Biomedical Engineering Reference
In-Depth Information
20
15
10
B
5
A
0
Figure 11.8
Two particles in a one-dimensional box
11.8 Finite Well
The infinite well occupies an important place in the history of molecular quantum mechan-
ics, and we will meet it again in Chapter 12. We discussed the finite well in Chapters 9 and
10, but from the viewpoint of classical mechanics. The finite well potential gives a rather
crude model for a diatomic molecule, and it is shown again in Figure 11.9. It keeps an
infinite potential for the close approach of the two nuclei, but allows for a 'valence region'
where the potential is constant.
The potential is infinite for
−∞ ≤
x
≤
0, zero (or a constant) for the valence region
where 0
. As usual, we divide up the
x
axis into
three regions and then match up ψ at the two boundaries. It is also necessary to match up
the first derivative of ψ at the
x
≤
x
≤
L
and a constant
D
for
L
≤
x
≤∞
=
L
boundary.
In region I, the potential is infinite and so ψ
=
=
0. This means that ψ
0 at the boundary
=
when
x
0.
In region II, the potential is constant (zero) and so we have
h
2
8π
2
m
d
2
ψ (
x
)
d
x
2
−
=
εψ (
x
)
This can be solved to give
A
sin
8π
2
m
ε
h
2
B
cos
8π
2
m
ε
h
2
ψ
=
x
+
x
