| San José State University Department of Economics |
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| applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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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 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.,
then
the representation of A becomes An axial vector can be considered a representation of a second order antisymmetric
tensor; i.e.,
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.
x → -x, y → -y,
z → -z,
(-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.
| B1 |
| B2 |
| B3 |
| = |
| 0 | B1 | B2 |
| -B1 | 0 | B3 |
| -B2 | -B3 | 0 |
This information can be summarized in the form of a table:
| Types of Scalars, Vectors and Tensors | |||
|---|---|---|---|
| Name | Origin | Examples | Transformation Law |
| Polar Vector | Position vectors | Displacement, Electric Field | E'i = ΣlirEr |
| Axial Vector | cross product of polar vectors | Magnetic Field | B'i= |det (l)|Σlir Br |
| Polar Tensor | relation between two vectors of the same type | Magnetic Permeability | P'i..j= Σlir ...lkwPr..w |
| Axial Tensor | Relation between two vectors of different types | Optical Gyration | Q'i..j=|det(l)|Σlir ...lkwQr..w |
| Psuedo-scalar | polar vector dotted with axial vector | Rotary Power of Optically Active Crystal |
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
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.,
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.)
Let B'1, B'2, ...,B'n.
We want to know how the coefficients such that
X = a1B1 + a2B2 +....+anBn.
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.,
X = a'1B'1 + a'2B'2 +....+a'nB'n.
Bi = ci1B'1 + ci2B'2
+...+ cinB'n
= ΣcijB'j
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:
X = (Σaicij)B'j
and thus
a'j = Σaicij
A' = AC.
The Labeling of Basis Vectors and the Components of Vectors