San José State University
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Clebsch-Gordan Coefficients

Clebsch-Gordan Coefficients

In order to understand what Clebsch-Gordan coefficients are and what they are used for, as opposed to dealing with their numerical values and computation, it is necessary to look at the nature of vectors and vector spaces.

What is a vector?

A three dimensional vector is represented in a particular coordinate system by a triplet of real numbers. But the vector is not that triplet of number, it is something whose representation as three numbers changes in a systematic way as the coordinate system changes.

What is ordinarily meant by the term vector is called a polar vector. There is also something called an axial vector, which is the vector (cross) product of two polar vectors. The difference between polar and axial vectors is revealed when we consider a transformation of the coordinate system that changes the handedness of the coordinate system.

Consider two vectors whose respresentation are:

A = (1, 2, 3) and B = (3, 4, 7).

Their cross product C=AxB is: (2, 2, -4).

If we invert the coordinate system; i.e.,

x → -x,  y → -y,  z → -z,

then the representation of A becomes
(-1, -2, -3) and likewise the representation of B becomes (-3, -4, -7). (If the original coordinate system was right-handed then the new coordinate system is left-handed.) If C were a true vector then its representation should go to (-2. -2, 4). But the cross product of the representations of A and B in the new coordinate system is (2, 2, -4). Thus representation of C = AxB did not go to (-2, -2, 4) as would be required if C were to be a true vector. For this reason C=AxB is called a pseudovector or axial vector.

An axial vector can be considered a representation of a second order antisymmetric tensor; i.e.,


This information can be summarized in the form of a table:

Types of Scalars, Vectors and Tensors
NameOriginExamplesTransformation Law
Polar VectorPosition vectorsDisplacement,
Electric Field
E'i = ΣlirEr
Axial Vectorcross product of polar vectors Magnetic Field B'i=
|det (l)|Σlir Br
Polar Tensorrelation between two vectors of the same type Magnetic Permeability P'i..j=
Σlir ...lkwPr..w
Axial TensorRelation between two vectors of different typesOptical Gyration Q'i..j=|det(l)|Σlir ...lkwQr..w
Psuedo-scalarpolar vector dotted with axial vectorRotary Power of Optically Active Crystal 

Bases for a Vector Space

A basis for a vector space is a set of vectors such that any vector in the space can be expressed as a linear combination of elements of the basis. The coefficients in the linear combination of basis lelements for a vector are the components of that vector with respect to the basis. Those components are not the vector itself; they are just the representation of that vector with respect to a particular basis.

For another basis there will be a different set of components. The essential character of a vector is how the representation changes as the basis changes. The change in the components of an arbitrary vector can be represented in terms of the components of the elements of one basis expressed in terms of the elements of the other basis.

This relationship is expressed algebraically as follows. Let B1, B2, ...,Bn be a basis for a vector space V. Let X be a vector in V. Then there exist coefficients a1, a2,..., an such that

X = a1B1 + a2B2 +....+anBn.

Let B'1, B'2, ...,B'n. We want to know how the coefficients such that

X = a'1B'1 + a'2B'2 +....+a'nB'n.

are related to the a1, a2,..., an such that The answer is simple. The first basis vectors can be expressed in terms of the second basis; i.e.,

Bi = ci1B'1 + ci2B'2 +...+ cinB'n
= ΣcijB'j

These representations of the Bi can now be substituted into the representation of X in terms of the Bi's. The result is an expression of X in terms of the B'i's; ile.,

X = (Σaicij)B'j
and thus
a'j = Σaicij

Thus if A and A' represent the coefficients of the representation of X in terms of the B and B', respectively, as row vectors and C is the coefficients of the coefficient for representing the B basis vectors in terms of the B' basis vectors then the relationship between A and A' can be expressed in matrix form as:

A' = AC.

The Labeling of Basis Vectors and the Components of Vectors

The elements of a set of basis vectors and hence the components of vectors are usually labeled with the positive integers, but there is no reason that this has to be. Any countable set will do. Since the basis set is often the eigenvectors of the Schroedinger equation, they can be labeled with their associated eigenvalues or some composite label which includes their eigenvalues.

(To be continued.)