Physics 361
Problem Set 8
due November 18, 2009
note:
These are both new problems and I could easily have misstated something. If you see something
ﬁshy, please drop me an email. —TW
P8.1 Localizing a Fermisurface electron
In class we gave various arguments that electrons excited near
the Fermi surface could form bound states under an arbitrarily
weak attraction. In this example we consider attraction to a ﬁxed point in space. The attractive potential
is a spherical square well
U
(
r
) of depth

V
and width ﬁxed
a
. We are interested in the limit of small
V
.
One way to demonstrate a bound state is to exhibit a localized wavefunction whose energy is lower than
without the potential. Our wavefunction must be made from available planewave states above the Fermi
surface, with wavevector
k
with
k > k
F
. In order to localize the wavefunction near the origin, we consider
a superposition
ψ
(
r
) of the form
ψ
(
r
)
≡
Z

k

>k
F
d
3
k
[
e
ik
·
r
+
e

ik
·
r
]
e

k

k
F

/±
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 Spring '10
 Pucker
 Physics, Energy, Potential Energy, Solid State Physics, 2m, bound state, excess kinetic energy

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