13 Addition of angular momenta

TFY4250/FY2045 Tillegg 13 - Addition of angular momenta 5
Conclusion and comments
(i) By measuring both S 1z and S 2z we can prepare the two-spin system in one of the
“old” states, |↑ 1 ↑ 2 〉, |↑ 1 ↓ 2 〉, |↓ 1 ↑ 2 〉, |↓ 1 ↓ 2 〉. In these states S 1z and S 2z have sharp values,
and so does S z .
(ii) For these “old” states S 2 is not sharp, except for |↑ 1 ↑ 2 〉 and |↓ 1 ↓ 2 〉, which happen
to be the upper and lower rungs in the triplet ladder. The reason that S 2 is not sharp for
the other “old” states (|↑ 1 ↓ 2 〉 and |↓ 1 ↑ 2 〉) is that Ŝ2 does not commute with Ŝ1z and Ŝ2z.
(iii) If we choose instead to measure S z and |S|, this spin system is prepared in one of
the “new” states, either in one of the triplet states
|1, 1〉 = |↑ 1 ↑ 2 〉,
|1, 0〉 = 1 √
2
( |↑ 1 ↓ 2 〉 + |↓ 1 ↑ 2 〉 ) ,
|1, −1〉 = |↓ 1 ↓ 2 〉,
(triplet)
(T13.13)
or in the singlet,
|0, 0〉 = 1 √
2
( |↑ 1 ↓ 2 〉 − |↓ 1 ↑ 2 〉 ) . (singlet) (T13.14)
Here S 1z and S 2z are unsharp, except in the two states |↑ 1 ↑ 2 〉 and |↓ 1 ↓ 2 〉. The figure below
gives an illustration of both the old and the new states.
Remember: The “old” states on the left are prepared by measuring S 1z and S 2z separately.
The “new” states on the right are prepared by measuring the size |S| and the z-component
S z of the total spin.
(iv) The singlet corresponds to S = 0, which means that the √ two spins S 1 and S 2 are
antiparallel. The triplet corresponds to an angle α = 2 arccos 2/3 ≈ 70.5 ◦ between the
two vectors S 1 and S 2 . This is the closest the two spins come to being parallel.