Lecture 11 Highlights
We considered the spin-orbit interaction in Hydrogen. The objective is to
calculate the āfine-structureā energy splitting due to the spin-orbit effect. The magnetic
moment of the electron interacts with the magnetic field created by the proton to produce
a small energy difference between the
L
r
parallel to S
r
and
L
r
anti-parallel to S
r
situations for the atom. This difference is due to āspin-orbit coupling.ā
As discussed in Griffiths page 271 the magnetic field experienced by the electron
is given (classically) by:
32
0
8
Brmc
Le
ĻĪµ
r
r=,
where e is the electronic charge, is the electron mass, cis the speed of light in vacuum,
and m
r
is the proton-electron distance. Note that the magnetic field is parallel to the
electron orbital angular momentum vector. To see why, consider things from the
protonās rest frame as the electron moves with velocity v
r
through the static electric field
E
rproduced by the proton. It will experience an effective magnetic field given by
E
c
v
B
r
r
r
Ćā= (from relativity). By comparing the direction of this field with the direction
of the angular momentum of the electron in its orbit about the proton, this argument
shows that
B
ris parallel to
L
r.
The magnetic moment of charged āspinningā particles is given by:
,
S
r
r
Ī³Ī¼
=
where
Ī³
is called the gyromagnetic ratio. It relates the gyration (or rotation) of the
particle (as embodied in S
r
) to the magnetic moment developed (
Ī¼
r
). A moving charge
creates a magnetic field. A charge moving in a āsmallā current loop can be treated as a
magnetic moment, or magnetic dipole, at least for distances large compared to the
diameter of the current loop. For the electron the gyromagnetic ratio is found to be
e
em
e
ā=
Ī³
to very good approximation. See Griffiths page 272 for a āderivationā of this
result. For heavier particles like the proton the gyromagnetic ratio is much smaller due to
the larger mass in the denominator and the fact that angular momentum is quantized and
of order hfor all particles. This fact allows us to ignore the interaction of the protonās
magnetic moment with the magnetic field created by the electron, at least for now.
The electronās magnetic moment experiences a torque due to its motion around
the proton. There is a perturbing interaction energy given by;
B
so
r
r
ā¢ā=Ī
Ī¼
which becomes;
LS
rmc
Le
S
m
e
so
r
r
r
rā¢
ā
ā
ā
ā
ā
ā
ā
ā
ā¢
ā
ā
ā
ā
ā
āāā=Ī ~
832
0
ĻĪµ
This new operator LS
r
r
ā¢has some interesting properties. It commutes with and
, but does not commute with
2
L
2
SS
r
or
L
r
(Homework 4). This means that S
r
and
L
r are no
1
