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Next: Addition
of Spin- Operators Up: Addition
of Angular Momenta Previous: Addition
of Angular Momenta
In general, if we have two angular momentum operators, 
and 
(where the 
could correspond to the sum of the orbital AM and the spin AM or the spin
AM of 2 electrons etc.), we would like to know the possible values of the
total AM 
can take where
Assuming that
and
correspond to distinct degrees of freedom (for example if they correspond
to the spin AM of 2 electron), they commute with each other:
Together with the AM commutation relations for
and
individually, the commutation relations for
are given by:
So all the properties of AM and their eigenstates discussed in the sections above also hold for the total AM.
We start with states 
and 
where the two quantum numbers 
and 
are fixed and the 
take the values 
. The corresponding eigenvalue equations are:
From these states we construct the product states
which are eigenfunctions of 
with eigenvalue 
but NOT eigenfunctions of 
since
for i=1,2. These product states are eigenstates of the operators
What we need to do is search for states in which
is also diagonal; that is, we seek eigenfunctions
of the four mutually commuting operators
with eigenvalues 
. At the same time we have to find the values taken by j (the corresponding
are then 
), and we have to represent 
as a linear combination of the product states above. Before considering
the general case, we first look at the addition of two spins and the addition
of an orbital AM to a spin.