Physics 361 Problem Set 8 due November 18, 2009 note: These are both new problems and I could easily have mis-stated something. If you see something ﬁshy, please drop me an email. —TW P8.1 Localizing a Fermi-surface 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 wave-function must be made from available plane-wave states above the Fermi surface, with wavevector k with k > k F . In order to localize the wave-function 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 | /± This
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