Biomedical Engineering Reference
In-Depth Information
Probabilities then become time independent since
ψ
∗
(
r
) exp
ψ (
r
) exp
2π
j
ε
t
h
2π
j
ε
t
h
Ψ
∗
(
r
,
t
) Ψ (
r
,
t
)
=
+
−
=
ψ
∗
(
r
) ψ (
r
)
11.3 Particles in Potential Wells
I have mentioned a number of model potentials in earlier chapters when dealing with
molecular dynamics and Monte Carlo. A model potential of historical interest in quantum
mechanical studies is the 'infinite well'.
11.3.1 One-Dimensional Infinite Well
A particle of mass
m
is constrained to a region 0
L
by a potential that is infinite
everywhere apart from this region, where it is equal to a constant
U
0
(Figure 11.1). This
might be a very crude model for the conjugated electrons in a polyene, or for the conduction
electrons in a metallic conductor. (In fact there is a technical problem with this particular
model potential caused by the infinities, as I will explain shortly.)
≤
x
≤
Region I
Region III
Region II
U
U
0
x
=
0
x
=
L
Figure 11.1
One-dimensional infinite well
To solve the problem, we examine the Schrödinger equation in each of the three
regions and then match up the wavefunction at the boundaries. The wavefunction must
be continuous at any boundary.
In regions I and III the potential is infinite and so the wavefunction must be zero, in order
to keep the Schrödinger equation finite. In region II the potential is a finite constant
U
0
and
we have
U
0
ψ (
x
)
h
2
8π
2
m
d
2
d
x
2
+
−
=
εψ (
x
)
