The Nature of the Cooper Pair - University of Liverpool

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We apply the relation Multiply � R 0 sin(kr) sin(k′ r)dr = � k ′ � �2k ′2 ak ′ me − E by sin(kr) and integrate, we get where R 2 a k � � � 2 k 2 me � R/2 for k = k ′ 0 otherwise sin(k ′ r) = −V (r) � ak ′ sin(k ′ r) − E � l � k ′ = − � ak ′Vkk ′, Vkk ′ = V0 0 sin(kr) sin(k′ r)dr. The upper limit is l instead of R because V (r) is zero for r > l. Superconductivity 37 k ′
The product of the waves sin(kr) and sin(k ′ r) would be oscillatory. If k and k ′ are not equal, then it oscillates between plus and minus, and the integral may cancel to zero. If k and k ′ are nearly equal, then the product is nearly always positive. This is likely to happen if, within the distance l, the two waves differ by less than half wavelength. This means or kl − k ′ l = π, k − k ′ = π/l. Superconductivity 38
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 and 34: APPENDIX: Solving for the Cooper pa
Page 35 and 36: The attractive potential is very sm
Page 37: Therefore and sin(kR) = 0, k = nπ
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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