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PHY4604
R. D. Field
Department of Physics
Chapter2_6.doc
University of Florida
The Infinite Square Well (4)
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
≤
L/2
and V(x) =
∞
if
x
≥
L/2
and V(x) = 0 for
L/2 < x < L/2
(Note that now
V(x) = V(x)
).
As before we have
0
)
(
=
x
for
x
≤
L/2
and
x
≥
L/2
For
L/2 < x < L/2
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
)
sin(
'
)
cos(
'
)
(
kx
B
kx
A
Be
Ae
x
ikx
ikx
+
=
+
=
−
where A' and B' are constants.
Boundary Conditions:
We require that
ψ
(x) be “squareintegrable” and
that it be continuous and “single valued”.
Thus at
x = L/2
(1)
0
)
2
/
sin(
'
)
2
/
cos(
'
)
2
/
(
=
+
=
=
kL
B
kL
A
L
x
.
At
x = L/2
we have
(2)
0
)
2
/
sin(
'
)
2
/
cos(
'
)
2
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.
 Spring '07
 FIELDS
 mechanics

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