

∂rk, with i, j, k = 1, 2, 3 = x, y, z, (2)

where εijk is the Levi-Civita antisymmetric tensor,

εijk =

0 if two or more indices are equal,1 if ijk is an even permutation of 1 2 3,−1 if ijk is an odd permutation of 1 2 3.

(3)

The nabla operator is: ∇r = (∇1,∇2,∇3) with ∇i = ∂/∂ri.It is well-known that the li satisfy the commutation relations

[li, lj] ≡ lilj − ljli = i∑

k

εijklk. (4)

Often one takes the latter equations as a definition. Three operators ji that satisfy thecommutation relations (4) are then by definition angular momentum operators.

The importance of angular momentum operators in quantum physics is due to the factthat they are constants of the motion (Hermitian operators that commute with the Hamil-tonian) in the case of isotropic interactions and also in the case of the motion of a quantumsystem in isotropic space. This will be shown in the next section, where we will derive therelation between 3-dimensional rotations and angular momentum operators.

In the third section we introduce abstract angular momentum operators acting on ab-stract spaces, and in the fourth we consider the decomposition of tensor products of thesespaces.

Then we will introduce irreducible tensor operators (angular momentum operators arean example of such operators) and discuss the celebrated Wigner-Eckart theorem.

In the sixth section we will discuss the concept of recoupling and introduce the 6j-symbol. In the final section we will show how matrix elements, appearing in the close-coupling approach to atom-diatom scattering, may be evaluated by an extension of the

1

Wigner-Eckart theorem and that the algebraic (easy) part of the calculation requires theevaluation of a 6j-symbol.

These notes form the content of a six hour lecture course for graduate students special-izing in theoretical chemistry. Since obviously six hours are not enough to cover all detailsof the program just outlined, most of the derivations are delegated to appendices and wereskipped during the lectures. Furthermore the exposition will be restricted to 3-dimensionalrotations; spin or half-integer quantum numbers will not be discussed.

II. THE CONNECTION BETWEEN ROTATIONS AND ANGULAR MOMENTA

In this section we will subsequently rotate: vectors, functions, operators, and N -particlesystems.

A. The rotation of vectors

A proper rotation of the real 3-dimensional Euclidean space is in one-to-one correspon-dence with a real orthogonal matrix R with unit determinant. That is, if the vector r ′ isobtained by rotating r, then

r′ = Rr, with RT = R

−1 and detR = 1. (5)

A product of two proper (unit determinant) orthogonal matrices is again a proper or-thogonal matrix. The inverse of such a matrix exists and is again orthogonal and has unitdeterminant. Matrix multiplication satisfies the associative law

((AB)C) = (A(BC)). (6)

These properties show that the (infinite) set of all orthogonal matrices form a group: thespecial (unit determinant) orthogonal group in three dimensions SO(3).

It is well-known that any rotation has a rotation axis. This fact was first proved geomet-rically by Leonhard Euler1, and is therefore known as Euler’s theorem. We will prove thisfact algebraically and notice to this end that it is equivalent to the statement that everyorthogonal matrix has an eigenvector with unit eigenvalue. Knowing the following rules forn × n determinants: detA

T = detA, detAB = detA detB and det(−A) = (−1)n detA, weconsider the secular equation of R with λ = 1,

det (R− 11) = det((R− 11)T

)= det

(RT (11− R)

)= detR

T det (11− R)

= det (11− R) = − det (R− 11) , (7)

so that det (R− 11) = 0 and λ = 1 is indeed a solution of the secular equation of R. Since anorthogonal matrix is a normal matrix, see Appendix A, it has three orthonormal eigenvectors.The product of eigenvalues of R being equal to detR = 1, the degeneracy of λ = 1 is either

1Novi Commentarii Academiae Scientiarium Petropolitanae 20, 189 (1776).

2

one or three. In the latter case R ≡ 11 and the rotation is trivial (any vector is an eigenvectorof 11). The former case proves Euler’s theorem.

We will designate the normalized eigenvector by n. Thus,

Rn = n with |n| = 1. (8)

In other words, any rotation of a vector r can be described as the rotation of r over an angleψ around an axis with direction n. Accordingly we write R as R(n, ψ).

Using the components of n, we define the matrix N by

N ≡

0 −n3 n2

n3 0 −n1

−n2 n1 0

, i.e. Nij = −

∑

k

εijknk and Nr = n× r. (9)

In Appendix B it is shown that

R(n, ψ) = 11 + sinψN + (1− cosψ)N2, (10)

where 11 is the 3× 3 unit matrix.

B. The rotation of functions

Consider an arbitrary function f(r). The rotation operator U(n, ψ) is defined as follows

U(n, ψ)f(r) ≡ f(R(n, ψ)−1r). (11)

[The inverse of the rotation matrix appears here to ensure that the map R(n, ψ) 7→ U(n, ψ) isa homomorphism, i.e., to conserve the order in the multiplication of operators and matrices.]

Let us next consider an infinitesimal rotation over an angle ∆ψ. By infinitesimal wemean that only terms linear in ∆ψ are to be retained: ∆ψ À (∆ψ)2. We will show inAppendix C that

U(n,∆ψ)f(r) = (1− i∆ψn · l)f(r), (12)

where l is defined in Eq. (1). Because of this relation one sometimes refers to n · l as thegenerator of an infinitesimal rotation.

Functions of operators can be defined by using the Taylor series of ‘normal’ functions.For instance, the exponential operator is defined by

e−iψn·l =∞∑

k=0

1

k!(−iψn · l)k. (13)

Formally one must prove that this series makes sense, i.e., that it converges with respect tosome criterion. We ignore these mathematical subtleties and we also assume that differentialequations satisfied by these operators can be solved in the same manner as for ‘normal’functions. It is easy to derive a differential equation for U(n, ψ). Since

3

U(n, ψ +∆ψ)f(r) = U(n, ψ)U(n,∆ψ)f(r) (14)

= U(n, ψ)(1− i∆ψn · l)f(x), (15)

(16)

we find

dU(n, ψ)

dψ= −iU(n, ψ)n · l, (17)

which, since U(n, 0) = 1, has the solution

U(n, ψ) = e−iψn·l. (18)

So, we now have an explicit expression of the rotation operator, defined in (11), which showsmost succinctly the connection with the orbital angular momentum operator l.

C. The rotation of an operator

We will consider the rotation of an operator H(r), where r indicates that H acts onfunctions of r. Writing Ψ′(r) = H(r)Ψ(r), we find

UH(r)Ψ(r) = UΨ′(r) = Ψ′(R−1r) = H(R−1r)Ψ(R−1r) = H(R−1r)UΨ(r), (19)

so that the rotated operator is given by

H(R−1r) = UH(r)U−1. (20)

Notice the difference with the corresponding Eq. (11) defining the rotation of functions.Suppose next that H(r) is invariant under rotation: H(r) = H(R−1r) for all R

−1, thenwe find from (20) that

UH(r)U−1 ≡ e−iψn·lH(r)eiψn·l = H(r). (21)

Upon expansion of the exponentials we find

e−iψn·lHeiψn·l = H + iψ[H,n · l] + · · · = H + iψ∑

i

ni[H, li] + · · · = H. (22)

Since it was assumed that Eq. (21) holds for any n and ψ, we find the very important resultthat U(n, ψ)HU(n, ψ)−1 = H if and only if [H, li] = 0 for i = 1, 2, 3.

Suppose H(r) is the Hamiltonian of a certain one-particle system, for instance a particlemoving in a spherical symmetric field due to a nucleus, or a particle moving in field-freespace, then H(r) is rotationally invariant. The unitary operators U(n, ψ) are symmetryoperators (unitary operators that commute with the Hamiltonian) and the li are constantsof the motion of such a system.

4

D. The rotation of an N-particle system

The theory of this section was so far derived for the rotation of one vector r, say thecoordinate of one electron. It is easy to generalize the theory to N particles. If we simulta-neously rotate all coordinate vectors r(α) (α = 1, . . . , N) around the same axis n and overthe same angle ψ, then the N -particle rotation operator still has the exponential form ofEq. (18), but the angular momentum operator is now given by

L =N∑

α=1

l(α), (23)

where l(α) = −ir(α)×∇(α). This is due to the fact that [li(α), lj(β)] = 0 (for α 6= β), sothat the N -particle rotation factorizes into a product of one-particle rotations,

e−iψn·L =N∏

α=1

e−iψn·l(α). (24)

If the N particles do not interact, the separate rotation of any coordinate vector is asymmetry operation, and the angular momenta l(α) are all constants of the motion. If, onthe other hand, the particles do interact via an isotropic interaction potential V (rαβ), withrαβ = |r(α) − r(β)|, then obviously only the same simultaneous rotation of all particlesconserves the interparticle distances and commutes with the Hamiltonian. Thus, in the caseof interacting particles only the rotation in Eq. (24) is a symmetry operation and only thetotal angular momentum L is a constant of the motion.

E. Exercises

1. We have seen that n is an eigenvector of R(n, ψ) with unit eigenvalue. What are theother two eigenvalues of this matrix?

