zhanhai said:
In my understanding, "deformation of lattice" is a classical description and may not be used in quantum mechanical explanation.
The phrase “lattice deformation” does not imply a system that is fully classical. The nuclei part of the lattice is much more massive then the valence electrons in the chemical bonds that hold the lattice together. Therefore, one can use an approximation of quantum mechanics where the nuclei and core electrons of the lattice move according to classical dynamics, but the electrons are behaving as a wave in quantum mechanics.
So an "explanation" can have "quantum" electrons and "classical" phonons. However, the composite particle called a Cooper pair is governed mostly by quantum mechanics. The Cooper pair is a quantum mechanical object because it is so large compared to the spacing between the Cooper pairs. Basically, the size of the Cooper pair is an indeterminacy of its position. So if you want to treat the Cooper pair as a quasiparticle, you have to take the quantum mechanics into account. So we have a "quantum" Cooper pair.
What you are possibly saying is that perturbation techniques aren’t entirely valid under the conditions of “lattice deformation”. However, there are other approimxations. The adiabatic approximation and the WKB approximation are valid under the conditions of a “slowly moving” lattice.
The electrons are moving far faster than the nuclei under these conditions. The indeterminacy in the position of the nuclei are much smaller than the undeterminacy of the valence electrons. Therefore, one can use “hybrid” mechanics where nuclei with core electrons are “classical particles” and both conduction electrons and valence electrons are “quantum waves”.
The distorted lattice, consisting of displaced nuclei, can be pictured as generate and electric field. A higher concentration of positive charges (the nuclei) are the source of an electric field that is moving outward from the points of greatest concentration of nuclei. The conduction electrons are in a potential that is caused by this electric field. So the electrons can “move” coherently with the deformed lattice.
Two electrons can move together toward the region where the positive charge density is greatest. However, the electrons are also pulling at the nuclei. They are making the regions of high density nuclei less dense. Although this is a self consistent picture classically, one can get more accuracy by assuming that the electrons at least are quantum mechanical.
So basically two electrons are interacting through the lattice deformation, otherwise called a phonon.
Here is a link that claims that variational and WKB approximation are valid with Cooper pairs. You can assume that the electrons are quantum mechanical.
http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)
“An example of this phenomenon may be found in conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs. When faced with such systems, one usually turns to other approximation schemes, such as the variational method and the WKB approximation. This is because there is no analogue of a bound particle in the unperturbed model and the energy of a soliton typically goes as the inverse of the expansion parameter. However, if we "integrate" over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the order of or in the perturbation parameter g. Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions (which typically blow up as the expansion parameter goes to zero).”
http://arxiv.org/ftp/arxiv/papers/1012/1012.0879.pdf
“ The binding energy of superconducting electrons dominates the superconducting transition temperature in the corresponding material. Under an electric field, superconducting electrons move coherently with lattice distortion wave and periodically exchange their excitation energy with chain lattice, that is, the superconducting electrons transfer periodically between their dynamic bound state and conducting state, so the superconducting electrons cannot be scattered by the chain lattice, and supercurrent persists in time. Thus, the intrinsic feature of superconductivity is to generate an oscillating current under a dc voltage.”
You mustn’t think that the Cooper pairs are individual particles. Actually, they are squashed together. Therefore, position of each electron is highly uncertain. So even if you think of the lattice deformation as classical, the electrons are not classical.
http://www.desy.de/f/students/lectures2009/schmueser1.pdf
“The binding energy of a Cooper pair turns out to be small, 104103 eV, so low temperatures are needed to preserve the binding in spite of the thermal motion. According to Heisenberg’s Uncertainty Principle a weak binding is equivalent to a large extension of the composite system, in this case the above-mentioned d = 100 1000 nm. As a consequence, the Cooper pairs in a superconductor overlap each other. In the space occupied by a Cooper pair there are about a million other Cooper pairs. Figure 22 gives an illustration. The situation is totally different from other composite systems like atomic nuclei or atoms which are tightly bound objects and well-separated from another. The strong overlap is an important prerequisite of the BCS theory because the Cooper pairs must change their partners frequently in order to provide a continuous binding.”
