Problem Set 4
Physics 341
Due November 19
Some abbreviations: S  Shankar.
1
. Understanding the spectrum for simple potentials is very important. So we will continue
with a problem from the midterm. Start by considering the case in the exam: an electron
of mass
m
in one dimension subject to a potential
V
(
x
) =
gδ
(
x
)
where
g
is a constant and
δ
(
x
) is the Dirac deltafunction. This system approximates a
highly localized potential around the point
x
= 0.
(i) For what sign(s) of
g
is there a quantum mechanical bound state?
(ii) Please ﬁnd a relation between the bound state energy
E
and
g
.
(iii) Now consider the double well potential
V
(
x
) =
gδ
(
x
+
a
) +
gδ
(
x

a
)
.
Again ﬁnd a relation between
E,g
and
a
for any bound states. What happens to the energy
E
of these states as
a
→
0?
2
. This problem is inspired by a discussion after lecture. For a repulsive square well
potential of width
a
and height
V
0
, we saw in lecture that perfect transmission is possible
as long as sin(
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 Spring '10
 Sghy
 Physics, mechanics, Boundary conditions, perfect transmission, bound state energy, repulsive square wells, normalized coherent state

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