2. Given the vector n′ = Sn with orthogonal S. Prove that R(n′, ψ) = SR(n, ψ)ST .

3. The tetrahedral group T consists of twelve proper (unit determinant) rotation matricesthat map the four vectors

111

,

−1−11

,

−11−1

and

1−1−1

(25)

onto each other. Give the twelve matrices. Hint: Use Eq. (10) for one 3-fold rotation,the result of the previous exercise and the fact that R

T = R−1.

4. Prove that R(n, ψ) = exp(ψN), where n and N are related by Eq. (9). Hint: use thatN

3 = −N.

5. Show by expanding and considering the first few terms that eAeB = eA+B if and only if[A,B] = 0.

5

III. EIGENSPACES OF j2 AND j3

We consider an abstract Hilbert space L and assume that it is invariant under theHermitian operators ji, i = 1, 2, 3 that satisfy the commutation relations (4), i.e.,

[ji, jj] = i∑

k

εijkjk.

In one of the first papers on quantum mechanics1 it was already shown that L decomposesinto a direct sum of subspaces V j

α spanned by orthonormal kets

|α, j,m 〉, m = −j, . . . , j, (26)

with

j2|α, j,m 〉 = j(j + 1)|α, j,m 〉, (27)

j3|α, j,m 〉 = m|α, j,m 〉, (28)

j±|α, j,m 〉 =√j(j + 1)−m(m± 1)|α, j,m± 1 〉, (29)

where j2 =∑i j

2i , and j± = j1 ± ij2 are the well-known step up/down operators. It may

happen that an eigenspace of j2, characterized by a certain quantum number j, occurs morethan once in L. The index α resolves this multiplicity. Having made this point, we will fromhere on suppress α in our notation.

The proof of the existence of basis (26) is very well-known. Briefly, the main argumentsare:

• As [j2, j3] = 0, we can find a common eigenvector | a, b 〉 of j2 and j3 with j2| a, b 〉 =a2| a, b 〉 and j3| a, b 〉 = b| a, b 〉. (Since it is easy to show that j2 has only non-negativereal eigenvalues, we write its eigenvalue as a squared number).

• Considering the commutation relations [j±, j3] = ±j± and [j2, j±] = 0, we find, thatj2j+| a, b 〉 = a2j+| a, b 〉 and j3j+| a, b 〉 = (b+ 1)j+| a, b 〉. Hence j+| a, b 〉 = | a, b+ 1 〉

• If we apply j+ now k + 1 times we obtain, using j†+ = j−, the ket | a, b + k + 1 〉 withnorm

〈 a, b+ k | j−j+ | a, b+ k 〉 = [a2 − (b+ k)(b+ k + 1)]〈 a, b+ k | a, b+ k 〉. (30)

Thus, if we let k increase, there comes a point that the norm on the left hand sidewould have to be negative (or zero), while the norm on the right hand side would stillbe positive. A negative norm is in contradiction with the fact that the ket belongs tothe Hilbert space L. Hence there must exist a value of the integer k, such that the ket| a, b+k 〉 6= 0, while | a, b+k+1 〉 = 0. Also a2 = (b+k)(b+k+1) for that value of k.

• Similarly l+1 times application of j− gives a zero ket | a, b− l− 1 〉 with | a, b− l 〉 6= 0and a2 = (b− l)(b− l − 1).

1M. Born, W. Heisenberg, and P. Jordan, Z. Phys. 35, 557–615 (1926).

6

• From the fact that a2 = (b + k)(b + k + 1) = (b − l)(b − l − 1) follows 2b = l − k, sothat b is integer or half-integer. This quantum number is traditionally designated bym. The maximum value of m will be designated by j. Hence a2 = j(j + 1).

• Requiring that | j,m 〉 and j±| j,m 〉 are normalized and fixing phases, we obtain thewell-known formula (29).

The spaces with half-integer j (and m) belong to a spin Hilbert space. In agreement withwhat was stated in the introduction, we will not consider half-integer j any further.

We can define a unitary rotation operator, as in Eq. (18)

U(n, ψ) ≡ e−iψn·j . (31)

An exponential operator is defined by its expansion, cf. Eq. (13). Since the eigenspace V j

of j2 spanned by the basis (26) is invariant under j1, j2, and j3, it is also invariant underU(n, ψ) and we can define its matrix with respect to that orthonormal basis

U(n, ψ)| j,m 〉 =j∑

m′=−j

| j,m′ 〉〈 j,m′ |U(n, ψ) | j,m 〉, m = −j, . . . , j. (32)

Often one writes

Djm′,m(n, ψ) ≡ 〈 j,m′ |U(n, ψ) | j,m 〉, (33)

the so-called Wigner D-matrix.The unitary D-matrices satisfy certain orthogonality relations. In order to explain this,

we note that the map R(n, ψ) 7→ Dj(n, ψ) is a representation of the group SO(3) carried by

V j. (Strictly speaking we must prove at this point that this map satisfies the homomorphismcondition, but we forgo this proof). The space V j is not only invariant under SO(3), butalso irreducible, which means that we cannot find a proper subspace of V j that is invariantunder {U(n, ψ)}. (We omit the proof of this fact, too). Next, we recall from elementarygroup theory the so-called great orthogonality relations for irreducible representations

∑

g∈G

Dλij(g)

−1Dµkl(g) = δλµδjkδil

|G|fλ

, (34)

where λ and µ label irreducible matrix representations of the finite group G of order |G|.The number fλ is the dimension of D

λ. These orthogonality relations can be generalized tocertain kinds of groups of infinite order, among which SO(3). In this generalization the sumover the finite group is replaced by integrals over the parameter space. Because a rotationaround n over the angle 2π − ψ is the same as the rotation over ψ around −n, we cover allof SO(3) if we restrict ψ to run from 0 to π and n over the unit sphere. So, if θ and φ arethe spherical polar angles of n, the parameter space of SO(3) is

0 ≤ ψ ≤ π, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π. (35)

It requires some knowledge of tensor analysis (or Lie group theory) to derive the propervolume element. Moreover, the actual derivation is tedious, so we give the result withoutproof:

7

dτ = 2(1− cosψ) sin θdψdθdφ hence |SO(3)| =∫ π

0

∫ π

0

∫ 2π

0dτ = 8π2. (36)

The Wigner D-matrices thus satisfy the orthogonality relations

∫ π

0

∫ π

0

∫ 2π

0Dj1m′

1m1

(n, ψ)∗Dj2m′

2m2

(n, ψ) dτ = δj1,j2δm′

1,m′

2δm1,m2

8π2

2j1 + 1, (37)

where we have used that the D-matrix is unitary

Dj1m′

1m1

(n, ψ)∗ = Dj1m1m′

1

(n, ψ)−1. (38)

A. Exercises

6. Prove by only using (4) the commutation relations: [j2, ji] = 0 for i = 1, 2, 3 and[j±, j3] = ±j±.

7. Show that j−j+ = j2 − j3(j3 + 1), again by only using (4).

8. Show from (10) that R(−n, 2π − ψ) = R(n, ψ).

IV. VECTOR COUPLING

In this section we will indicate operators with a hat (as A), in order to distinguish clearlybetween the operators and their quantum numbers.

Consider two different spaces V j1 and V j2 spanned by | j1,m1 〉, m1 = −j1, . . . , j1, and| j2,m2 〉, m2 = −j2, . . . , j2, respectively. The tensor product space V j1⊗V j2 is by definitionthe space spanned by the product kets | j1,m1 〉 ⊗ | j2,m2 〉 ≡ | j1m1; j2m2 〉. The productspace is invariant under the operators Ji ≡ ji ⊗ 1 + 1 ⊗ ji, i = 1, 2, 3. This means thatthe elements of V j1 ⊗ V j2 are mapped onto elements in the space under the action of theoperators, so that we can diagonalize J2 = J2

1 + J22 + J2

3 and J3 on the product space. [Thereason why we would like to do this was pointed out above: it may be that J2 is a constantof the motion and that j2(1) and j2(2) are not]. The two-particle operators Ji obviouslysatisfy the angular momentum commutation relations.

A remark on the notation is in order at this point. In quantum chemistry one usuallywrites a two-particle orbital product as ψ(1)φ(2). In mathematics one would write moreoften ψ ⊗ φ, i.e., the orbital product is an element of a tensor product space. Notice thatψ ⊗ φ 6= φ ⊗ ψ just as ψ(1)φ(2) 6= φ(1)ψ(2). Operators on a tensor product space may betensor product operators themselves, written as A⊗ B. Their action is defined by

(A⊗ B)ψ ⊗ φ ≡ (Aψ)⊗ (Bφ). (39)

The matrix of this operator is a Kronecker product matrix. In quantum chemistry one wouldwrite this expression as

A(1)B(2)ψ(1)φ(2) = (Aψ(1))(Bφ(2)). (40)

8

Clearly the operator ji ⊗ 1 + 1 ⊗ ji ≡ ji(1) + ji(2). Note that the terms commute, whichwas already used to arrive at Eq. (24). We will use the quantum chemical as well as tensornotation, whichever is the most convenient.

