Place an atom in a magnetic field, and (as Zeeman discovered) its spectral
lines will split. Pais's book *Inward Bound* gives fascinating details
of the experimental history.

For some atoms, spectral lines split in three under a magnetic field. This
is known as the *normal Zeeman effect*. For other atoms, the spectrum
displays a more complex pattern of splittings, known as the *anomalous Zeeman effect*. When the magnetic field becomes strong enough,
however, some lines merge back together, and the anomalous Zeeman splitting
coalesces smoothly into a normal splitting-- this is called the *Paschen-Back effect*. Young Heisenberg wrestled with the anomalous Zeeman
effect, as a student in Sommerfeld's seminar on atomic spectra (see
Cassidy's biography *Uncertainty*.)

The full explanation of all this phenomena presented quite a puzzle to the quantum pioneers. Even today, fitting all the pieces together can be confusing. In the next few sections I will try to assemble this puzzle.

I treat the Zeeman effect in such gory detail for two reasons. First, historical interest: the Zeeman effect gave the first hints of the complexities of quantum spin. Many of the strange ``spin facts'' I stated earlier were first discovered empirically, from poring over spectra and noticing patterns.

Second, ``gestalt'': the very complexity of the Zeeman effect means that any explanation of it must bring together several aspects of quantum mechanics.

We can get started by comparing figures 1 and 2. We see that spectral lines will split if energy levels move up or down (i.e., if the energy levels of quantum states change). To progress further, one needs a formula for the energy levels of the states of an atom-- or rather, a formula for the change in energy level due to the magnetic field.

Say we write
, where is the energy with no
magnetic field, and is the energy due to the magnetic field.
(I will add quantum-number subscripts (like we had with ) in a
moment.) Here is a formula for that accounts for the normal
Zeeman effect:

where and are another pair of quantum numbers to be discussed below. The complicated fraction built out of , , and on the right hand side is known as the Landé -factor.

So there are four pieces to the puzzle:

- What is the meaning of the six quantum numbers , , , , , and ?
- How do the ``normal'' and ``anomalous'' formulas (formulas 1 and 2) account for the details of the Zeeman splitting?
- Where do the normal and anomalous Zeeman formulas come from?
- How does the anomalous Zeeman formula turn into the normal Zeeman formula in the Paschen-Back limit of strong magnetic fields?

© 2001 Michael Weiss