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HOMEWORK ASSIGNMENT 9
PHYS852 Quantum Mechanics II, Spring 2008
1. Use the LippmanSchwinger equation:

ψ
i
=

ψ
0
i
+
GV

ψ
i
(1)
to solve the onedimensional problem of resonant tunneling through two deltapotentials. Take
ψ
0
(
x
) =
e
ikx
and
V
(
x
) =
g
[
δ
(
x
) +
δ
(
x

L
)]
.
(2)
a.) Express (1) as an integral equation for
ψ
(
x
) and use it to ﬁnd the general solution
ψ
(
x
) for
arbitrary
k
and
g
.
b.) use your result from a.) to compute the transmission probability
T
=

t

2
as a function of
k
.
c.) Consider an inﬁnite squarewell
V
(
x
) = 0 from
x
= 0 to
x
=
L
, and
V
(
x
) =
∞
otherwise. Show
that the bound states are of the form
ψ
n
(
x
)
∝
sin(
k
n
x
), where
k
n
=
nπ/L
.
d.) Under what conditions does the transmission probability have a maximum at or near the
k
n
corresponding to the squarewell bound states?
2. Swave resonance scattering: use the method of boundary conditions to solve the 3d resonant scat
tering problem of a spherical deltashell potential, given by
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This note was uploaded on 11/26/2010 for the course PHYSICS PHYS 852 taught by Professor Michaelmoore during the Spring '10 term at Michigan State University.
 Spring '10
 MichaelMoore
 mechanics, Work

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