As the reasoning by which we decomposed a Hilbert space is valid for any angularmomentum, we know that the product space must contain eigenkets | (j1j2)JM 〉, M =−J, . . . , J of J2 and J3. The pair (j1j2) indicates that the eigenket is formed as a linearcombination of product kets | j1m1; j2m2 〉.

The question now arises which eigenvalues J(J +1) of J2 we may expect in the productspace. This problem was solved (in a different context) by two German mathematicians:R.F.A. Clebsch1 and P.A. Gordan2. The quantum number J occurs only once and it satisfiesthe triangle condition

|j1 − j2| ≤ J ≤ j1 + j2. (41)

The proof of the triangle condition is given in Appendix D. By using the resolution of theidentity on V j1 ⊗ V j2 we can write

| (j1j2)JM 〉 =j1∑

m1=−j1

j2∑

m2=−j2

| j1m1; j2m2 〉〈 j1m1; j2m2 | (j1j2)JM 〉. (42)

This is the Clebsch-Gordan series and the coefficients 〈 j1m1; j2m2 | (j1j2)JM 〉 are known asClebsch-Gordan (CG) coefficients. Since {| (j1j2)JM 〉} is also an orthonormal basis of theproduct space V j1 ⊗ V j2 we can write

| j1m1; j2m2 〉 =j1+j2∑

J=|j1−j2|

J∑

M=−J

| (j1j2)JM 〉〈 (j1j2)JM | j1m1; j2m2 〉. (43)

In Appendix E we give an explicit expression (without proof) for the CG-coefficients, whichshows that they are real numbers. Since the CG-coefficients give a transformation be-tween orthonormal bases, they constitute a unitary matrix. Moreover, they are real sothat 〈 j1m1; j2m2 | (j1j2)JM 〉 = 〈 (j1j2)JM | j1m1; j2m2 〉. Some more properties of the CG-coefficients are given in Appendix E. In the following we will usually drop the quantumnumbers j1 and j2 standing next to JM in the CG-coefficients because they are superfluous.

The symmetry relations of CG-coefficients (see Appendix E) are awkward and difficultto remember. Wigner3 introduced the following coefficients, 3j-symbols, that have morepleasant symmetry properties:

(j1 j2 j3m1 m2 m3

)≡ (−1)j1−j2−m3

√2j3 + 1

〈 j1m1; j2m2 | j3,−m3 〉. (44)

From the results in Appendix E it follows easily that

1Theorie der binaren algebraische Formen, Teubner, Leipzig, (1872).2Uber das Formensystem binarer Formen, Teubner, Leipzig, (1875).3Unpublished notes (1940) reprinted in: Quantum Theory of Angular Momentum, L.C. Bieden-

harn and H. van Dam, Academic, New York (1965).

9

(j1 j2 j3m1 m2 m3

)= (−1)j1+j2+j3

(j2 j1 j3m2 m1 m3

)=(j3 j1 j2m3 m1 m2

), (45)

i.e., the 3j-symbol is invariant under a cyclic (even) permutation of its columns and obtainsthe phase (−1)j1+j2+j3 upon an odd permutation (transposition) of any two of its columns.Further

(j1 j2 j3−m1 −m2 −m3

)= (−1)j1+j2+j3

(j1 j2 j3m1 m2 m3

), (46)

and finally we find directly from Eq. (44) that the 3j-symbol vanishes unless m1+m2+m3 =0.

It is of importance to see what happens to the matrix of the rotation operator (31),or rather to its two-particle equivalent, which describes the simultaneous rotation of twoparticles over the same axis and around the same angle

U(n, ψ) ≡ e−iψn·J = e−iψn·(j(1)+j(2)) ≡ e−iψn·j ⊗ e−iψn·j , (47)

when we decompose the product space V j1 ⊗ V j2 into eigenspaces of J2. The matrix〈 J ′M ′ |U(n, ψ) | JM 〉 is diagonal in J and J ′ because the spaces spanned by | JM 〉 areinvariant under J1, J2 and J3. [Observe that we suppressed (j1j2) in | JM 〉. It is under-stood that | JM 〉 belongs to V j1⊗V j2 for a fixed pair (j1, j2). The running indices describingbases are either the pair m1,m2, or the pair J,M ]. Introducing twice the resolution of theidentity, we find

〈 J ′M ′ | e−iψn·J | JM 〉 =∑

m1m2m′

1m′

2

〈 J ′M ′ | j1m′1; j2m

′2 〉〈 j1m′

1; j2m′2 | e−iψn·J | j1m1; j2m2 〉

×〈 j1m1; j2m2 | JM 〉 (48)

from which follows,

∑

m1m2m′

1m′

2

〈 J ′M ′ | j1m′1; j2m

′2 〉〈 j1m′

1 | e−iψn·j(1) | j1m1 〉〈 j2m′2 | e−iψn·j(2) | j2m2 〉

×〈 j1m1; j2m2 | JM 〉 = δJ ′J 〈 JM ′ | e−iψn·J | JM 〉. (49)

This may be rewritten as

∑

m1m2

m′

1m

′

2

〈 J ′M ′ | j1m′1; j2m

′2 〉(Dj1(n, ψ)⊗ D

j2(n, ψ))m′

1m′

2;m1m2

〈 j1m1; j2m2 | JM 〉

= δJ ′J DJM ′M(n, ψ). (50)

Equation (50) shows that the Kronecker product matrix Dj1(n, ψ) ⊗ D

j2(n, ψ), which inprinciple is a completely filled matrix, is block-diagonalized by means of a unitary similaritytransformation by CG-coefficients. The blocks on the diagonal in the matrix on the lefthand side are labeled by J = |j1 − j2|, . . . , j1 + j2, block J being of dimension 2J + 1. It isalso easy to go the other way,

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〈 j1m′1; j2m

′2 | e−iψn·J | j1m1; j2m2 〉 =

∑

JMM ′

〈 j1m′1; j2m

′2 | JM ′ 〉 (51)

×〈 JM ′ | e−iψn·J | JM 〉〈 JM | j1m1; j2m2 〉.

This may be rewritten as(Dj1(n, ψ)⊗ D

j2(n, ψ))m′

1m′

2;m1m2

=∑

JMM ′

〈 j1m′1; j2m

′2 | JM ′ 〉 DJ

M ′M(n, ψ) (52)

×〈 JM | j1m1; j2m2 〉.

We are now in a position to show the following useful relation, which we will need in theproof of the Wigner-Eckart theorem:

∫Dj1m′

1m1

(n, ψ)∗Dj2m′

2m2

(n, ψ)Dj3m′

3m3

(n, ψ)dτ =8π2

2j1 + 1〈 j1m′

1 | j2m′2; j3m

′3 〉 (53)

×〈 j1m1 | j2m2; j3m3 〉,

where the integral and its volume element are defined in Eq. (36). To show this relation wefirst use Eq. (52), so that (suppressing integration variables) we get

∫(Dj1

m′

1m1

)∗Dj2m′

2m2Dj3m′

3m3dτ =

∑

JM ′M

〈 j2m′2; j3m

′3 | JM ′ 〉〈 JM | j2m2; j3m3 〉 (54)

×∫

(Dj1m′

1m1

)∗DJM ′Mdτ

By the use of (37) we find

∫(Dj1

m′

1m1

)∗DJM ′Mdτ = δj1Jδm′

1M ′δm1M

8π2

2J + 1, (55)

so that∫

(Dj1m′

1m1

)∗Dj2m′

2m2Dj3m′

3m3dτ =

∑

JM ′M

δj1Jδm′

1M ′δm1M

8π2

2J + 1

×〈 j2m′2; j3m

′3 | JM ′ 〉〈 JM | j2m2; j3m3 〉

=8π2

2j1 + 1〈 j2m′

2; j3m′3 | j1m′

1 〉〈 j1m1 | j2m2; j3m3 〉, (56)

which proves Eq. (53).

A. Exercises

9. Evaluate 〈 20 | 11; 1− 1 〉 by the explicit expression in Appendix E.

10. Decompose explicitly the space V 1⊗V 1 by the use of step-down operators. Hint: thisproduct space is invariant under the transposition P12. This observation may be usedin the required orthogonalizations.

11. Prove Eqs. (45) and (46) from the relations in Appendix E.

11

V. THE WIGNER-ECKART THEOREM

The Wigner-Eckart theorem1 is a tool to evaluate matrix elements of tensor operators,and so we start this section with the introduction of such operators. Let us first consider asimple example, namely a tensor operator of rank one, also known as a vector operator. Wehave seen in Eq. (20) that an arbitrary one-particle operator Q(x) rotates as follows

Q(R(n, ψ)−1x) = U(n, ψ)Q(x)U(n, ψ)†. (57)

Thus, each component of the coordinate operator transforms in this way, and we can write

R(n, ψ)−1x = U(n, ψ)xU(n, ψ)†, (58)

or

U(n, ψ)xiU(n, ψ)† =

3∑

j=1

xjRji(n, ψ). (59)

This equation serves as the definition of a vector operator: a vector operator consists of threecomponents that upon rotation of the system transform among themselves as in Eq. (59).Another example is ∇r = (∇1,∇2,∇3). Recall that a vector product x × y rotates as avector

(Rx)× (Ry) = (detR) R(x× y) (60)

and for a proper rotation detR = 1. It is then evident that the angular momentum definedas a vector product in Eq. (1) is another example of a vector operator.

