PHY 4604 Fall 2009 – Homework 4
Due by 5:00 p.m. on Monday, October 19. Please turn in your homework in class
before the deadline, or else bring it to NPB 2162. (You may push it under the
door if NPB 2162 is unoccupied.)
No credit will be available for homework submitted
after the start of class on Friday, October 24.
Answer all three questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
You may find useful the following:
cos
2
x
+ sin
2
x
= 1
cos
2
x

sin
2
x
= cos 2
x
2 sin
x
cos
x
= sin 2
x
sin(
x
±
y
) = sin
x
cos
y
±
cos
x
sin
y
1. Consider a piecewise constant potential
V
(
x
).
(a) Show that in any region where
V
(
x
) =
V
j
= constant, the general stationary state
ψ
(
x
) =
A
j
exp(
ik
j
x
) +
B
j
exp(

ik
j
x
) can be rewritten in the equivalent form
ψ
(
x
) =
C
j
cos
k
j
x
+
D
j
sin
k
j
x.
(1)
Provide expressions for
C
j
and
D
j
in terms of
A
j
and
B
j
.
(b) Writing
C
=

C

e
iγ
and
D
=

D

e
iδ
where
γ
and
δ
are real, find an expression
for the probability density
ρ
(
x
) in the stationary state specified in Eq. (1). You
should find that
ρ
(
x
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 Fall '07
 Field
 mechanics, Work, wave function, bound state, Probability amplitude

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