Chemistry Reference
In-Depth Information
1.6 Expectation values and the momentum operator
Schrödinger's equation can inprinciple be solved for an arbitrary potential,
V
, giving a set of allowed energy levels
E
n
with associated wavefunc-
tions,
2
with the probability distribution
of the particle, and the particle has a 100 per cent chance (probability
ψ
n
. As we wish to associate
|
ψ
(
x
)
|
n
1)
of being somewhere along the
x
-axis, it is customary to 'normalise' the
wavefunction
=
ψ
(
x
)
so that
n
∞
−∞
|
ψ
2
d
x
(
x
)
|
=
1
(1.26)
n
and the probability of finding a particle in the
n
th state between
x
and
x
+
d
x
is then given by
2
d
x
(
)
=|
ψ
(
)
|
P
n
x
d
x
x
(1.27)
n
as illustrated in fig. 1.5. The
expectation
(or average) value
x
n
of the posi-
tion
x
for a particle with wavefunction
ψ
(
x
)
, is then found by evaluating
n
the integral
∞
2
d
x
x
n
=
x
|
ψ
(
x
)
|
(1.28a)
n
−∞
which can also be written as
∞
−∞
ψ
n
(
x
n
=
x
)
x
ψ
(
x
)
d
x
(1.28b)
n
Although both forms of eq. (1.28) give the same result, it can be shown that
the second form is the correct expression to use. The expectation value for
an arbitrary function
G
(
x
)
is then given by
∞
−∞
ψ
n
(
G
n
=
x
)
G
(
x
)ψ
(
x
)
d
x
(1.29)
n
P
n
(
x
)
xx
+
d
x
2
Figure 1.5
Plot of the probability distribution function,
P
n
(
x
)
=|
ψ
n
(
x
)
|
for a nor-
malised wavefunction,
ψ
n
(
x
). The total area under the curve,
−∞
P
n
(
x
)d
x
,
equals 1, and the probability of finding the particle between
x
and
x
+d
x
is
equal to the area of the shaded region.
