Chapter 4
The Hilbert Space of States
and the Dirac Notation
The single defining relation for a wave packet is its “square integrability”,
i.e., its finite energy.
The space of complex functions with finite energy
denoted by
L
2
(
R
3
) in three dimensions, and by
L
2
(
R
) in one dimension,
1
, is
the home of all quantum mechanical wave functions; it is known as a Hilbert
space denoted by
H
and it is technically defined as a complete innerproduct
space. We shall now describe the Hilbert space of states
H
, in one spatial
dimension (generalization to three spatial dimensions is straightforward),
whose defining property is their finite energy, namely,
∞
Z
∞

ψ
(
x
)

2
dx <
∞
.
4.1
Innerproduct and Normed Vector Spaces
The most basic algebraic structure on
H
is the vector space property that
we summarize in the following form
2
: if
ψ
1
, ψ
2
∈
H
, then so is the arbitrary
1
R
is
the
real
line,
i.e.,
{
x
:
∞
< x <
+
∞}
,
and
R
3
is
the
three
dimen
sional
Euclidean
space,
i.e.,
{
(
x, y, z
) :
∞
< x, y, z <
+
∞}
.
The
complex
plane
{
x
+
iy
:
∞
< x, y <
+
∞}
is denoted by
C
.
2
For the more mathematically inclined here is the full definition of a vector space
V
,
whose members are denoted by
v
, defined over a field
F
with a binary operation called
addition, and a scalar multiplication by the members of the field satisfying the following
axioms: to every pair
v
1
and
v
2
of vectors corresponds a vector
v

1 +
v
2
∈
V
that is
called their sum. Addition of vectors is commutative and associative. There is a unique
vector
0
such that
v
+
0
=
v
for all
v
∈
V
.
For every vector
v
∈
V
there is a unique
vector

v
such that
v
+ (

x
) =
0
. For every vector
v
∈
V
and every scalar
α
∈
F
there
corresponds a vector
α
v
∈
V
46
Chapter 4.
The Hilbert Space of States and the Dirac Notation
linear combination
c
1
ψ
1
+
c
2
ψ
2
for complex constants
c
1
, c
2
∈
C
.
To define convergence in a vector space we must introduce a norm. A
normed vector space is equipped with a nonnegative real valued function
known as the norm such that
k
ψ
k ≥
0
∀
ψ
∈
H
k
ψ
k
= 0
⇔
ψ
= 0
k
cψ
k
=

c
 k
ψ
k
k
ψ
1
+
ψ
2
k ≤ k
ψ
1
k
+
k
ψ
2
k
,
∀
ψ
1
, ψ
2
∈
H
An inner product can be used to project vectors (functions) in arbitrary
directions, in addition to introducing the concept of orthogonality between
vectors (functions). For any two functions
ψ
1
(
x
) and
ψ
2
(
x
) (we are special
izing to one spatial dimension for brevity) in
H
we define the inner product
h
ψ
1
, ψ
2
i ≡
∞
Z
∞
ψ
*
1
(
x
)
ψ
2
(
x
)
dx.
The inner product can be defined more abstractly as a function that maps
any two members
ψ
1
and
ψ
2
of
H
to a complex number denoted by
h
ψ
1
, ψ
2
i
such that:
h
ψ
1
, ψ
2
i
=
h
ψ
2
, ψ
1
i
*
h
ψ
1
, cψ
2
i
=
c
h
ψ
1
, ψ
2
i
h
ψ, ψ
i
is nonnegative and is 0 if and only if
ψ
= 0.