It is often convenient to work with spherical vectors, defined by

(x1, x0, x−1) ≡ (x, y, z) S with S =1√2

−1 0 1−i 0 −i0√2 0

. (61)

Observe that this is the same transformation as between real and complex atomic p-orbitals,and indeed, the spherical components of a vector are proportional to r Y 1

m(θ, φ), whereY 1m(θ, φ) is a spherical harmonic function and m = 1, 0,−1. Recall that spherical harmonic

functions satisfy the relations (27), (28), and (29), where the angular momentum is theorbital angular momentum of Eq. (1).

If we compute the matrix of U(n, ψ) on basis of Y 1m(θ, φ), with m = 1, 0,−1, we obtain

the matrix D1(n, ψ). If we perform the transformation (61) on this matrix, we get R(n, ψ),

i.e.,

S D1(n, ψ) S

† = R(n, ψ) (62)

1C. Eckart, Rev. Mod. Phys. 2, 305 (1930); E. Wigner, Gruppentheorie und ihre Anwendung auf

die Quantenmechanik der Atomspektren, Vieweg, Braunschweig, 1931. Translated by J.J. Griffin

as: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic,

New York (1959).

12

and so,(S†

R(n, ψ) S

)m′m

= D1m′m(n, ψ). (63)

Using this equation and Eq. (61), we can rewrite Eq. (59) in terms of spherical components,

U(n, ψ) xm U(n, ψ)† =1∑

m=−1

xm′D1m′m(n, ψ). (64)

We now come to the definition of a spherical tensor operator, which is defined withrespect to a set of axes attached to a quantum mechanical system (an atom, a molecule,etc.). If the 2j + 1 operators T jm transform among themselves as

U(n, ψ)T jmU(n, ψ)† =

j∑

m=−j

T jm′Djm′m(n, ψ). (65)

upon rotation of the system, then the set {T jm|m = −j, . . . , j} is called an irreducible spher-ical tensor operator. The adjective ‘irreducible’ refers to the fact that the D-matrices con-stitute an irreducible matrix representation of SO(3).

A very well-known example of a tensor operator is the electric multipole operator

M lm ≡

√4π

2l + 1

N∑

α=1

ZαrlαY

lm(θα, φα), (66)

where rα, θα and φα are the spherical polar coordinates of particle α and Zα is its charge.From the fact that spherical harmonics are eigenfunctions of l2 and l3 follows

U(n, ψ)|Y lm 〉 =

l∑

m′=−l

|Y lm′ 〉Dl

m′m(n, ψ), (67)

and considered as a one-particle multiplicative (local) operator a spherical harmonic satisfiestherefore (65). Hence the multipole operator is a tensor operator, provided we rotate all Nparticles simultaneously.

A rank-zero tensor operator has by definition j = 0. Since D0 is unity, we see that its

definition (65) coincides with Eq. (21), where a rotationally invariant operator was defined.One also refers to rank-zero operators as scalar operators in analogy to rank-one operators,which are vector operators.

The following is the famous Wigner-Eckart theorem:

〈 γ′j′m′ |T kq | γjm 〉 = 〈 j ′m′ | kq; jm 〉〈 γ ′j′ | |T k| | γj 〉, (68)

which states that the (2j ′ + 1)(2k + 1)(2j + 1) different matrix elements are all given by aCG-coefficient (the algebraic part) times a so-called reduced matrix element 〈 γ ′j′ | |T j| | γj 〉,which contains the dynamics of the problem and hence is the difficult part. We introducedthe extra quantum numbers γ and γ ′ to remind us of the fact that in general j and m arenot sufficient to label a state.

The proof of the Wigner-Eckart theorem is simple. Since U(n, ψ) is unitary, we have

13

C ≡ 〈 γ′j′m′ |T kq | γjm 〉 = 〈U(n, ψ)γ ′j′m′ |U(n, ψ)T kq U(n, ψ)† |U(n, ψ)γjm 〉. (69)

We may integrate both sides of this equation over SO(3). On the left hand side this givessimply 8π2C. In the matrix element on the right hand side we first let U(n, ψ) act in bra,ket, and on the operator and obtain

8π2C =∑

µ′q′µ

〈 γ′j′µ′ |T kq′ | γjµ 〉∫Dj′

µ′m′(n, ψ)∗Dkq′q(n, ψ)D

jµm(n, ψ)dτ . (70)

When we substitute Eq. (53) into Eq. (70) and define the following quantity

〈 γ′j′ | |T k| | γj 〉 ≡ 1

2j′ + 1

∑

µ′q′µ

〈 j′µ′ | kq′; jµ 〉〈 γ′j′µ′ |T kq′ | γjµ 〉, (71)

which obviously does not depend on µ′, q′ or µ, we get

C = 〈 j ′m′ | kq; jm 〉〈 γ ′j′ | |T k| | γj 〉. (72)

This proves the Wigner-Eckart theorem.The Clebsch-Gordan coefficient arising in the case of a scalar operator is,

〈 j′m′ | 00; jm 〉 = δj′jδm′m. (73)

So, we see that only states of the same j and m quantum numbers mix under a scalaroperator. An example of such an operator is the Hamiltonian H of an atom,

〈 γ′j′m′ |H | γjm 〉 = δj′jδm′m〈 γ′j | |H| | γj 〉 (74)

and we see here a confirmation of the fact that j2 is a constant of the motion of an atom,as the Hamiltonian matrix on basis of | γjm 〉 is obviously block diagonal. The blocks arelabeled by j and the quantum numbers γ ′ and γ label the rows and columns of the blocks.Notice also that the matrix element of a scalar operator is independent of m.

As another application of the Wigner-Eckart theorem (68) we discuss the Gaunt series1.This series expresses a coupled product of two spherical harmonics depending on the sameangles as a spherical harmonic,

∑

m1m2

Y l1m1

(θ, φ)Y l2m2

(θ, φ)〈 l1m1; l2m2 |LM 〉 =[(2l1 + 1)(2l2 + 1)

4π(2L+ 1)

] 1

2

〈 l10; l20 |L0 〉Y LM(θ, φ).

(75)

The Gaunt series resembles the CG-series, but differs in the fact that the CG-series pertainsto different coordinates, for instance the coordinate vectors of two different particles. Toprove Eq. (75), we observe that Y λ

µ (θ, φ) is complete, hence we may write

∑

m1m2

Y l1m1

(θ, φ)Y l2m2

(θ, φ)〈 l1m1; l2m2 |LM 〉 =∑

λµ

CLMλµ Y λ

µ (θφ), (76)

1J.A. Gaunt, Trans. Roy. Soc. A228, 151 (1929).

14

where CLMλµ is an expansion coefficient. Project both sides with 〈Y l3

m3| and we get by the

orthonormality of the spherical harmonics

∑

m1m2

〈Y l3m3|Y l1

m1|Y l2

m2〉〈 l1m1; l2m2 |LM 〉 = CLM

l3m3. (77)

Apply the Wigner-Eckart theorem:

〈Y l3m3|Y l1

m1|Y l2

m2〉 = 〈 l3 | |l1| | l2 〉 〈 l3m3 | l1m1; l2m2 〉 (78)

and use

∑

m1m2

〈 l3m3 | l1m1; l2m2 〉〈 l1m1; l2m2 |LM 〉 = δl3Lδm3M . (79)

Then from Eq. (77)

CLMl3m3

= 〈L | |l1| | l2 〉δl3Lδm3M , (80)

which substituted into (76) gives

∑

m1m2

Y l1m1

(θ, φ)Y l2m2

(θ, φ)〈 l1m1; l2m2 |LM 〉 = 〈L | |l1| | l2 〉Y LM(θ, φ). (81)

In order to determine the reduced matrix element we use the fact that

Y lm(0, 0) =

[2l + 1

4π

] 1

2

δm0, (82)

and insert this into the left and the right hand side of (81). Thus

[(2l1 + 1)(2l2 + 1)

16π2

] 1

2

〈 l10; l20 |L0 〉 = 〈L | |l1| | l2 〉[2L+ 1

4π

] 1

2

, (83)

and so

〈L | |l1| | l2 〉 =[(2l1 + 1)(2l2 + 1)

4π(2L+ 1)

] 1

2

〈L0 | l10; l20 〉, (84)

which, after substitution into (81), proves Eq. (75).The integral in Eq. (78) is often written in terms of 3j-symbols

〈Y l1m1|Y l2

m2|Y l3

m3〉 = (−)m1

[(2l1 + 1)(2l2 + 1)(2l3 + 1)

4π

] 1

2(l1 l2 l30 0 0

)(l1 l2 l3−m1 m2 m3

).

