Chemistry Reference
In-Depth Information
The asterisk refers to the complex conjugate of the wavefunction. If the wavefunction
contains a real part
a
and
an
imaginary part
b
, then the complex conjugate is the same
function but with i (i
=
√
−
1) replaced by
−
i. Then,
ψ
∗
ψ
=
(
a
−
i
b
)(
a
+
i
b
)
=
a
2
+
b
2
=
2
; note that although the wavefunction may contain imaginary numbers, the probability
will always be purely a real number.
Clearly, the bond must have some value of
x
, and so if we look over all possible values
of
x
the probability of finding the correct bond extension is unity. This is often referred to
as the normalization condition and can be stated mathematically as
|
ψ
|
∞
ψ
∗
ψ
d
x
=
1
(A6.29)
−∞
For the ground-state wavefunction of the bond vibration we have found a real function (i.e.
no imaginary part), and so the integrand is simply the wavefunction squared:
∞
N
2
exp(
−
α
x
2
)d
x
=
1
(A6.30)
−∞
This integral is the total area under the Gaussian curve, a standard result, so we can write
N
2
π
α
α
π
1
/
4
=
1
and,
N
=
(A6.31)
which allows us to complete the trial wavefunction definition:
exp
α
π
1
/
4
−
α
x
2
2
1
ψ
0
=
with
α
=
μ
m
k
(A6.32)
This function is plotted as state
n
= 0 in Figure A6.2a. The line showing the level for
the zero-point energy is also used as the zero line for the wavefunction plot. Where this
Energy
E
(a)
(b)
V
(
x
)
n
= 3
n
= 3
n
= 2
n
= 2
n
= 1
n
= 1
n
= 0
n
= 0
x
x
Figure A6.2
(a) The first four wavefunctions of the harmonic oscillator; (b) the squares of
the wavefunctions, which give the probability of finding the bond at a particular extension x.
