: The simple example revisited : lecture_8 : A simple example of

# The general problem

The problem of adding two arbitrary angular moment and amounts to finding a unitary transformation from the set of basis vectors defined by the tensor products of the individual eigenstates of and and and to the eigenstates of and , where

The individual eigenstates of 1 and 2 satisfy

We may also define raising and lowering operators according to

We then define the tensor product basis vectors as

and we seek a transformation to a set basis set denoted that satisfies

The method of obtaining the transformation is simply to expand the new basis vectors in terms of the old:

The coefficients

are the general Clebsch-Gordan coefficients. In principle, they can be determined by the programmatic procedure outlined in the last section applied to the arbitrary angular momenta. Note that

unless , which restricts the summations in the above expansion considerably.

Although a general formula exists for Clebsch-Gordon coefficients (see below), let us first examine some of the properties of the coefficients that are useful in constructing the unitary transformation:

1.
The Clebsch-Gordon coefficients are real:

2.
Orthogonality:

This can be seen by recognizing that

so that if we insert an identity operator in the inner product in the form

the orthogonality relation results. A similar orthogonality relation is

which can also be proved starting from

and inserting identity in the form

Note that the label is not a fixed label like and . This is because different total values can result. The minimum value of is clearly , while its maximum is , and we need to sum over these in the completeness relation.

3.
Recursion relation:

which can be derived starting from

and taking matrix elements of both sides between the new and old basis vectors:

On the left side, acts on to produce the term on the left in the recursion relation. On the right, the operators and operate to the left as and . However,

and hence produce the opposite action as the on the left.

Finally, the general formula for the Clebsch-Gordan coefficients is

where , and the summation runs over all values for which all of the factorial arguments are greater than or equal to 0.

This formula is rather cumbersome to work with, so it is useful to deduce some special cases. These are as follows:

i.

ii.
if or and , then

iii.
If ,

: The simple example revisited : lecture_8 : A simple example of
Mark Tuckerman 平成17年2月18日