Quantum Mechanics
Problem Sheet 5
(Solutions are to be handed in for marking before the lecture on Monday, 21 Feb 2005)
1.
φ
(
x
) is any of the oddparity solutions to the stationary Schr¨
odinger equation for a Fnite square well
V
(
x
) =

V
0
for

L
2
< x <
L
2
,
0
for

x

>
L
2
.
(a) Show that the solutions for the bound states (
E <
0) can be written as
φ
(
x
) =
F
e

κx
for
x >
L
2
B
sin(
kx
)
for 0
≤
x
≤
L
2

φ
(

x
)
for
x <
0
,
where
F
and
B
are constants,
κ
≡
√

2
mE/
¯
h
, and
k
≡
p
2
m
(
E
+
V
0
)
/
¯
h
.
(b) Sketch the form of the lowestenergy oddparity wave function
φ
(
x
).
(c) Using the continuity conditions on
φ
(
x
) and d
φ/
d
x
, show that
y
=

z
cot
z
y
2
+
z
2
=
R
2
,
where
z
≡
kL/
2,
y
≡
κL/
2, and
R
≡
√
2
mV
0
L/
(2¯
h
).
(Because of the symmetry of the wave function you need to consider the continuity conditions only
at
x
=

L/
2
or at
x
=
L/
2
, but not both. If you consider them at both points you just get twice the
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 Spring '02
 POCANIC
 Physics, mechanics, 2m, 2 m, 2 L, 2 2m, stationary Schr¨dinger equation

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