The Nature of the Cooper Pair - University of Liverpool

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Let us see what the wavefunction looks like. Ψ = �ω �D 0 1 ε + ∆ sin kr It is a function of r only, so it is spherically symmetric. It is a sum of many sine functions. The coefficient decreases with energy. When the energy ε goes from 0 to ∆, the coefficient decreases from 1/∆ to 1/(2∆). So we may suppose that the terms are only significant in the energy range from 0 to ∆. As energy changes, k changes. When r is large enough, sin kr starts to go out of phase for different wavevectors k. Superconductivity 21
They will start interfering destructively, and the sum Ψ will decrease. This gives us a way of estimating the size of the wavefunction. Consider the minimum and maximum k. These correspond to the energies ε = 0 and 1/∆, relative to E F . The absolute energies are E F and E F + ∆. Consider the radius r = ρ at which the sine functions for these two values of k first get out of phase by π radians. This gives the largest cancellation, and we may take this radius ρ as an estimate of the size of the wavefunction. Superconductivity 22
Page 1 and 2: Statistical and Low Temperature Phy
Page 3 and 4: When an electron passes in between
Page 5 and 6: Using the small change approximatio
Page 7 and 8: The attractive force is the Coulomb
Page 9 and 10: Recall that the Debye frequency ω
Page 11 and 12: The displacement then gives us the
Page 13 and 14: We shall approximately represent th
Page 15 and 16: We have seen that two electrons mus
Page 17 and 18: ... recall the quantisation conditi
Page 19 and 20: The Schrodinger equation is then of
Page 21: Here, we just state the solution to
Page 25 and 26: Using ∆k.ρ = π we can solve for
Page 27 and 28: The bound pair of electrons is the
Page 29 and 30: The number of Cooper pairs is there
Page 31 and 32: So, as in superfluid helium-4, we m
Page 33 and 34: APPENDIX: Solving for the Cooper pa
Page 35 and 36: The attractive potential is very sm
Page 37 and 38: Therefore and sin(kR) = 0, k = nπ
Page 39 and 40: The product of the waves sin(kr) an
Page 41 and 42: Return to the Schrodinger’s equat
Page 43 and 44: So in energy variable, the density
Page 45 and 46: However, ε is 2�ω D, then the u
Page 47 and 48: We have found both aε and ∆. Thi

The Nature of the Cooper Pair - University of Liverpool

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