Description of the Condensed State

The ground state of a free electron gas corresponds to complete filling of the one-electron energy levels of wavevector k and energy ħ²k²/2m up to a certain energy E F  =  ћ 2 k F 2 / 2 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497032/d2b0510a-8f84-4d3a-8776-dff0770e5b14/content/eq458.tif"/> (the Fermi energy). However, in the presence of an attractive interaction, no matter how weak, this state becomes unstable (Cooper, 1957). The instability can be understood by considering two particular electrons of coordinates r ₁ and r ₂, the other electrons still being treated as a free electron gas. The only effect of this electron gas is to forbid the two electrons to occupy all states k < k_F by the exclusion principle. Let ψ(r ₁, r ₂) be the wavefunction of the two electrons. Consider only states where the center of gravity of the pair (r ₁, r ₂) is at rest; ψ is then only a function of r₁ – r ₂. Expand ψ in plane waves (4-1) ψ ( r 1  −  r 2 )  =  ∑ k g ( k ) e i k ⋅ ( r 1  −  r 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497032/d2b0510a-8f84-4d3a-8776-dff0770e5b14/content/eq459.tif"/>