Download 2 1 2 3 2 5 2 4 1 2 2 1 1 3 5 4 1 2 2 1 1 4 1 2 2 1 2 2 1 2 1 2 2 2 1 2 1

Note on the addition theorem of two angular momenta
N. Chandrakumar
The results of the theorem on addition of angular momenta state that when two
angular momenta j1 and j2 are added to form the resultant angular momentum j, the
eigenvalues of j2 correspond to the quantum numbers j = j1+j2, j1+j21, j1+j22, ,
|j1j2|+2, |j1j2|+1, | j1j2|. This result may be easily understood.
It is crucial to recognize that because j=j1+j2, we have jz=j1z+j2z, and therefore
their eigenvalues satisfy the relation: m = m1+m2.
Now, since the largest eigenvalue m (ie, j) equals the sum of the largest
eigenvalues m1 and m2, the largest quantum number j is clearly j1+j2, since mi’s can
only be from the set of numbers ji to +ji in steps of 1.
We can find the degeneracy of a given value of m by noting in how many
ways it can be expressed by the sum m1+m2. However, such degeneracy of m can only
arise from a number of different allowed j values for the resultant, because each
allowed value of m arises exactly once for a given value of j. This permits us to
immediately infer that the next allowed value of the resultant quantum number j is
j1+j21, since the value m = j1+j21 can arise from the two combinations m1 = j1 and
m2 = j21, as well as m1 = j11 and m2 = j2
It is also clear that the highest possible degeneracy in the value of m is 2j2+1,
where we take j2  j1, without loss of generality. Therefore there are precisely 2j2+1
possible values of j.
Thus, one readily infers that the series of allowed values of j are: j1+j2, j1+j21,
j1+j22, etc. We can also easily see that the series must terminate with |j1j2| (which is
the same as j1j2 if j2  j1).
How many states does the system have? In the j1, j2 representation the system
clearly has (2j1+1)( 2j2+1) states. The same number of states must be found after
adding the two angular momenta j1 and j2 to form the resultant j.
We have listed (2j2+1) values of j, starting from j1+j2 and ending with j1j2.
We recall that the members of the set of values of j increase in steps of 1 starting from
the lowest allowed value. This implies that the number of states for each allowed
value of j in the set increases in steps of 2. The total number of states associated with
this set of values of j is thus given by:
(2k  1)  (2k  3)  (2k  5)    (2k  4 j2  1)
Here, k = j1j2. The summation of this arithmetic progression, with constant
difference 2, and initial and final terms as given above, is seen to be:
 2k  1   2k  3   2k  5      2k  4 j2  1
 2k  2 j2  1  1  3  5     4 j2  1 
2 j 1
 2  j1  j2  2 j2  1   2  1   4 j2  1 
2 j 1
 2  j1  j2  2 j2  1  2  2   2 j2  1
  2 j2  1  2 j1  2 j2  2 j2  1
  2 j1  1 2 j2  1
2
2
It is thus established that the series of allowed resultant angular momentum
quantum numbers must terminate with |j1j2| and thus the set of values of the resultant
angular momentum quantum number is: j = j1+j2, j1+j21, j1+j22, , |j1j2|+2,
|j1j2|+1, | j1j2|.