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PHY4604
R. D. Field
Department of Physics
Chapter2_3.doc
University of Florida
The Infinite Square Well (1)
Particle in a OneDimensional Box:
Consider the
solution of
)
(
)
(
)
(
)
(
2
2
2
2
x
E
x
x
V
dx
x
d
m
ψ
=
+
−
h
,
where
h
/
)
(
)
,
(
iEt
e
x
t
x
−
=
Ψ
,
for the case V(x) =
∞
if
x
≤
0
and V(x) =
∞
if
x
≥
L
and V(x) = 0 for
0 < x <
L
.
For
x
≤
0
and
x
≥
L
we have
0
)
(
)
(
)
(
)
(
)
(
)
(
2
2
2
2
=
→
=
+
−
∞
→
x
x
x
V
E
x
dx
x
d
x
mV
V
h
.
For
0 < x < L
we have
)
(
)
(
2
)
(
2
2
2
2
x
k
x
mE
dx
x
d
−
=
−
=
h
where
2
2
h
mE
k
=
.
The most general solution is
ikx
ikx
Be
Ae
x
−
+
=
)
(
where A and B are constants.
Bounday Conditions:
We require that
ψ
(x) be “squareintegrable” and
that it be continuous and “single valued”.
Thus at
x = 0
0
)
0
(
=
+
=
=
B
A
x
and hence
)
sin(
2
)
(
)
(
kx
iA
e
e
A
x
ikx
ikx
=
−
=
−
.
At
x = L
we have
0
)
sin(
2
)
(
=
=
=
kL
iA
L
x
which implies that
kL = n
π
with
n = 1, 2, 3,.
..
Energy Levels:
We see that only certain values of
k
are allowed which
means that only the following energies are allowed:
0
2
2
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 Spring '07
 FIELDS
 mechanics

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