Thus there is unit area under the graph of
δ
(
x
), which is remarkable, given
that the function vanishes for
x
negationslash
= 0! Although the name of
δ
(
x
) includes
the word ‘function’, this object is not really a function because we cannot
assign it a value at the origin. It is best considered to be the limit of a series
of functions that all have unit area under their graphs but become more and
more sharply peaked around the origin (see Figure 2.1).
The analogue of equation (1.31) is
integraldisplay
d
x

ψ
(
x
)

2
= 1
,
(2
.
43)
which expresses the physical requirement that there is unit probability of
finding the particle at
some
value of
x
.
The analogue of equation (2.2) is
I
=
integraldisplay
d
x

x
)(
x

.
(2
.
44)
4
The analogy would be clearer if we wrote
a
(
x
) for
ψ
(
x
), but for historical reasons
the letter
ψ
is hard to avoid in this context.
2.3
Position representation
25
Figure 2.1
A series of Gaussians of unit area.
The Dirac delta function is the limit of
this series of functions as the dispersion tends to zero.
It is instructive to check that the operator that is defined by the right side
of this equation really is the identity operator. Applying the operator to an
arbitrary state

ψ
)
we find
I

ψ
)
=
integraldisplay
d
x

x
)(
x

ψ
)
(2
.
45)
By equations (2.37) and (2.38) the expression on the right of this equation
is

ψ
)
, so
I
is indeed the identity operator.
When we multiply (2.45) by
(
φ

on the left, we obtain an important
formula
(
φ

ψ
)
=
integraldisplay
d
x
(
φ

x
)(
x

ψ
)
=
integraldisplay
d
xφ
∗
(
x
)
ψ
(
x
)
,
(2
.
46)
where the second equality uses equations (2.38) and (2.39). Many practical
problems reduce to the evaluation of an amplitude such as
(
φ

ψ
)
. The expres
sion on the right of equation (2.46) is a well defined integral that evaluates
to the desired number.
By analogy with equation (2.5), the
position operator
is
ˆ
x
=
integraldisplay
d
xx

x
)(
x

.
(2
.
47)
After applying ˆ
x
to a ket

ψ
)
we have a ket

φ
)
= ˆ
x

ψ
)
whose wavefunction
φ
(
x
′
) =
(
x
′

ˆ
x

ψ
)
is
φ
(
x
′
) =
(
x
′

ˆ
x

ψ
)
=
integraldisplay
d
xx
(
x
′

x
)(
x

ψ
)
=
integraldisplay
d
xxδ
(
x
−
x
′
)
ψ
(
x
) =
x
′
ψ
(
x
′
)
,
(2
.
48)
where we have used equations (2.38) and (2.40). Equation (2.48) states that
the operator ˆ
x
simply multiplies the wavefunction
ψ
(
x
) by its argument.
In the position representation, operators turn functions of
x
into other
functions of
x
. An easy way of making a new function out of an old one is
to differentiate it. So consider the operator ˆ
p
that is defined by
(
x

ˆ
p

ψ
)
= (ˆ
pψ
)(
x
) =
−
i¯
h
∂ψ
∂x
.
(2
.
49)
In Box 2.2 we show that the factor i ensures that ˆ
p
is a Hermitian operator.
The factor ¯
h
ensures that ˆ
p
has the dimensions of momentum:
5
we will find
5
Planck’s constant
h
= 2
π
¯
h
has dimensions of distance
×
momentum, or, equivalently,
energy
×
time, or, most simply, angular momentum.
26
Chapter 2: Operators, measurement and time evolution
Box 2.2:
Proof that ˆ
p
is Hermitian
We have to show that for any states

φ
)
and

ψ
)
,
(
ψ

ˆ
p

φ
)
= (
(
φ

ˆ
p

ψ
)
)
∗
. We
use equation (2.49) to write the left side of this equation in the position
representation:
(
ψ

ˆ
p

φ
)
=
−
i¯
h
integraldisplay
d
xψ
∗
(
x
)
∂φ
∂x
.
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 Spring '15
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 Physics, mechanics, The Land, David Skinner, probability amplitudes, James Binney, Physics of Quantum Mechanics