Next: General Case Up: Addition of Angular Momenta Previous: Addition of Spin- Operators
Next: General
Case Up: Addition
of Angular Momenta Previous: Addition
of Spin- Operators
and Orbital AMStarting with 
and 
, define the total AM as:
The eigenstates of the operators 
are given by:
where 
. From these states one can form 2(2l+1) product states:
However, these states are NOT eigenstates of the total AM, 
. We therefore seek eigenstates of 
to be obtained by forming linear combinations of the above product states.
To this end, we will make use of the results proved earlier:
and
Denote the eigenstates of
by 
(we could have added a fourth index 
but this is understood as we are concerned with electron-spin here). Presumably,
j, the quantum number to
has the values:
(This would give the right number of states since 
). First consider the case 
and the eigenket at the top of the ladder:
Applying 
we have:
Applying
we have:
This shows that 
is an eigenstate of
with the eigenvalues 
. To obtain all other states just apply the lowering operator:
repeatedly to
.
truein
truein
Hence, we have:
By repeated application of 
one obtains the general result (which can be verified by mathematical induction
with respect to 
):
where
takes half integer values in the range 
. The eigenstates corresponding to 
are orthogonal to all the states derived above and where 
are given by
(This can be verified using the same approach as above).