(85)

This integral is known as a Gaunt coefficient. From the symmetry relation (46) followsimmediately that the matrix element vanishes if l1 + l2 + l3 is odd. Here we see that theWigner-Eckart theorem furnishes a selection rule, and, indeed, the providing of selectionrules is one of the important applications of the theorem.

15

A. Exercise

12. Given that

(1 j j0 −j j

)= −

√j

(2j + 1)(j + 1)compute

(1 j j1 −j j − 1

). (86)

Hints:(i) From Eq.(61) follows that

j11 ≡ −

1√2(j1 + ij2), j1

0 ≡ j3, and j1−1 ≡

1√2(j1 − ij2) (87)

are the components of a vector operator.(ii) Compute the reduced matrix element 〈 j | |j1| | j 〉 from 〈 jj | j1

0 | jj 〉 and the given3j-symbol.(iii) Use Eq. (29) and the Wigner-Eckart theorem to compute 〈 j, j − 1 | j1

−1 | j, j 〉.

VI. RECOUPLING AND 6j-SYMBOLS

Consider the tensor product space V j1 ⊗ V j2 ⊗ V j3 spanned by

| j1m1; j2m2; j3m3 〉 ≡ | j1m1 〉 ⊗ | j2m2 〉 ⊗ | j3m3 〉,

where the kets satisfy Eqs. (27), (28), and (29). This space is invariant under

Ji = ji(1) + ji(2) + ji(3) ≡ ji ⊗ 1⊗ 1 + 1⊗ ji ⊗ 1 + 1⊗ 1⊗ ji with i = 1, 2, 3 = x, y, z.

We may diagonalize J2 and J3 on this space, which can be done most easily by repeatedClebsch-Gordan coupling.

There are three essentially different bases of the total space that may be obtained thisway. We can couple first j1 and j2 to angular momentum j12 and then couple this with j3to J . The basis so obtained consists of kets | ((j1j2)j12j3)JM 〉. We can also couple first j2and j3 and then j1, which leads to | (j1(j2j3)j23)JM 〉 and finally we may couple first j1 andj3 to j13 and then j2. If the order within one pair-coupling is permuted, we do not obtainan essentially new basis, but one which has only a different phase (cf. the third symmetryrelation of CG-coefficients in Appendix E). The three different bases of V j1 ⊗ V j2 ⊗ V j3 areorthonormal, and all three give a resolution of the identity. Thus, for instance,

| ((j1j2)j12j3)JM 〉 =∑

j23

| (j1(j2j3)j23)JM 〉〈 (j1(j2j3)j23)JM | ((j1j2)j12j3)JM 〉. (88)

The Fourier coefficient (overlap matrix element) in Eq. (88) is a recoupling coefficient. Wecan look upon this coefficient as a matrix element of the unit operator, which is a scalaroperator, so that by the Wigner-Eckart theorem the recoupling coefficient is diagonal in Jand M and does not depend on M; therefore we will drop M in the recoupling coefficient.

By inserting the definition of the coupled kets into the left and right hand side of Eq. (88)we can relate the recoupling coefficient to CG-coefficients,

16

∑

m1m2m3m12

| j1m1; j2m2; j3m3 〉〈 j1m1; j2m2 | j12m12 〉〈 j12m12; j3m3 | JM 〉

=∑

j23

〈 (j1(j2j3)j23)J | ((j1j2)j12j3)J 〉∑

m1m2m3m23

| j1m1; j2m2; j3m3 〉

×〈 j2m2; j3m3 | j23m23 〉〈 j1m1; j23m23 | JM 〉. (89)

Equating the coefficients of the product kets gives

∑

m12

〈 j1m1; j2m2 | j12m12 〉〈 j12m12; j3m3 | JM 〉 =∑

j23m23

〈 (j1(j2j3)j23)J | ((j1j2)j12j3)J 〉

×〈 j2m2; j3m3 | j23m23 〉〈 j1m1; j23m23 | JM 〉. (90)

Multiply both sides with 〈 j ′23m′23 | j2m2; j3m3 〉 and sum over m2 and m3, use the unitarity

of CG-coefficients, i.e.,

∑

m2m3

〈 j′23m′23 | j2m2; j3m3 〉〈 j2m2; j3m3 | j23m23 〉 = δj′

23j23δm′

23m23

, (91)

then multiply both sides by 〈 J ′M ′ | j1m1; j′23m

′23 〉 sum over m1 and m′

23 and use again theunitarity, we then find (dropping primes)

〈 (j1(j2j3)j23)J | ((j1j2)j12j3)J 〉 =∑

m1m2m3m12m23

〈 JM | j1m1; j23m23 〉〈 j23m23 | j2m2; j3m3 〉

×〈 j1m1; j2m2 | j12m12 〉〈 j12m12; j3m3 | JM 〉. (92)

Since the recoupling coefficient is independent of M , we may sum the right hand side overM , provided we divide by 2J + 1. A 6j-symbol is proportional to the recoupling coefficient,but somewhat more symmetric:

{j3 j12 Jj1 j23 j2

}= (−1)j1+j2+j3+J [(2j12 + 1)(2j23 + 1)]−

1

2 〈 (j1(j2j3)j23)J | ((j1j2)j12j3)J 〉.

(93)

An alternative notation that is often used, is the W -coefficient of Racah:

W (j1j2j3j4; j5j6) ≡ (−1)j1+j2+j3+j4

{j1 j2 j5j4 j3 j6

}(94)

By Eq. (92) a 6j-symbol is given as a sum of products of four CG-coefficients, or alternatively,as a sum of products of four 3j-symbols. Since the m quantum numbers in Eq. (92) satisfylinear relations, e.g. m12 = m1 +m2, this equation contains in fact only a double sum. EachCG-coefficient is given by a single summation (cf. Appendix E) and hence a 6j-symbol isdefined as a six-fold summation. Racah1 was able to reduce this six-fold sum to a single sumby a series of substitutions that were so incredibly ingenious that no-one tried to explainthis ever since. We will not make an attempt either, and simply give Racah’s formula inAppendix F.

1G. Racah, Phys. Rev. 62, 438 (1942).

17

The 6j-symbol satisfies two sets of symmetry relations that are easily derived by the useof Jucys’s angular momentum diagrams. Because of time limitations we will not give thederivation, but simply state the results. First, any of the six possible permutations of itscolumns leaves a 6j-symbol invariant, thus

{j1 j2 j3j4 j5 j6

}={j2 j1 j3j5 j4 j6

}={j2 j3 j1j5 j6 j4

}, etc. . (95)

Second, any two elements in the upper row may be interchanged with the elements under-neath them:

{j1 j2 j3j4 j5 j6

}={j4 j5 j3j1 j2 j6

}={j4 j2 j6j1 j5 j3

}={j1 j5 j6j4 j2 j3

}. (96)

The four triangular conditions which must be satisfied by the six angular momenta in the6j-symbol may be illustrated in the following way:

d d d

d

¡¡

d

@@d

d

@@d d

d d

¡¡

d (97)

At the same time these diagrams illustrate the second set of symmetry relations obeyed bythe 6j-symbol.

VII. ATOM-DIATOM SCATTERING

As discussed above, one often meets the case of two subsystems with different configu-ration spaces, that are in eigenstates of j2(1) and j2(2) and that are coupled

| (j1j2)j12m12 〉 ≡∑

m1m2

| j1m1 〉| j2m2 〉〈 j1m2; j2m2 | j12m12 〉 with m12 = m1 +m2, (98)

cf. Eq. (42). An obvious (for the quantum chemist) example is formed by two atomicelectrons in orbitals with quantum numbers l1 and l2, which are Russell-Saunders coupledto a total angular momentum L.

A somewhat less obvious example is offered by a scattering complex consisting of adiatom and an atom. In this section we will discuss the use of the Wigner-Eckart theoremto compute the angular matrix elements that arise in the coupled channel (close coupling)approach to scattering. In Fig. 1 we have drawn the coordinates of this complex, the so-calledJacobi coordinates.

Fig. 1

The coordinates relevant in atom-diatomscattering. The vector ~r is along the diatomand has a length equal to the bondlength.It has the coordinate vector r with respectto space-fixed axes, labeled by x, y and z.The vector ~R points from the center of massof the diatom to the atom. Its length is theintersystem distance. This vector has coor-dinate vector R.

18

The rotational wave function of a diatomic molecule is Y lm(r), where r is a unit vector

along ~r. Notice that there is a one-to-one correspondence between r and the spherical polarangles θ and φ of r, which explains the notation Y l

m(r). The relative motion of the atom andthe diatom is described by spherical waves depending on R, i.e., by the functions Y L

M(R).

