PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 13
Topics Covered:
Spin
Please note that the physics of spin1/2 particles will ±gure heavily in both the ±nal exam for 851, as well
as the QM subject exam.
Spin
1
/
2
:
The Hilbert space of a spin1/2 particle is the tensor product between the in±nite dimen
sional ‘motional’ Hilbert space
H
(
r
)
and a twodimensional ‘spin’ Hilbert space,
H
(
s
)
. The spin Hilbert
space is de±ned by three noncommuting observables,
S
x
,
S
y
, and
S
z
. These operators satisfy angular
momentum commutation relations, so that simultaneous eigenstates of
S
2
=
S
2
x
+
S
2
y
+
S
2
z
and
S
z
exist.
According to the general theory of angular momentum, these states can be designated by two quantum
numbers,
s
, and
m
s
, where
s
must be either an integer or half integer, and
m
s
∈ {
s, s
−
1
, . . . ,
−
s
}
. The
theory of spin says that for a given particle, the value of
s
is ±xed. A spin1/2 particle has
s
= 1
/
2, so
that
m
s
∈ {−
1
/
2
,
1
/
2
}
. Since
s
never changes, we can label the two eigenstates of
S
z
as
 ↑
z
a
and
 ↓
z
a
,
where
 ↑
z
a
=

s
=
1
2
, m
s
=
1
2
a
and
 ↓
z
a
=

s
=
1
2
, m
s
=
−
1
2
a
, so that
S
z
 ↑
z
a
=
p
2
 ↑
z
a
(1)
S
z
 ↓
z
a
=
−
p
2
 ↓
z
a
(2)
and
S
2
 ↑
z
a
=
3
p
4
 ↑
z
a
(3)
S
2
 ↓
z
a
=
3
p
4
 ↓
z
a
.
(4)
As eigenstates of an observable, these states must satisfy the orthonormality conditions
A↑
z
 ↑
z
a
= 1,