Chemistry Reference
In-Depth Information
Replacing
in eq. (1.2) by the wavenumber
k
, we find that the momentum,
p
, of a wave is directly related to its wavenumber by
λ
p
=
hk
/
2
π
=
k
(1.13)
where we introduce
. Substituting eq. (1.13) in (1.12), we derive
Heisenberg's uncertainty principle, one of the most widely quoted results
in quantum mechanics:
=
h
/
2
π
x
p
≥
/
2
(1.14)
namely that it is impossible to know the exact position and exact momen-
tum of an object at the same time.
A similar expression can be found relating energy
E
and time
t
.We
saw that the energy
E
is related to a frequency
ν
by
E
=
h
ν
. The uncer-
tainty in measuring a frequency
ν
depends on the time
t
over which the
measurement is made
ν
∼
1
/
t
(1.15)
so that the uncertainty in energy
E
=
h
ν
∼
h
/
t
, which can be
re-arranged to suggest
h
. When the derivation is carried out
more rigorously, we recover a result similar to that linking momentum
and position, namely
E
t
∼
E
t
≥
/
2
(1.16)
so that it is also impossible to determine the energy exactly at a given
moment of time.
1.5 Schrödinger's equation
Although it is impossible to
derive
the equation which determines the
form of the wavefunction
, Schrödinger was nevertheless able to deduce
or postulate its form. We discussed above how
(
x
,
t
)
may be given by
a complex function. We assume we can choose
A
e
−
i
(ω
t
−
kx
)
(
x
,
t
)
=
(1.17)
ω
for a wave propagating in the
x
-direction with angular frequency
and
wavenumber
k
. Using eqs (1.1) and (1.2), we can rewrite
ω
and
k
, and hence
in terms of the energy
E
and momentum
p
, respectively:
A
e
−
i
/
(
Et
−
px
)
(
x
,
t
)
=
(1.18)
We can then take the partial derivative of the wavefunction with respect
to time,
t
, and find
∂
∂
i
E
=−
t