The total angular momentum of the complex is ~J = ~l + ~L, where ~l is a vector operatorwith components −ir ×∇r with respect to the space-fixed frame and the vector operator~L, depending on R, is defined similarly. Remember that ~J is a constant of the motion ifand only if the Hamiltonian is invariant under a simultaneous rotation of ~r and ~R, that is,under a rotation of the total collision complex. Because we assume the complex to move inan isotropic space (no external fields), this is the case. Notice that a rotation of only ~r, (or~R) changes the interaction energy, and hence ~l and ~L are not constants of the motion, atleast at distances where the interaction is appreciable.1

The atom-diatom interaction can be expanded in terms of Legendre functions Pλ(cos γ)

depending on the angle γ between ~r and ~R. The famous spherical harmonic addition theoremrelates Pλ(cos γ) to spherical harmonics depending on the space-fixed coordinates θ, φ and

Θ, Φ of ~r and ~R, respectively:

Pλ(cos γ) =4π

2λ+ 1

λ∑

µ=−λ

(−1)µY λ−µ(θ, φ)Y

λµ (Θ,Φ). (99)

To prove Eq. (99), we first note that the right hand side of this equation is a rotationalinvariant. Indeed, since in the usual Condon & Shortley phase convention (−1)µY λ

−µ = (Y λµ )∗

and because the Wigner D-matrices are unitary, the quantity∑µ |Y λ

µ |2 is an invariant.Alternatively, one can look upon the right hand side of Eq. (99) as being proportionalto a CG coupling of two spherical harmonics to total L = 0, because 〈λµ;λµ′ | 00 〉 =δµ,−µ′(−1)λ−µ[2λ + 1]−1/2. Since the right hand side of Eq. (99) is invariant under rotationof the collision complex, the spherical harmonic addition theorem follows easily by rotatingthe whole complex, such that ~r ′ is along the space-fixed z-axis, i.e., r′ = (0, 0, r). Inserting(82) we get, using the rotational invariance,

4π

2λ+ 1

λ∑

µ=−λ

(−1)µY λ−µ(θ, φ)Y

λµ (Θ,Φ) =

[4π

2λ+ 1

] 1

2

Y λ0 (Θ′,Φ′) = Pλ(cosΘ

′). (100)

The angle Θ′ is the angle that the rotated vector ~R′ makes with the z-axis and hence isequal to γ.

In the coupled channel approach one projects the Schrodinger equation by a coupled setof spherical harmonics

1The attentive reader may note that the interaction is invariant under any rotation of ~R around

~r. However, some Coriolis terms arising in the kinetic energy operator are not invariant under this

rotation, and hence ~r · ~L is not a constant of the motion. When we neglect these small Coriolis

terms, as one does in the coupled state approximation in scattering theory, we obtain axial rotation

symmetry with infinitesimal generator ~r · ~L. In the coupled channel approximation the Coriolis

terms are not neglected.

19

| (lL)JM 〉 =∑

mM

Y lm(θ, φ)Y

LM(Θ,Φ)〈 lm;LM | JM 〉, (101)

and since J and M are good quantum numbers we meet only matrix elements diagonal inJ and M

〈 (l′L′)J M |∑

µ

(−1)µY λ−µ(r) Y

λµ (R) | (lL)J M 〉 = 〈 JM | 00; JM 〉〈 (l′L′)J | |Pλ| | (lL)J 〉.

(102)

As 〈 JM | 00; JM 〉 = 1, the matrix elements are independent of M . The reduced matrixelement can be evaluated by the Wigner-Eckart theorem. In doing so one meets algebraiccoefficients: the 6j-symbols.

Before evaluating the matrix element (102), we will treat general tensor operators andconsider

C ≡ 〈 (j ′1j′2)J ′M ′ |∑

µ

(−1)µT λ−µSλµ | (j1j2)JM 〉. (103)

Remember that the scalar operator is diagonal in J and M . The tensor operator T λµ acts

on the first subsystem and Sλµ on the second. We use Eq. (42) in bra and ket and theWigner-Eckart theorem [Eq. (68)] to write C,

C =∑

µm′

1m′

2m1m2

(−1)µ〈 JM | j1m1; j2m2 〉〈 j′2m′2 |λµ; j2m2 〉〈 j′1m′

1 |λ,−µ; j1m1 〉

×〈 j′1m′1; j

′2m

′2 | JM 〉〈 j ′1 | |T λ| | j1 〉〈 j′2 | |Sλ| | j2 〉. (104)

Substitute

〈 j′2m′2 |λµ; j2m2 〉 = (−1)j2−j′2−µ

[2j′2 + 1

2j2 + 1

] 1

2

〈 j2m2 |λ,−µ; j ′2m′2 〉, (105)

and

〈 j′1m′1 |λ,−µ; j1m1 〉 = (−)j1+λ−j′

1〈 j1m1;λ,−µ | j ′1m′1 〉 (106)

then we find, while replacing µ by −µ,

C =

[2j′2 + 1

2j2 + 1

] 1

2

〈 j′1 | |T λ| | j1 〉〈 j′2 | |Sλ| | j2 〉(−1)j1+j2−j′

2−j′

1+λ

×∑

µm′

1m

′

2m1m2

〈 JM | j1m1; j2m2 〉〈 j2m2 |λµ; j ′2m′2 〉〈 j1m1;λµ | j ′1m′

1 〉〈 j′1m′1; j

′2m

′2 | JM 〉

= (−1)j1+j2−j′2−j′

1+λ

[2j′2 + 1

2j2 + 1

] 1

2

〈 j′1 | |T λ| | j1 〉〈 j′2 | |Sλ| | j2 〉

×〈 (j1(λj ′2)j2)J | ((j1λ)j ′1j′2)J 〉, (107)

where we used Eq. (92) for the recoupling coefficient. Introducing the 6j-symbol [Eq. (93)]we finally find for the matrix element (103)

20

C = (−1)j2−j′1−J [(2j ′1 + 1)(2j ′2 + 1)]1

2 〈 j′1 | |T λ| | j1 〉〈 j′2 | |Sλ| | j2 〉{j′2 j′1 Jj1 j2 λ

}. (108)

In the case of atom-diatom scattering we have the tensor operators T λµ = Y λµ (θ, φ) and

Sλµ = Y λµ (Θ,Φ), cf. Eq. (99). The reduced matrix elements are given by (84), so that in total

Eq. (102) becomes

〈 (l′L′)J ′ M ′ |∑

µ

(−1)µY λ−µ(r) Y

λµ (R) | (lL)J M 〉 = δJ ′JδM ′M(−1)L−l′−J 2λ+ 1

4π(109)

×[(2l + 1)(2L+ 1)]1

2 〈λ0; l0 | l′0 〉〈λ0;L0 |L′0 〉{L′ l′ Jl L λ

}.

The right hand side of this equation is known as a Percival-Seaton coefficient1.

A. Exercise

13. Prove the spherical expansion of a plane wave:

eik·r = 4π∞∑

l=0

iljl(kr)l∑

m=−l

(−1)mY lm(r)Y

l−m(k), (110)

where jl(kr) is a spherical Bessel function of the first kind defined by

jl(ρ) ≡ (−ρ)l(1

ρ

d

dρ

)lsin ρ

ρ. (111)

Hints:(i) Use that Legendre functions Pl(x) are complete on the interval (−1, 1).(ii) Use that

∫ 1−1 Pl′(x)Pl(x)dx = δll′2/(2l + 1).

(iii) Use the integral representation of the Bessel function:

jl(kr) =1

2il∫ π

0eikr cos θPl(cos θ) sin θdθ. (112)

(iv) Use the spherical harmonic addition theorem.

1I.C. Percival and M.J. Seaton, Proc. Camb. Phil. Soc. 53, 654 (1957).

21

APPENDIX A: EULER’S THEOREM

Basically Euler’s theorem states that any rotation is around a certain axis. We willgeneralize this statement to unitary matrices. Remember from linear algebra that an n× nmatrix A can be unitarily diagonalized, (has n orthonormal eigenvectors) if and only if it isnormal, that is, if A satisfies

A†A = AA

†. (A1)

The most important examples of normal matrices are Hermitian matrices: A† = A and

unitary matrices: A† = A

−1. Recall that a real unitary matrix is usually referred to as‘orthogonal matrix’.

Consider the n×n unitary or orthogonal matrix A and its unitary eigenvector matrix U:

U†

A U = D with D = diag(λ1, . . . , λn). (A2)

Since

U†

A U

(U†

A U

)†= D D

† = D D∗ = 11, (A3)

we find that λiλ∗i = 1 for i = 1, . . . , n. All eigenvalues of unitary and orthogonal matrices

lie on the unit circle in the complex plane.Recall further the fundamental theorem of algebra: a polynomial Pn(x) of degree n in the

variable x has n roots zi. These roots may be real or complex. Suppose that the coefficientsof Pn(x) are all real. If zi is a root, that is Pn(zi) = 0, then obviously Pn(zi)

∗ = Pn(z∗i ) = 0

and z∗i is also a root. Thus, the fundamental theorem of algebra has as a corollary that theroots of a polynomial with real coefficients are either real or appear in complex conjugatepairs.

Consider now the secular problem of an orthogonal matrix O: det (O− λ11) = 0. Thisequation has n complex roots which, since the elements of O are real, appear in complexconjugate pairs and lie on the unit circle in the complex plane. Obviously, detO =

∏i λi

is real and hence is ±1. We find that there are two possibilities for a proper (det = +1)orthogonal matrix:

(i) The dimension n is even, then the eigenvalues 1 are degenerate of even degree andthere is no single invariant vector. (Since zero is even, zero degeneracy is not excludedand the matrix may not have a unit eigenvalue at all in this case).

