The BCS-model of superconductivity is based on the following Hamiltonian: HBCS = (p)apΣaa²-plat+ P.P P.O (5) where the s

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The BCS-model of superconductivity is based on the following Hamiltonian: HBCS = (p)apΣaa²-plat+ P.P P.O (5) where the sums over momenta are limited to the spherical layer around the Fermi surface by condition F (p) < EF + hup, where (p) is the energy of the electron, F = (PF) is the Fermi energy and wp is the Debye frequency. In the normal (non-superconducting) state, all thermal averages of operators of the type (aa) and (atat) are equal to zero, because these operators have no diagonal matrix elements. The situation, however, is different in the superconducting state. 1 a.) Below the superconducting transition temperature Te, the thermal average (a-p.ap.t). is not equal to zero. Find (a-papt) for T=0. Hint: Use Bogoliubov transformation. b.) Given the result of part (a), find the real space anomalous average (ar,.art) as a function of r r₁ r₂. Obtain the general integral expression, try to go as far as possible with analytical approximations and sketch the overall behaviour as a function of r.