PHY 4604 Fall 2008 – Practice Questions for Exam 2 (With Solutions)
1. A particle of mass
m
moves in one dimension under the potential
V
(
x
)=
(
∞
for
x<
0
,
V
0
aδ
(
x

a
)
for
x>
0
,
where
V
0
and
a
are positive quantities.
(a) Write down the general form of a stationarystate wave function of energy
E>
0
in the regions
0, 0
<x<a
,and
x>a
.
Solution
: The general form of the stationarystate wave function is
ψ
(
x
0
for
0
,
A
1
e
ikx
+
B
1
e

ikx
for 0
,
A
2
e
ikx
+
B
2
e

ikx
for
,
where
k
=
+
√
2
mE/
~
.
(b) By applying the appropriate boundary conditions, express the stationarystate
wave function from (a) in terms of just one unknown amplitude.
Solution
: Wemustimposecontinuityof
ψ
at
x
=0
,where
V
(
x
) jumps to inﬁnity:
A
1
+
B
1
⇒
A
1
=

B
1
⇒
ψ
(
x
S
sin
kx
for 0
.
Given this, it simpliﬁes matters slightly to rewrite
ψ
(
x
C
cos
+
D
sin
for
.
Then we must impose continuity of
ψ
at
x
=
a
:
C
cos
ka
+
D
sin
=
S
sin
D
+
C
cot
=
S.
(1)
Finally, we must enforce Δ
ψ
0
(
a
)=(2
mV
0
a/
~
2
)
ψ
(
a
):

kC
sin
+
kD
cos
=
kS
cos
+
2
mV
0
a
~
2
S
sin
D

C
tan
=
S
(1 +
α
tan
)
,
(2)
where
α
=2
mV
0
a/
(
~
2
k
). The solution to Eqs. (1) and (2) is
C
=

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 Fall '07
 Field
 mechanics, Mass

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