Biomedical Engineering Reference
In-Depth Information
U
Region
II
I
III
D
0
L
0
x
Figure 11.9
The finite box
where
A
and
B
are constants of integration. We can eliminate the
B
term straight away by
the argument of Section 11.3.
In region III we have
D
ψ (
x
)
h
2
8π
2
m
d
2
d
x
2
+
−
=
εψ (
x
)
For now, let us interest ourselves only in those solutions where the energy is less than or
equal to
D
. Such solutions are called
bound states
. We will shortly see that solutions exist
where the energy is greater than
D
and these are called
unbound states
. I can rewrite the
Schrödinger equation for region III as
d
2
ψ
d
x
2
h
2
8π
2
m
(
D
−
−
ε) ψ
=
0
The standard solution to this second-order differential equation is
E
exp
8π
2
m
(
D
x
F
exp
x
8π
2
m
(
D
−
ε)
−
ε)
ψ
=
+
−
h
2
h
2
where
E
and
F
are constants of integration.
We now have to apply boundary conditions for the bound state solutions where ε is
less than
D
. If we want the wavefunction to be zero at infinity, then
E
must be zero. The
wavefunction and its first derivativemust also be equal at the boundary
x
=
L
. Thus we have
