From the components of two vectors, we can construct a matrix by the
outer product. For vectors |v
1
i = [a
1
a
2
…a
n
]
T
and |v
2
i = [b
1
b
2
…b
n
]
T
, this
is denoted by |v
1
ihv
2
| and represented by a matrix given by
|v
1
ihv
2
| =
a
1
a
2
.
.
.
a
n
h
b
∗
1
b
∗
2
… b
∗
n
i
=
a
1
b
∗
1
a
1
b
∗
2
… a
1
b
∗
n
a
2
b
∗
1
a
2
b
∗
2
… a
2
b
∗
n
.
.
.
.
.
. …
.
.
.
a
n
b
∗
1
a
n
b
∗
2
… a
n
b
∗
n
(3.12)
Operators on a Hilbert space can be represented in terms of outer products
of the basis vectors of the space: a matrix A with matrix elements A
ij
is the
expansion
A ≡
X
i,j
A
ij
|iihj|.
Conversely, in the above basis, a matrix A has elements
A
ij
= hi|A|ji,
The Essentials of Quantum Mechanics 47
where i is the row index and j is the column index. The space of matrices
is thus a linear vector space with basis “vectors” given by the matrices |iihj|
composed of the outer products of the basis vectors of the Hilbert space. For
example, in 2 dimensions, the basis matrices will be
|0ih0| =
“
1
0
#
h
1 0
i
=
“
1 0
0 0
#
,
|0ih1| =
“
1
0
#
h
0 1
i
=
“
0 1
0 0
#
,
|1ih0| =
“
0
1
#
h
1 0
i
=
“
0 0
1 0
#
,
|1ih1| =
“
0
1
#
h
0 1
i
=
“
0 0
0 1
#
.
(3.13)
A 2×2 matrix is represented as
A = A
00
|0ih0| + A
01
|0ih1| + A
10
|1ih0| + A
11
|1ih1|
= A
00
“
1 0
0 0
#
+ A
01
“
0 1
0 0
#
+ A
10
“
0 0
1 0
#
+ A
11
“
0 0
0 1
#
=
“
A
00
A
01
A
10
A
11
#
.
The spectral theorem can then be expressed in the form
ˆ
A =
X
i
a
i
|a
i
iha
i
| (3.14)
where the a
i
are the eigenvalues of
ˆ
A corresponding to its eigenvectors |a
i
i.
Note how we use the eigenvalue itself as the label for the corresponding eigen-
state! The set of eigenvalues is called the spectrum of the operator and this
equation is called the spectral resolution of the operator.
Example 3.2.3. Matrix representation for the spin operators in the compu-
tational basis:{|0i, |1i} = {|↑i
z
, |↓i
z
}:
ˆ
S
z
=
~
2
|0ih0| −
~
2
|1ih1| =
~
2
“
1 0
0 −1
#
.
To construct the representation for
ˆ
S
x
in this basis we first note that it is
diagonal in the basis of its own eigenvectors:
ˆ
S
x
=
~
2
|↑i
x
h↑|
x
−
~
2
|↓i
x
h↓|
x
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