(ii) The dimension n is odd, then the eigenvalues 1 are degenerate of odd degree, so thereis at least one invariant vector.

If we return to the case n = 3 then we see that the eigenvalue 1 of R either occurs withmultiplicity 3, and R ≡ 11, or with multiplicity 1 and then the corresponding eigenvector n

is uniquely determined. This proves Euler’s theorem.

22

APPENDIX B: PROOF OF EQUATION (10)

Decompose r into a component parallel to the invariant unit vector n and a componentx⊥ perpendicular to it:

r = (r · n)n + x⊥ with x⊥ = r − (r · n)n. (B1)

The vectors x⊥ , y⊥ ≡ n × r , and n form a right-handed frame. The vector n has unitlength by definition and the vectors x⊥ and y⊥ both have the length |r|2 − (n · r)2 (whichis not necessarily unity). When we rotate r around n its component along n is unaffectedand its perpendicular component transforms as

x⊥ → cosψx⊥ + sinψy⊥. (B2)

Hence,

R(n, ψ)r = cosψ[r − (r · n)n] + sinψn× r + (r · n)n. (B3)

We have already seen that

n× r = N r, (B4)

where N is given in Eq. (9). The dyadic product n ⊗ n is a matrix with i, j element equalto ninj. Evidently,

(r · n)n = n⊗ nr. (B5)

By direct calculation one shows that

N2 = n⊗ n− 11. (B6)

By substituting (B4), (B5) and (B6) into (B3) we obtain finally

R(n, ψ)r = [11 + sinψN + (1− cosψ)N2]r, (B7)

from which Eq. (10) follows, because r is arbitrary.

APPENDIX C: PROOF OF EQUATION (12)

Let us write the infinitesimally rotated function as follows

f(r; ∆ψ) ≡ U(n,∆ψ)f(r) = f(r′), (C1)

where r′ ≡ R(n,∆ψ)−1r. Since ∆ψ is infinitesimal, f(r; ∆ψ) can be written in the followingtwo-term Taylor series

f(r; ∆ψ) = f(r; 0) + ∆ψ

(df(r;ψ)

dψ

)

ψ=0

. (C2)

23

Because of (C1) we have f(r; 0) = f(r). In order to evaluate the derivative in Eq. (C2) weapply the chain rule to the rightmost term of Eq. (C1)

(df(r;ψ)

dψ

)

ψ=0

=

(df(r′)

dψ

)

ψ=0

=∑

i

(∂f

∂r′i

)

ψ=0

(dr′idψ

)

ψ=0

. (C3)

Obviously, (∂f/∂r′i)ψ=0 = ∂f/∂ri. Using Eq. (10) we find

(dr′

dψ

)

ψ=0

=

(dR(n, ψ)−1

dψ

)

ψ=0

r = −N r. (C4)

Invoking Eq. (9), we now find

(df(r;ψ)

dψ

)

ψ=0

= −∑

ij

Nijrj∂f

∂ri=∑

ijk

εijknkrj∂

∂rif(r). (C5)

Because of Eq. (2) this equation can be rewritten in terms of the lk,

(df(r;ψ)

dψ

)

ψ=0

= −i∑

k

nklkf(r), (C6)

and substitution of this result into (C2) finally gives

f(r; ∆ψ) = f(r)− i∆ψn · lf(r), (C7)

which is the required result.

APPENDIX D: PROOF OF THE TRIANGULAR CONDITION

We shall show that

|j1 − j2| ≤ J ≤ j1 + j2. (D1)

As stated in the main text, the (2j1 + 1)(2j2 + 1) dimensional product space V j1 ⊗ V j2

spanned by | j1,m1; j2,m2 〉, (m1 = −j1, . . . , j1, and m2 = −j2, . . . , j2), is invariant under J2

and J3. This means that both operators can be diagonalized simultaneously on this space.Product kets are automatically eigenkets of J3 with eigenvalue M = m1 + m2, so that itsuffices to diagonalize J2 on the eigensubspaces of J3 contained in V j1⊗V j2 . We will discusshow, in principle, bases of the eigenspaces of J 2 can be constructed.

In order to see which subspaces occur we consider Fig. 2, where we give an example forj1 = 3 and j2 = 2.

24

Fig. 2

Schematic representation of the basis of the product spacewith j1 = 3 and j2 = 2. Each dot represents a productket | 3,m1; 2,m2 〉. The quantum numbers m1 and m2 aregiven on the lefthand side and on the top of the diagram,respectively. On the righthand side and at the bottom ofthe diagram we find M = m1 +m2.

The hooks represent eigenspaces of J2, see text for de-tails.

Evidently, there is only one product ket | j1, j1; j2, j2 〉 of maximum M , Mmax = j1 + j2,i.e., this eigenspace of J3 is one dimensional and the eigenproblem of J 2 on this spaceis also one-dimensional. In other words, | j1, j1; j2, j2 〉 is automatically an eigenvector ofJ2. In Fig. 2 this is the ket in the upper righthand corner with Mmax = 5. Acting withJ+ ≡ j+(1) + j+(2) onto this ket, we generate zero, and hence | j1, j1; j2, j2 〉 is indeed anupper rung. The J quantum number of a ladder being equal to its highest M quantumnumber, this ladder is characterized by the quantum number J = Mmax = j1 + j2, [i.e., allrungs are eigenkets of J2 with eigenvalue (j1 + j2)(j1 + j2 + 1)]. The lower rungs may begenerated by repeated action of J−.

In Fig. 2 we see that the second highest eigenvalue of J3 (M = 4) occurs twice, so thatthe corresponding eigenproblem of J2 is of dimension two. However, we already found oneeigenvector, namely | j1 + j2, j1 + j2− 1 〉 ≡ J−| j1 + j2, j1 + j2 〉, so that a diagonalization ofJ2 is unnecessary. (Notice parenthetically that we use the notation | JM 〉 for the coupledket, suppressing j1 and j2). We only have to find a ket in the two-dimensional space that isorthogonal to | j1 + j2, j1 + j2 − 1 〉; this will be an eigenfunction of J 2.

It is easy to see that this orthogonalized function will again be the highest M stateof a ladder of eigenkets of J2. Indeed, write briefly |ψ 〉 = | j1 + j2, j1 + j2 − 1 〉 and theorthogonalized function as |φ 〉, then action of J+ onto |φ 〉 must give an eigenfunction ofJ3 with eigenvalue Mmax, i.e., a function proportional to J+|ψ 〉, or zero. The former case isexcluded because

〈 J+φ | J+ψ 〉 = 〈φ | J−J+ |ψ 〉 = 〈φ | J2 − J3(J3 + 1) |ψ 〉 = 0, (D2)

which follows from |ψ 〉 being an eigenfunction of J 2 and J3, together with the orthogonality〈φ |ψ 〉 = 0. Hence J+|φ 〉 = 0 does not have a non-zero component along J+|ψ 〉. Thehighest rung |φ 〉 having M = j1 + j2 − 1, the whole ladder generated by acting with J− ischaracterized by J = j1+j2−1. Since the eigenspace of J3 withMmax−1 is two-dimensional,the eigenvalue j1 + j2 − 1 occurs only once in the product space.

The third highest eigenspace of J3 (M = 3 in the example) is of dimension three. Inthis space we already found two eigenvectors of J 2 by the laddering technique. Orthogonal-izing an arbitrary product ket in this space onto both these functions, we will find anothereigenfunction of J2. It is easily shown that this function is the highest rung of a ladder withJ =Mmax − 2, so that this quantum number also occurs once only.

25

We may continue this way and thus prove the triangular conditions. Graphically we mayeasily convince ourselves that these relations hold by reinterpreting the dots in Fig. 2. Thenumber of product kets is the same as the number coupled states (eigenstates of J 2 and J3),

dim(V j1 ⊗ V j2) = (2j1 + 1)(2j2 + 1) =j1∑

m1=−j1

j2∑

m2=−j2

1 =j1+j2∑

J=|j1−j2|

J∑

M=−J

1 =j1+j2∑

J=|j1−j2|

(2J + 1).

(D3)

[This equation is easily proved by realizing that∑J(2J+1) is the sum of an arithmetic series

and by using the well-known sum formula for such a series]. Thus, we may let a dot standfor a coupled state, for instance the dot with coordinates m1 = 2,m2 = 2, will representthe coupled state J−| 3, 3; 2, 2 〉 ≡ | 5, 4 〉 and the dot with m1 = 3,m2 = 1 the state | 4, 4 〉orthogonal to it. Repeatedly acting with J− onto the highest M ket we generate the borderof the diagram, i.e., a hook consisting of the rightmost column and the bottom row. Actingwith J− onto coupled state with coordinates m1 = 3,m2 = 1 we again generate a hook: theladder with J = j1 + j2 − 1, and so on. Only the last eigenvalue J = |j1 − j2| = 1 is nota hook, but the topmost part of the leftmost column. Alternatively, one can look now atequation (D3) as a change of summation variables. The sum over m1 and m2 covers the gridof Fig. 2 row after row, whereas the sum over J runs over the hooks and the sum over M iswithin hook J . The latter double sum also covers the grid exactly once.

