Editing BCS theory (section)

===Details===
BCS theory starts from the assumption that there is some attraction between electrons, which can overcome the [[Coulomb repulsion]]. In most materials (in low temperature superconductors), this attraction is brought about indirectly by the coupling of electrons to the [[crystal lattice]] (as explained above). However, the results of BCS theory do ''not'' depend on the origin of the attractive interaction. For instance, Cooper pairs have been observed in [[Ultracold atom|ultracold gases]] of [[fermion]]s where a homogeneous [[magnetic field]] has been tuned to their [[Feshbach resonance]]. The original results of BCS (discussed below) described an [[Atomic orbital|s-wave]] superconducting state, which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the [[Atomic orbital|d-wave]] high-temperature superconductors.

Extensions of BCS theory exist to describe these other cases, although they are insufficient to completely describe the observed features of high-temperature superconductivity.

BCS is able to give an approximation for the quantum-mechanical many-body state of the system of (attractively interacting) electrons inside the metal. This state is now known as the BCS state. In the normal state of a metal, electrons move independently, whereas in the BCS state, they are bound into Cooper pairs by the attractive interaction. The BCS formalism is based on the reduced potential for the electrons' attraction. Within this potential, a variational [[ansatz]] for the wave function is proposed. This ansatz was later shown to be exact in the dense limit of pairs. Note that the continuous crossover between the dilute and dense regimes of attracting pairs of fermions is still an open problem, which now attracts a lot of attention within the field of ultracold gases.