Next: About this document Up: Angular Momentum Previous: Addition of Spin- and
Next: About
this document Up: Angular
Momentum Previous: Addition
of Spin- and
Two arbitrary AM vectors, 
and 
have AM eigenstates are given by 
and 
, respectively. Form the total AM vector 
. We then have available, two different complete sets of commuting AM operators
, with eigenstates 
and 
with eigenstates 
. Both sets of eigenkets form a complete orthonormal system, which implies
that they can be expanded in terms of each other. For example the expansion
of
in the basis 
is:
The (real) coefficients 
are called Clebsch-Gordon coefficients. (Note that because the z-component
of the AM is conserved, the only composite states that contribute to the
above sum are those for which 
). In a commonly applied terminology, one refers to
as an eigenfunction in the coupled representation and to 
as an eigenfunction in the the un-coupled representation. The allowed values
that j can take (given 
and 
) are
with the possible values of m being:
A general formula for the Clebsch-Gordon coefficients has been derived by Wigner (using Group Theory) and numerical tables of these coefficients can be found in books dealing with atomic spectroscopy.