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ps4 - Physics 880.06 Problem Set 4 This problem set is due...

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Physics 880.06: Problem Set 4 This problem set is due by BEGINNING OF CLASS on Tuesday, May 10, 2011. This problem set consists of just a single problem, which is worth 25 points. 1. The BCS equation for the gap function Δ k at temperature T = 0 is Δ k = - 1 2 summationdisplay k Δ k V k , k radicalBig Δ 2 k + ξ 2 k , (1) where ξ k = ǫ k - E F , and ǫ k is the energy of a non-interacting electron of wave vector k . (a). Show that if V k , k = - A k A k , where A k is a specified function of k , then the above equation is solved by a gap function of the form Δ k = A k Δ, where Δ is independent of k , and find an equation for Δ. (b). Now suppose specifically that our superconductor is two-dimensional, and that A k has the form A k = A 0 cos(2 φ ) (2) if | ǫ k - E F | < ¯ c , and A k = 0 (3) otherwise. Here ¯ c is a cutoff energy, ǫ k = ¯ h 2 k 2 / (2 m ), and we have written k = ( k cos φ,k sin φ ). Obtain an integral expression which de- termines Δ in this case, but you need not solve this to obtain Δ ex- plicitly.
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