The Nature of the Cooper Pair - University of Liverpool

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The Schrodinger equation has the same form as that for the ground state electron of the hydrogen atom: − �2 1 2µ r2 d R dr � r 2dR dr � + V (r)R = ER. except that the electron mass is replaced by µ, the reduced mass me/2. The Schrodinger equation can be rewritten in the familiar 1D form by using a new function ψ(r) = rR(r). Note this this is r times the actual wavefunction R(r). The Schrodinger equation then becomes − �2 2µ Superconductivity 33 d2ψ + V (r)ψ = Eψ. dr2
The attractive potential is very small. It is known in quantum mechanics that a bound state is not possible if this is so. http://en.wikipedia.org/wiki/Finite_potential_well However, the electrons here are not in free space. Those at the Fermi energy are not allowed to have smaller energies, because all the states are occupied. In the 1950s, Leon Cooper showed that when this happens, a bound state becomes possible. Superconductivity 34
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 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: APPENDIX: Solving for the Cooper pa
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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