In order to further explain the structure of the hydrogen atom, one needs
to consider that the electron not only has orbital angular momentum
, but also intrinsic angular momentum , called
*spin*. There is an associated spin operator , as well as
operators and , just as with . Usually
written in matrix form, these operators yield results analogous to
and when acting on the wavefunction ,

where *s* and are quantum numbers defining the magnitude of the spin
angular momentum and its projection onto the *z*-axis, respectively. For
an electron *s*=1/2 always, and hence the electron can have
.

Associated with this angular momentum is an intrinsic magnetic dipole moment

where

is a fundamental unit of magnetic moment called the *Bohr magneton*.
The number is called the *spin gyromagnetic ratio* of the electron,
expected from Dirac theory to be exactly 2 but known experimentally to be
. This is to be compared to the magnetic dipole moment
associated with the orbit of the electron,

where is the *orbital gyromagnetic ratio* of the electron.
That is, the electron creates essentially twice as much
dipole moment per unit spin angular momentum as it does per unit
orbital angular momentum. One expects these magnetic dipoles to interact,
and this interaction constitutes the spin-orbit effect.

**Figure 2:** On the left, an electron moves around the nucleus in a Bohr orbit.
On the right, as seen by the electron, the nucleus is in a circular orbit.

The interaction is most easily analyzed in the rest frame of the electron,
as shown in figure 2. The electron sees the nucleus moving
around it with speed *v* in a circular orbit of radius *r*, producing a
magnetic field

In terms of the electron orbital angular momentum *L*=*mrv*, the field may
be written

The spin dipole of the electron has potential energy of orientation in this magnetic field given by

However, the electron is not in an inertial frame of reference. In
transforming back into an inertial frame, a relativistic effect
known as *Thomas precession* is introduced, resulting in a factor of
1/2 in the interaction energy. With this, the Hamiltonian of the spin-orbit
interaction is written

With this term added to the Hamiltonian, the operators and
no longer commute with the Hamiltonian, and hence the projections
of and onto the *z*-axis are not conserved
quantities. However, one can define the total angular momentum operator

It can be shown that the corresponding operators and
*do* commute with this new Hamiltonian. Physically what happens
is that the dipoles associated with the angular momentum vectors
and
exert equal and opposite torques on each other, and hence they couple together
and precess uniformly around their sum in such a way that the
projection of on *z*-axis remains fixed. The operators
and acting on yield

where *j* has possible values

For a hydrogenic atom *s*=1/2, and hence the only
allowed values are *j*=*l*-1/2,*l*+1/2, except for *l*=0, where only
*j*=1/2 is possible. Figure 3 illustrates spin-orbit
coupling for particular values of *l*, *j*, and .

**Figure 3:** Spin-orbit coupling for a typical case of *s*=1/2, *l*=2,
*j*=5/2, , showing how and precess about
.

Since the coupling is weak and hence the interaction energy is small relative to the principle energy splittings, it is sufficient to calculate the energy correction by first-order perturbation theory using the previously found wavefunctions. The energy correction is then

The value of is easily found by calculating

and hence when acting on ,

One then needs to calulate the expectation of , which is more complicated. The answer is

where the value *s*=1/2 has been included.

Mon Oct 26 15:21:07 EST 1998