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PHY4221
Quantum Mechanics I
Spring 2004
1.3 Representations
In the preceding section we have developed the basic mathematical framework as
used in Dirac’s formalism of quantum mechanics. We are now ready to apply the
abstract theory to tackle real physical systems. To achieve this goal, it is sometimes
more convenient to replace the abstract quantities by the more familiar mathematical
structures and to work in terms of them. This procedure is similar to using coordinates
in geometry, and has the advantage of giving one greater mathematical power for
solving particular problems. The way in which the abstract quantities are to be
replaced by more familiar mathematical structures is not unique, there being many
possible ways corresponding to the many systems of coordinates one can have in
geometry. Each of these ways is called a
representation
. When one has a particular
problem to work out in quantum mechanics, one can minimize the labour by using a
representation in which the problem becomes as simple as possible.
1.3.1 Momentum Operator in the Coordinate Space
Consider the eigenvalue equation of the position operator ˆ
q
ˆ
q

q
i
=
q

q
i
(1)
where

q
i
is the eigen ket vector corresponding to the eigenvalue
q
. Since the eigen
value
q
assumes a value between
∞
and
∞
, the position operator ˆ
q
is said to have
a continuous eigen spectrum. The complete set of eigenvectors
{
q
i}
are found to be
mutually orthogonal and normalized to a Dirac delta function:
h
q
0

q
i
=
δ
(
q
0

q
).
Proof:
In terms of the basis
{
q
i}
, an arbitrary normalized ket vector

ψ
i
in the Hilbert space can be expressed as

ψ
i
=
Z
∞
∞
dq

q
ih
q

ψ
i
(2)
1
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Z
∞
∞
dq

q
ih
q

=
ˆ
I .
(3)
From Eq.(2), we obtain the following two equivalent expressions
h
ψ

ψ
i
=
Z
∞
∞
dq
Z
∞
∞
dq
0
h
ψ

q
0
ih
q
0

q
ih
q

ψ
i
(4)
and
h
ψ

ψ
i
=
Z
∞
∞
dq
h
ψ

q
ih
q

ψ
i
.
(5)
Comparing the above two diﬀerent, yet equivalent, expressions of
h
ψ

ψ
i
yields
h
ψ

q
i
=
Z
∞
∞
dq
0
h
ψ

q
0
ih
q
0

q
i
,
(6)
which in turn implies the result
h
q
0

q
i
=
δ
(
q
0

q
)
.
(
Q.E.D.
)
(7)
Next, let us introduce the operator
ˆ
T
²
associated with shifting the eigenvalue
q
by
a constant
²
:
ˆ
T
²

q
i
=

q
+
²
i
.
(8)
The action of this operator upon an arbitrary normalized ket vector
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