Addition of Angular Momenta

correspond to eigenstates of J 2 , J z , L 2 and S 2 where the total angular momentum is
defined as ⃗ J ≡ ⃗ L + ⃗ S. That is, the new kets should satisfy
J 2 |jm>= j(j + 1)¯h 2 |jm>
J z |jm>= m¯h|jm>
L 2 |jm>= l(l + 1)¯h 2 |jm>
and
S 2 |jm>= s(s + 1)¯h 2 |jm> .
The new vectors can be constructed from the direct product basis
|jm>=
l∑
s∑
m l =−l m s =−s
a lsml m s ;jm|lsm l m s >
and the problem reduces to finding the linear combination coefficients a lsml m s ;jm =<
lsm l m s |jm> which are called Clebsch-Gordan coefficients. They have been tabulated in
may books and are also preprogrammed in various mathematical software packages.
Solution for Two Spin 1/2 Particles:
The general method for finding eigenstates of the total angular momentum is a bit
involved and it is worthwhile to illustrate the essence of the problem in the simple but
important case of two spin 1 angular momenta. Here a straight forward application of
2
linear algebra will do the job. Later, this same problem will be solved using the general
method, and finally the general solution will be presented.
The total spin angular momentum is S ⃗ ≡ S ⃗ 1 + S ⃗ 2 . We want to find a set of states
|sm > such that S ⃗2 |sm >= s(s + 1)¯h 2 |sm > and S z |sm >= m¯h|sm >. We will express
them as linear combinations of the direct products of eigenstates of S ⃗ 1 and eigenstates of
⃗S 2 :
|sm>= a | + +> + b | + −> + c | − +> + d | − −> .
The direct product states satisfy
S 1z | + +> = ¯h 2 | + +>
S 1z | + −> = ¯h 2 | + −>
S 1z | − +> = −¯h 2 | − +>
S 1z | − −> = −¯h 2 | − −>
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