Experiments with Supersonic Beams as a Source of Cold Atoms

field. In the L − S coupling regime (as opposed to jj coupling which is more common
for heavier atoms) the magnetic moment of an atom is
μatom = −μB gl
li + gssi = −μB gL L + gS
S , (4.4)
i
where μB is the Bohr magneton, and gL and gS are the “g-factors” for the orbital and
spin angular momenta respectively. Thus the Hamiltonian becomes
HB = μB
gL L + gS
S · B. (4.5)
The manner in which the magnetic moment is calculated depends on the ratio of this
Hamiltonian to the Hamiltonian which produces the fine structure splitting.
4.1.1 The Zeeman Effect
If the interaction due to the Hamiltonian in equation 4.5 is much smaller
than the fine structure splitting, then J will remain a good quantum number. In
this regime, called the Zeeman regime, the individual orbital angular momentum
vector L and the spin angular momentum vector S precess around the sum of these
vectors, the total angular momentum vector J. This in turn precesses around the
external magnetic field, where the projection of J in the direction of the magnetic
field (quantization axis) determines the magnetic moment of the atom. In this picture,
as L and S precess around J, their projections along the direction of the magnetic
field are rapidly changing, meaning that these projections (Lz and Sz) arenotgood
quantum numbers. However, since the projection of J along the quantization axis
remains constant, the eigenvalues of the operator Jz will remain constant, making mJ
a good quantum number. This situation is illustrated in figure 4.1.
Since mJ is a good quantum number, but not mL and mS, the equation for
the Hamiltonian (equation 4.5) must be expressed in terms of J. Thisisdoneusing
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