Chem 120A
Spin Algebra and Statistics
04/06/07
Spring 2007
Lecture 30
Reading: Ratner/Schatz Chs. 7, 8
Atkins/Friedman Ch. 7.87.11
This lecture contains details of the derivation of the eigenvalues of general angular momentum, in addition
to the material discussed in class. This is additional material for those interested to see where the eigenvalues
of
S
2
,
S
z
,
L
2
and
L
z
come from.
1
Recap and more about Spin
Elementary particles and composite particles carry an intrinsic angular momentum called “spin” =
S
, which
is related to the intrinsic magnetic moment by
μ
=

ge
2
m
S
. Here
g
(the gfactor) is a unitless factor. For
electrons,
g
≈
2.
Let’s briefly look at the similarities and differences between spin angular momentum and orbital angular
momentum.
For orbital angular momentum:
μ
=

e
2
m
L
For spin angular momentum:
μ
=

ge
2
m
S
Orbital angular momentum has a classical analog where the magnetic moment
μ
comes from the electron
spinning about some axis (see Figure 1)
Figure 1: A charge moving,
q
, moving at a velocity,
v
in a loop of radius,
r
, produces a magnetic moment,
μ
. The righthand rule determines the direction of the magnetic moment
The intrinsic angular momentum of an electron, on the other hand, has nothing to do with its orbital motion,
but it does lead to an intrinsic
μ
. This is a relativistic effect that can be derived from relativistic quantum
mechanics Again, and we can’t stress this enough, electron spin is not orbital angular momentum in the
classical sense. Experiments tell us, however, that we can take most general properties we derive for the
QM operator
L
=
r
×
p
=
ˆ
L
x
i
+
ˆ
L
y
j
+
ˆ
L
z
k
and we can simply apply them to the operator
S
=
S
x
i
+
˜
S
y
j
+
˜
S
z
k
.
Finally the orbital angular momentum of an electron is described by physical coordinates,
ψ
(
r
,
θ
,
φ
)
. The
wavefunction for spin is represented only by a single quantum number. As we will see shortly
ψ
(
m
s
) =
±
1
2
for electrons
From last time we saw that to understand the behavior of an electron’s intrinsic magnetic moment
μ
(which
is an observable we can measure), then we must understand the behavior of its intrinsic angular momentum
=
S
. Mathmatically, spin angular momentum,
S
, can be described in terms as orbital angular momentum,
L
.
Here we will give a unified treatment of angular momentum which holds for both
L
and
S
, thus we will use
the symbol for general angular momentum,
J
Classically, angular momentum is
J
=
r
×
p
=
ˆ
J
x
i
+
ˆ
J
y
j
+
ˆ
J
z
k
where
i
,
j
,
k
are the usual cartesian unit vectors.
To understand angular momentum in QM, we turn the classical observables into
operators
and study the
Chem 120A, Spring 2007, Lecture 30
1
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“algebra” of
J
=
r
×
p
in QM.
The four important operators associated with angular momentum are:
ˆ
J
x
,
ˆ
J
y
,
ˆ
J
z
, and
J
2
=
ˆ
J
x
2
+
ˆ
J
y
2
+
ˆ
J
z
2
.
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 Spring '07
 Whaley
 Physical chemistry, pH, Angular Momentum, Fermion, JZ

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