PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 13
Topics Covered:
Spin
Please note that the physics of spin1/2 particles will figure heavily in both the final 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 infinite dimen
sional ‘motional’ Hilbert space
H
(
r
)
and a twodimensional ‘spin’ Hilbert space,
H
(
s
)
.
The spin Hilbert
space is defined 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 fixed. 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
)
and
 ↓
z
)
,
where
 ↑
z
)
=

s
=
1
2
,m
s
=
1
2
)
and
 ↓
z
)
=

s
=
1
2
,m
s
=
−
1
2
)
, so that
S
z
 ↑
z
)
=
planckover2pi1
2
 ↑
z
)
(1)
S
z
 ↓
z
)
=
−
planckover2pi1
2
 ↓
z
)
(2)
and
S
2
 ↑
z
)
=
3
planckover2pi1
2
4
 ↑
z
)
(3)
S
2
 ↓
z
)
=
3
planckover2pi1
2
4
 ↓
z
)
.
(4)
As eigenstates of an observable, these states must satisfy the orthonormality conditions
(↑
z
 ↑
z
)
= 1,
(↓
z
 ↓
z
)
= 1, and
(↑
z
 ↓
z
)
=
(↓
z
 ↑
z
)
= 0. The two vectors
{ ↑)
,
 ↓)}
must therefore form a complete
basis that spans
H
(
s
)
, so that
I
=
 ↑
z
)(↑
z

+
 ↓
z
)(↓
z

.
(5)
1
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1. In this problem you will derive the 2
×
2 matrix representations of the three spin observables from
first principles:
(a) In the basis
{ ↑
z
)
,
 ↓
z
)}
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 Spring '11
 Smith
 Physics, Work, sz, momentum commutation relations

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