*Where do the normal and anomalous Zeeman formulas come from?*

First, a bit of classical physics. Imagine an ``infinitesimally small''
bar magnet - this is called a *magnetic dipole*. A vector (say
) represents the strength and direction of the dipole;
is known as the *magnetic moment*.

Suppose we place the dipole in a magnetic field **B**. The field will
exert a torque on the dipole, trying to flip it into alignment (so **B**
and are parallel). Twisting the dipole out of alignment
takes energy. In other words, the dipole has a potential energy that
depends on its orientation with respect to the field. Classical physics
says that this ``magnetic'' energy is given by
.

A tiny current loop acts like a dipole, with a magnetic moment
perpendicular to the plane of the loop. The current loop possesses angular
momentum (after all, the current consists of moving charges, which have
mass). According to classical physics, the magnetic moment is proportional
to the angular momentum:
constant times **J**. The
constant depends on the charge/mass ratio of the moving charges making up
the current.

We now apply all this to an electron in an atom in a magnetic field.
Thanks to its orbital angular momentum **l**, it has a ``magnetic''
energy proportional to
; likewise, the spin
contributes energy proportional to
. Adding up the
**l**'s and **s**'s for all the electrons in the atom, we might expect
a formula like this for the total ``magnetic'' energy:

Instead, the correct formula is:

Let us accept this formula, noting but not discussing the mysterious factor 2. If we pick the -axis in the direction of

or equivalently:

since .

Without the enigmatic 2, we'd simply have on the right hand side of formula 7. Without on the right hand side, the normal and anomalous Zeeman formulas would be the same-- as we will see.

How does the electron maintain its orientation in the magnetic field-- why
doesn't it snap into alignment? Or in ``old quantum'' terms, how can we
have stationary states with **S** (or **L**) out of alignment with the
field? This was a legitimate question for the old quantum theory, and it
had a good answer. The spinning electron acts like a gyroscope. The field
exerts a torque on the electron, and like any gyroscope, the electron
precesses. (The same argument holds for **L**.)

We can get the ``anomalous Zeeman'' formula from formula 7 with a little vector algebra, and a crucial assumption:

Assume that in any stationary quantum state,This is calledSandLeach precess aroundJ, so that the projection ofSonJdoesn't change, but the component ofSperpendicular toJrotates uniformly (ditto forL).

Where does spin-orbit coupling come from? Naively expressed, from a torque
trying to make **L** and **S** point opposite to each other (i.e.,
anti-parallel). Goudsmit and Uhlenbeck imagined it this way. Suppose
there is only a single electron. If we ride along an electron's orbit
right next to it, but not spinning ourselves, we will appear to see the
nucleus orbiting *us* (shades of Ptolemy!) Effectively we have a
positive current circulating around us; this will generate a magnetic field
proportional to **L**. The electron has a magnetic moment proportional
to **S**, and hence we get an energy term proportional to
. (Because of the positive charge on the nucleus, the signs
come out so that spin-orbit coupling trys to make **L** and **S**
anti-parallel.)

Life is more complicated in multi-electron atoms.
is
only an approximation, applicable when the electron-electron forces
dominate the spin-orbit forces. In this case, one can approximate the
totality of electrons as a single electron ``smear'', with total orbital
angular momentum **L**, and total spin **S**. For ``light'' atoms,
this works pretty well. For ``heavy'' atoms, a different approximation
works better.

A classical model has emerged. The vectors **L** and **S** feel a
torque trying to make them anti-parallel; when the magnetic field **B**
is turned on, **L** and **S** each feel a torque trying to align them
with **B**. We have, in other words, ``perturbation terms'' occuring
somewhere in the total energy formula:

Classical mechanics would next try to compute the ``trajectories'' of the
vectors **L** and **S** (i.e., how they vary with time). If , then **J** is constant (conservation of angular momentum), and
**L** and **S** precess about **J**. If the field is weak, then (to
a good approximation) **J** precesses about **B**, while **L** and
**S** still precess about **J**. **J** is no longer constant, since
it is subject to an external torque. If the field is strong, then **L**
and **S** precess about **B** (again to a good approximation), because
the magnetic torque swamps the spin-orbit torque.

The ``old quantum'' prescription says to find the energy levels of these stationary states.

As it happens, the energy of a stationary state depends primarily on terms
other than those in formula 8. We have terms analogous to the
of hydrogen; then terms depending on the magnitude of **L** and
**S** (i.e., terms depending on and ); only after that does
formula 8 come into play.

The spin-orbit coupling contributes a perturbation that depends on the
angle between **L** and **S** (and hence indirectly on the magnitude of
**J**). So when , we expect to find bunches of energy levels
with the same and grouped closely together, separated slightly by
an energy difference depending on . This is an aspect of the famous
*fine structure* of the spectrum.^{12} If
, then the fine structure disappears. (Low-lying energy levels of
atoms with even numbers of electrons usually have .)

Without a field, we have complete rotational symmetry in the energy
formula. The spin-orbit perturbation depends only on the *magnitude*
of **J**, not on its direction (on , not on ). With a field, the
symmetry is broken and the degeneracy lifted.

We have perturbations depending on and . With a strong field (or if ), we confidently replace these dot products with and , for the dot products don't change and must be quantized (i.e., they are stationary). We have seen already how this leads to the normal Zeeman effect. Note also another regularity: spectra without fine-structure have , and hence display normal Zeeman splitting.

With a weak field, we express the perturbation in terms of and . The first term we replace with . Were this all, we would again see a normal Zeeman effect, because and satisfy the same selection rule.

Turn to the
perturbation. Because **S** precesses
about **J**, the *average* value of
is the same
as

where is the projection of

The rest is vector algebra. You may skip to the next section if you like-- no new concepts appear.

First derive a formula for
, in terms of
quantum numbers. We have:

Rearrange this equation and replace with (and likewise for

where I've used an arrow instead of an equal sign to indicate ``quantization''-- classical quantities on the left, quantum numbers on the right. From the modern perspective, the left hand side is an operator, and the right hand side is one of its eigenvalues. (Formula 9 looks tantalizingly close to the Landé g-factor, but we're not quite there yet.)

Next derive a formula for
, the projection of **S** on
**J**:

``Quantizing'' the term in parentheses:

In other words, is this fraction times

Replace the term with the ``average'' value of , which is , which is the above fraction times :

And this reduces to our anomalous Zeeman formula, formula 2.

© 2001 Michael Weiss