Experiments with Supersonic Beams as a Source of Cold Atoms

B
Figure 4.1: A pictorial illustration of the angular momentum coupling of L and S
which takes place in the low field limit. The spin and orbital angular momenta are
coupled by the spin-orbit interaction, and as such L and S precess around the total
angular momentum J. This in turn precesses around the magnetic field B. As L and
S precess, their projection along B changes, meaning that mL and mS are not good
quantum numbers, but the projection of J along B remains constant and so mJ is a
good quantum number.
the relations
and
J
Save = S · J
J 2
Lave = L · J
J 2
L
S
J (4.6)
J. (4.7)
These provide the needed projections onto J, and can be calculated using
J 2 + S 2 − L 2
and
S · J =
2
J 2 + L 2 − S 2
L · J =
2
Using the above equations gives the Hamiltonian
HB = μBgL J · B
+ μBgS J · B
J(J+1)+L(L+1)−S(S+1)
2J(J+1)
J(J+1)+S(S+1)−L(L+1)
59
2J(J+1)
(4.8)
. (4.9)
.
(4.10)