APPENDIX E: CLEBSCH-GORDAN COEFFICIENTS

The expression below for the CG-coefficients, due to Van der Waerden1 is the mostsymmetric one of the various existing forms. Since its derivation is highly non-trivial, it willnot be presented.

〈 jm | j1m1; j2m2 〉 = δm,m1+m2∆(j1, j2, j) (E1)

×∑

t

(−1)t [(2j + 1)(j1 +m1)!(j1 −m1)!(j2 +m2)!(j2 −m2)!(j +m)!(j −m)!]1

2

t!(j1 + j2 − j − t)!(j1 −m1 − t)!(j2 +m2 − t)!

× 1

(j − j2 +m1 + t)!(j − j1 −m2 + t)!,

where

∆(j1, j2, j) =

[(j1 + j2 − j)!(j1 − j2 + j)!(−j1 + j2 + j)!

(j1 + j2 + j + 1)!

] 1

2

, (E2)

and the sum runs over all values of t which do not lead to negative factorials. This expressionand all relations in this appendix are valid for integer and half-integer indices.

This expression immediately implies that the CG-coefficients satisfy the following sym-metry properties

1B.L. van der Waerden, Die Gruppentheoretische Methode in der Quantenmechanik, Springer,

Berlin (1932).

26

1. 〈 jm | j1m1; j2m2 〉 = 〈 j1m1; j2m2 | jm 〉,

2. 〈 jm | j1m1; j2m2 〉 = (−1)j1+j2−j〈 j,−m | j1,−m1; j2,−m2 〉,

3. 〈 jm | j1m1; j2m2 〉 = (−1)j1+j2−j〈 jm | j2m2; j1m1 〉,

4. 〈 jm | j1m1; j2m2 〉 =[(2j + 1)

(2j2 + 1)

] 1

2

〈 j2m2 | jm;ϑ(j1m1) 〉,

where the time-reversal operator ϑ acts as follows:

|ϑ(j1m1) 〉 ≡ ϑ| j1m1 〉 = (−1)j1−m1 | j1,−m1 〉.

5. For the special case when j = m = 0 the CG-coefficients are equal to

〈 0 0 | j1m1; j2m2 〉 = δj1j2δm1,−m2(−1)j1−m1 (2j1 + 1)−

1

2 .

Proof:

1. The CG-coefficients are defined as an inner product; their explicit form shows thatthey are real. Hence

〈 jm | j1m1; j2m2 〉 = 〈 j1m1; j2m2 | jm 〉∗ = 〈 j1m1; j2m2 | jm 〉.

2. From the unitarity of the time reversal operator ϑ and the realness of CG-coefficientsfollows that

〈 jm | j1m1; j2m2 〉 = 〈 jm |ϑ†ϑ| j1m1; j2m2 〉 = 〈ϑ(jm) |ϑ(j1m1);ϑ(j2m2) 〉= (−1)j+j1+j2(−1)−(m+m1+m2)〈 j,−m | j1,−m1; j2,−m2 〉. (E3)

Since m = m1 +m2 and m is a half-integer if and only if j is a half-integer, we findthat

(−1)−(m+m1+m2) = (−1)−2m = (−1)−2j.

3. The explicit form for CG-coefficients, (E1), is invariant with respect to a simultaneousinterchange j1 ↔ j2 and m1 ↔ −m2, which implies that m = m1 + m2 → −m andthus

〈 jm | j1m1; j2m2 〉 = 〈 j,−m | j2,−m2; j1,−m1 〉 = (−1)j1+j2−j〈 jm | j2m2; j1m1 〉. (E4)

4. Make simultaneous replacements j ↔ j2 and m↔ −m2 in the explicit expression forCG-coefficients, Eq. (E1), and change the summation variable t to j1−m1−t. Observe

that except for the first factor (2j+1)1

2 and the phase (−1)t they leave the expression(E1) invariant. Thus

27

〈 jm | j1m1; j2m2 〉 =[(2j + 1)

(2j2 + 1)

] 1

2

(−1)j1−m1〈 j2,−m2 | j1m1; j,−m 〉

=

[(2j + 1)

(2j2 + 1)

] 1

2

(−1)j1−m1〈 j2m2 | jm; j1,−m1 〉

=

[(2j + 1)

(2j2 + 1)

] 1

2

〈 j2m2 | jm;ϑ(j1m1) 〉. (E5)

5. Follows by applying the previous rule to the identity

〈 j1m1 | j2m2; 00 〉 = δj1j2 δm1m2.

APPENDIX F: THE 6J-SYMBOL

The following relation is due to Racah:

{j1 j2 j3j′1 j′2 j′3

}= ∆(j1j2j3)∆(j1j

′2j′3)∆(j ′1j2j

′3)∆(j ′1j

′2j3)

∑

t

(−1)t(t+ 1)!

×[(t− j1 − j2 − j3)!(t− j1 − j′2 − j′3)!(t− j ′1 − j2 − j′3)!

×(t− j ′1 − j′2 − j3)!(j1 + j2 + j′1 + j′2 − t)!×(j1 + j3 + j′1 + j′3 − t)!(j2 + j3 + j′2 + j′3 − t)!

]−1, (F1)

where ∆(abc) is defined in Eq. (E2). The sum over t is restricted by the requirement thatthe factorials occurring in (F1) are non-negative.

APPENDIX G: FURTHER READING

The first (1931) book on the subject is still highly recommendable: E. Wigner, loc. cit..It gives also an introduction to group and representation theory.

Around 1960 the following books appeared, several of which saw new editions in themeantime :

1. A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton U.P., Prince-ton, N.J. (1957). A good introduction, but errs on the distinction between active andpassive rotations.

2. M.E. Rose, Elementary Theory of Angular Momentum, Wiley, New York, 1957. Doesnot follow Wigner’s convention, Eq. (11), but has an antihomomorphic definition.

3. U. Fano and G. Racah, Irreducible Tensorial Sets, Academic, New York, (1959).G. Racah, together with E. Wigner, created the subject. Short readable book withemphasis on the distinction between standard and contrastandard sets, (also knownas co- and contravariant tensors).

28

4. D.M. Brink and G.R. Satchler, Angular Momentum, Oxford U.P. London, 1962. Aconcise short work, very nice as a reference once one understands the material. Hasalso a chapter on Jucys diagrams.

5. A.P. Jucys, I.B. Levinson, and V.V. Vanagas, Mathematical Apparatus of the Theoryof Angular Momentum. Translated from the Russian by A. Sen and A.R. Sen, IsraelProgram for Scientific Translations, Jerusalem, (1962). Jucys was a Lithuanian and didnot spell his name in Cyrillic characters, nevertheless his name is often transcribed asYutsis. This book is noteworthy for (i) introduction of angular momentum diagrams,which are inspired by, but not the same as, Feynman/Goldstone/Hugenholtz diagrams.And (ii) discussion of high 3n-j symbols (recoupling coefficients).

6. L.C. Biedenharn and H. van Dam, Quantum Theory of Angular Momentum, Aca-demic, New York (1965). A collection of reprints, with the most important ones beingthe beautiful 1940 paper by Wigner on the representations of simply reducible groupsand the 1953 paper by J. Schwinger, in which he introduces into group theory bosoncreation and annihilation operators as a computational tool. Both papers are otherwiseunpublished.

In 1981 the definitive book on the subject appeared: L.C. Biedenharn and J.D. LouckAngular Momentum in Quantum Physics, Addison-Wesley, Reading, (1981). This is a verycomplete and authoritative work, not very easy, but worth studying carefully. Since it is socomplete, it is easy to overlook some of the basic material in it, such as the Wigner-Eckarttheorem for coupled tensor operators, etc.

A reference work is D.A. Varshalovich, A.N. Moskalev, and V.K. Khershonskii, Quan-tum Theory of Angular Momentum, translated from the Russian, World Scientific, Singa-pore (1988). This book has no text, only 508 pages of formulas, all pertaining to angularmomentum theory. Be careful, though, we did find some errors.

Also the book by Zare must be mentioned, R.N. Zare, Angular Momentum, Wiley,New York (1988), since it is very popular among beginners. However, at some places thepresentation is wrong. Consider Eqs. (1.67) and (3.58), which give for instance for J = 1 andM = 0 the erroneous result | 1, 0 〉 = αβ. In contrast, the spinfunctions on p. 114 correctly

read | 1, 0 〉 =(α(1)β(2) + β(1)α(2)

)/√2. Equations (1.67) and (3.58) are correct only

for commuting variables, such as complex numbers or boson second-quantized operators;one-electron spinfunctions do not belong to this class.

Finally, it must be pointed out that A. Messiah, Quantum Mechanics, Vol. II, NorthHolland, Amsterdam, (1965) has a very good introduction in Chapter XIII and a goodsummary in Appendix C. All sorts of arbitrary conventions enter the quantum theory ofangular momentum, but if you follow Messiah on this you cannot go wrong.

29