PHYSICS 4455 — QUANTUM MECHANICS
Problem Set 7 — Solutions.
The finite potential well in one dimension.
Consider a structureless, nonrelativistic particle of mass
m
in one dimension, moving in
the presence of a finite square well potential:
V
(
x
) =
−
V
0

x
 <
L
0

x
 ≥
L
.
Just like the semiinfinite square well, there are bound and scattering states associated with
this one, naturally for
E
<
0and
E
>
0, respectively. We assume that
V
0
>
0.
1.
Let’s find the energies and wavefunctions for all the bound states. Let’s divide the real
axis into three regions: Region (1) with
x
≤−
L
, region (2) with
−
L
<
x
<
L
, and region (3)
with
x
≥
L
. In regions (1) and (3) the Schrödinger equation takes the form:
−
2
2
m
ϕ
(
x
)=
E
ϕ(
x
)
(1) and (3)
andinregion(2)wehave
−
2
2
m
d
2
dx
2
−
V
0
ϕ(
x
)=
E
ϕ(
x
)
(2)
It is natural to assume that
E
cannot be smaller than
−
V
0
. Looking for bound states, we will
restrict ourselves to the energy range
−
V
0
<
E
<
0. We also introduce
λ
2
=−
2
m
2
E
>
0and
k
2
=
2
m
2
(
V
0
+
E
) >
0
both of which are confined to the range
0
<λ
2
,
k
2
<
2
m
2
V
o
For future reference, let us relate
λ
and
k
:
λ=
2
m
2
V
o
−
k
2
So, rewriting the Schrödinger equation, we have
0
=
d
2
dx
2
−λ
2
ϕ(
x
)=
0
(1) and (3)
which has solutions
e
±λ
x
and
0
=
d
2
dx
2
+
k
2
ϕ(
x
)=
0(
2
)
which has solutions
e
±
ikx
(note that we need only the positive roots,
λ=+ −
E
,and
k
=+
V
0
+
E
). For bound states, we demand that the wave function vanishes at
x
→±∞
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 Spring '09
 Mucciolo
 mechanics, Mass, 2m, 3L, 2L, Liboff, ϕ

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