Chem 120A
Spin Statistics/Approximation Methods
04/09/07
Spring 2007
Lecture 31
Reading: Ratner Schatz Chapter 9
1
Spin
our notation later on. integer or half integer. we can label our kets as
b
. (The letters
j
and
m
are standard
notation, but
a
and
b
are not.) We will use the standard
±
±
j
,
m
²
notation from now on. For a given value of
j
,
m
can take on values of

j
,

j
+
1
,

j
+
2
...
j

2
,
j

1
,
j
. A few examples are shown in Figure 3
specialize the above arguments to orbital angular momentum, we would simply replace all
J
’s with
L
’s and
all the
j
’s with
l
’s. For spin angular momentum, we replace all
J
’s with
S
’s, all the
j
’s with
s
’s, and all the
m
’s with
m
s
’s as shown below.
Last time we looked at the algebra of angular momentum in quantum mechanics and then specialized to spin
angular momentum. Remember that every fundamental particle has a particular value of “
s
”. That value of
s
then determines the possible values of
m
s
. We then gave the fundamental theorem of spin statistics:
If
s
is an integer = 1, 2, 3,.
... then the particle is a boson. Identical bosons are symmetric under pairwise
exchange:
ψ
(
1
,
2
) =
ψ
(
2
,
1
)
.
If
s
is a halfinteger = 1/2, 3/2, 5/2.
.. then the particle is a fermion. Identical fermions are antisymmetric
under pairwise exchange:
ψ
(
1
,
2
) =

ψ
(
2
,
1
)
.
Let’s look at two identical fermions, for instance two electrons in an He atom. Electrons have
s
=
1
/
2,
therefore
m
s
has possible values of
+
1
/
2 and

1
/
2. The total wavefunction has a spatial part (see Lectures
20,21,24) and a spin part.
Ψ
(
r
1
,
r
2
,
s
1
,
s
2
) =
ψ
(
r
1
,
r
2
)
φ
(
s
1
,
s
2
)
(1)
where
ψ
(
r
1
,
r
2
)
describes the spatial part of the two electrons and
φ
(
s
1
,
s
2
)
describes the spin angular mo
mentum of the two electrons.
Above we stated that the wavefunction describing the two particles must be antisymmetric overall. If the
spatial part of the wavefuntion is symmetric, then the spin part must be antisymmetric. If the spin part
is symmetric, then the spatial part must be antisymmetric (remember that the product of a symmetric and
antisymmetric function is overall antisymmetric). We will use a
+
subscript for symmetric wavefunctions