93. Within the centre of mass, the motion of a Cooper pair is non-zero. Two electrons interact by the potential Vk,k'. T

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93 Within The Centre Of Mass The Motion Of A Cooper Pair Is Non Zero Two Electrons Interact By The Potential Vk K T 1 (88.96 KiB) Viewed 38 times

93. Within the centre of mass, the motion of a Cooper pair is non-zero. Two electrons interact by the potential Vk,k'. They scatter with an initial momentum of k, -k and a final momentum of k', -k'. In this system, r, r', k, k' are vectors and that the energy dispersion is Eq h2q2 2m for the wave numbers q = Q, k, k' and their respective masses. (a) Derive the pair wavefunction (R,r) = eiQ-R *Σ Ikelkr for this system, knowing that the centre of mass has the momentum ħQ, where Q = k1 + kz. Assume both electrons have the same mass, i.e. m1 = m2. Hint: review the central potential problem in Quantum Mechanics. (b) The gap equation for zero momentum within the centre of mass frame is 1 == 1 2Ek-E V - Ε k>ke where k = (ki - k2)/2. = Show that if you add the centre of mass kinetic energy term f2/2M to the Hamiltonian, the Schrodinger equation in momentum space will have a term €Q9k added to it. What will the gap equation become now that there is momentum? Hint: follow the derivation of the gap equation for a zero momentum Cooper pair.