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PHYSICS 4455 — QUANTUM MECHANICS
Problem Set 8 — due 10
/
27
/
2005, in class.
“No room to wiggle”  exercises with
δ
function potentials
1.
Starting with the solution for the bound states in HW7, set
V
0
=∆
/
L
and take the limit
L
→
0.
Notice that, although the depth of the well gets larger and larger (i.e., the potential becomes
more and more “attractive”), there is now less and less room for the wave function to
wiggle. As a result, show (a one liner!) that there are no excited bound states, i.e., only the
lowest one survives. What is its energy, in terms of
∆
and
m
(since there is no
L
left!)?
Notice that, in this limit, although there is no width to the potential, its integral
∫
V
(
x
)
dx
remains positive. Thus, this limiting
V
is known as the delta function potential. Write down
the explicit mathematical expression for
V
(
x
)
.
2.
Get the same result for the bound state energies as above, by following the following
shortcut.
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This note was uploaded on 11/09/2009 for the course PHY 4604 taught by Professor Mucciolo during the Spring '09 term at University of Central Florida.
 Spring '09
 Mucciolo
 mechanics

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