The Nature of the Cooper Pair - University of Liverpool

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We now want to find out if two electrons can attract in the way just described, and form a bound state - a Cooper pair. A bound state refers to a wavefunction that has a finite size, like the wavefunction of an electron in an atom. This means that the value of the wavefunction falls off to zero after some distance. In contrast, a free electron is not in a bound state. Its wavefunction may be e ikx , which has the same amplitude at any distance. Since we treat the potential as effectively spherical, we can use the same form of the Schrodinger equation as for the hydrogen atom. Superconductivity 13
We have seen that two electrons must travel in opposite directions, and the paths must be very close, before they can experience the attractive potential. Electrons are at the Fermi energy have a wavelength of the order of d, the spacing between ions. We can see this by comparing the kinetic energy with E F : � 2 k 2 F 2me = �2 2me � � 3π2 2/3 N k F is the wavevector at 2π/λ F the Fermi energy, where λ F is the wavelength. Superconductivity 14 V .
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: We shall approximately represent th
Page 17 and 18: ... recall the quantisation conditi
Page 19 and 20: The Schrodinger equation is then of
Page 21 and 22: Here, we just state the solution to
Page 23 and 24: They will start interfering destruc
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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