PHY 4604 Fall 2009 – Homework 5
Due at the start of class on Monday, November 2. No credit will be available
for homework submitted after the start of class on Friday, November 6.
Answer both 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
tanh
x
=
1
coth
x
=
e
x

e

x
e
x
+
e

x
’
(
x
for

x
±
1
sgn
x
for

x
²
1
.
1. As a greatly simpliﬁed model of a diatomic molecule, consider a particle of mass
m
moving in the onedimensional potential
V
(
x
)=

v
~
2
2
ma
h
δ
±
x

a
2
²
+
δ
±
x
+
a
2
²i
,
where
a>
0 is the bond length (i.e., the distance between the two nuclei), and
v>
0isa
dimensionless real number that parametrizes the strength of the shortrange attractive
potential centered on each nucleus. This question concerns possible bound states of
this potential, i.e., stationary states having energies
E
=

~
2
K
2
/
2
m<
0.
Since
V
(

x
)=
V
(
x
), any such bound state can be classiﬁed as being even or odd under
x
→
x
.
(a) Write down the general form of an even boundstate wave function. Apply the
appropriate boundary conditions to obtain an equation relating
Ka
and
v
,wh
ich
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 Fall '07
 Field
 mechanics, Work, Ω, bound state, appropriate boundary conditions, outerproduct, eigenstates.

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