When considering the Zeeman effect, it is easiest first to consider the
hydrogen atom without hyperfine structure. Then is a good quantum
number, and the atom has a 2*j*+1 degeneracy associated with the
different possible values of . In the presence of an external
magnetic field, these different states will have different energies
due to having different orientations of the magnetic dipoles in the
external field. The splitting of these energy levels is called the
Zeeman effect.

Figure 8 illustrates the geometry of the Zeeman effect.

**Figure 8:** Geometry of the Zeeman effect. On the left, the total dipole
moment precesses around the total angular momentum
. On the right, precesses much more slowly
about the magnetic field.

The total magnetic dipole moment of the electron is

where and have been used. Because of the difference in the orbital and spin gyromagnetic ratios of the electron, this is not in general parallel to

So, as and precess about , the total
dipole moment also precesses about . Assuming
the external field to be in the *z* direction, this
field causes to precess about the *z*-axis. Typical internal
magnetic fields in the hydrogen atom can be shown to be of the order
1 Tesla. If the external field is much weaker than 1 Tesla, which it is
for almost all practical purposes, then the precession of
around the *z*-axis will take place much more slowly than the precession
of around . The Hamiltonian
of the Zeeman effect is

where is the projection of the dipole moment onto the direction of
the field, the *z*-axis. Because of the difference in the precession rates,
it is reasonable to evaluate by first evaluating the projection of
onto , called , and then evaluating the
projection of this onto , thus
giving some average projection of onto .
First, the projection of onto is

Then

Evaluating the dot product using again that , this becomes

So when first order perturbation theory is applied, the energy shift is

where

is called the Landé *g* factor for the particular state being
considered. Note that if *s*=0, then *j*=*l* so *g*=1, and if *l*=0,
*j*=*s* so *g*=2. The Landé *g* factor thus gives some effective
gyromagnetic ratio for the electron when the total dipole moment
is partially from orbital angular momentum and partially from spin.
From equation 97, it can be seen that the energy shift
caused by the Zeeman effect is linear in *B* and , so for
a set of states with particular values of *n*, *l*, and *j*, the individual
states with different will be equally spaced in energy, separated
by . However, the spacing will in general be different for
a set of states with different *n*, *l*, and *j* due to the difference in the
Landé *g* factor.

Including hyperfine structure with the Zeeman effect is more difficult, since the field associated with the proton magnetic dipole moment is weak, and hence it does not take a particularly strong external field to make the Zeeman effect comparable in magnitude to the strength of the hyperfine interactions. The approximation of small external field is thus not practical when discussing the Zeeman splitting of hyperfine structure. However, it can be treated, and the result for the most important case of the Zeeman splitting of the hyperfine levels in the ground state of hydrogen is shown in figure 9. The degeneracy of the triplet state is lifted, the three states of having different energies in the external field. Notice how the splitting is linear for small external field, but then deviates as the field gets larger. The ``21 cm'' transitions shown on the right will have slightly different energies, and measuring the amount of this splitting is a good tool for radio astronomers to measure magnetic fields in the interstellar medium.

**Figure 9:** On left, Zeeman splitting of the hyperfine levels in the ground
state of hydrogen. On right, some possible transitions
between these states.

Mon Oct 26 15:21:07 EST 1998